Moduli Spaces of Singular Yamabe Metrics Rafe Mazzeo mazzeo@math.stanford.edu Daniel Pollack pollack@math.uchicago.edu Karen Uhlenbeck uhlen@math.utexas.edu Mazzeo_Rafe | Pollack_Daniel | Uhlenbeck_Karen points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension $k$. For a generic set of nearby conformal classes the moduli space is shown to be a $k$-dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

]]> Complete, conformally flat metrics of constant positive scalar curvature on the complement of points in the -sphere, were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension . For a generic set of nearby conformal classes the moduli space is shown to be a -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds. ACF ACF L. Andersson, P.T. Chrusciel, and H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal data for Einstein's Field Equations , Comm. Math. Phys. 149 (1992), 587-612. 93i:53040 AMc AMc P. Aviles and R. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds , Duke Math. Jour. 56 No. 2 (1988), 395-398. 89b:58224 AKS AKS P. Aviles, N. Korevaar and R. Schoen, The symmetry of constant scalar curvature metrics near point singularities , preprint. B B K. Grosse-Brauckmann, New surfaces of constant mean curvature Math. Z. (to appear). CGS CGS L. Caffarelli, B. Gidas and J. Spruck, Asymptotic Symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth , Comm. Pure Appl. Math. 42 (1989), 271-297. 90c:35075 D D C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constant , J. Math. Pure Appl. 6 (1841), 309-320. Fe Fe H. Federer, Geometric measure theory , Springer-Verlag, New York, 1969. 41:1976 FMc FMc D.L. Finn and R.C. McOwen, Singularities and asymptotics for the equation g u -u q Su , Indiana Univ. Math. J. 42 (1993), 1487--1523. 95g:58216 FM1 FM1 A. Fischer and J. Marsden, Deformations of the scalar curvature , Duke Math. J. 42 (1975), 519-547. 52:1804 FM2 FM2 ---, Linearization stability of nonlinear partial differential equations , Proceedings of Symposia in Pure Math., vol. 27, 1975, pp. 219-263. 52:4337 Fo1 Fo1 R. H. Fowler, The form near infinity of real continuous solutions of a certain differential equation of the second order , Quart. J. Pure Appl. Math. 45 (1914), 289--349. Fo2 Fo2 ---, Further studies of Emden's and similar differential equations , Quart. J. Math. Oxford Series 2 (1931), 259--287. Kap Kap N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space , Ann. of Math. 131 (1990), 239-330. 93a:53007a K K T. Kato, Perturbation theory for linear operators , second ed., Springer-Verlag, Berlin, 1976. 53:11389 Ko Ko O. Kobayashi, A differential equation arising from the scalar curvature function , J. Math. Soc. Japan 34 (1982), 665-675. 84a:53046 KK KK N. Korevaar and R. Kusner, The global structure of constant mean curvature surfaces , Invent. Math. 114 (1993), 311-332. 95f:53015 KKMS KKMS ---,---, W. Meeks III, and B. Solomon, Constant mean curvature surfaces in hyperbolic space , Amer. J. of Math. 114 (1992), 1-43. 92k:53116 KKS KKS ---, --- and B. Solomon, The structure of complete embedded surfaces with constant mean curvature , J. Differential Geometry 30 (1989), 465-503. 90g:53011 KMP KMP R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces , Geom. and Functional Analysis (to appear). L L J. Lafontaine, Sur la geometrie d'une generalisation de l'equation differentielle d'Obata , J. Math. pures et appl. 62 (1983), 63-72. 84i:53047 Li Li A. Lichnerowicz, Propagateurs et commutateurs en relativite generale , Inst. Hautes Etudes Sci. Publ. Math. No. 10 (1961). 28:967 LN LN C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations , Contributions to Analysis, Academic Press, New York, 1974, pp. 245-272. 50:10543 M1 M1 R. Mazzeo, Regularity for the singular Yamabe equation , Indiana Univ. Math. J. 40 (1991), 1277-1299. 92k:53071 M2 M2 ---, Elliptic theory of differential edge operators I , Comm. in P.D.E. 16 (1991), 1615-1664. 93d:58152 MP MP --- and F. Pacard, A new construction of singular solutions for a semilinear elliptic equation , To appear, J. Differential Geometry. MPU MPU ---, D. Pollack and K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics , Preprint. MS MS --- and N. Smale, Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere , J. Differential Geometry 34 (1991), 581-621. 92i:53034 Mc Mc R. McOwen, Singularities and the conformal scalar curvature equation , Geometric Analysis and Nonlinear PDE (I. Bakelman, eds.), Marcel Decker, 1993. 94f:53076 Me Me R. Melrose, The Atiyah-Patodi-Singer index theorem , AK Peters Ltd., Wellesley, MA, 1993. 95:17 O O M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere , J. Math. Soc. Japan 14 (1962), 333-340. 25:5479 Pa Pa F. Pacard, The Yamabe problem on subdomains of even dimensional spheres , preprint. P1 P1 D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation , Comm. Anal. and Geom. 1 (1993), 347-414. 94m:58051 P2 P2 ---, Compactness results for complete metrics of constant positive scalar curvature on subdomains of n , Indiana Univ. Math. J. 42 (1993), 1441-1456. 95c:53052 RS RS M. Reed and B. Simon, Methods of modern mathematical physics , vol. IV, Academic Press, 1978. 58:12429c S1 S1 R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature , J. Differential Geometry 20 (1984), 479-495. 86i:58137 S2 S2 ---, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation , Comm. Pure and Appl. Math. XLI (1988), 317-392. 89e:58119 S3 S3 ---, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics , Topics in Calculus of Variations; Lecture Notes in Mathematics 1365 (M. Giaquinta, eds.), Springer-Verlag, 1989, pp. 120-154. 89k:58051 S4 S4 ---, On the number of constant scalar curvature metrics in a conformal class , Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds.), Wiley, 1991, pp. 311-320. 93b:53001 SY SY --- and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math. 92 (1988), 47-71. 89c:58139 T T C. Taubes, Gauge theory on asymptotically periodic 4-manifolds , J. Differential Geometry 25 (1987), 363-430. 88g:58176

[ACF]
L. Andersson, P.T. Chru\'{s}ciel, and H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal data for Einstein's Field Equations, Comm. Math. Phys. 149 (1992), 587-612. MR 93i:53040

[AMc]
P. Aviles and R. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. Jour. 56 No. 2 (1988), 395-398. MR 89b:58224

[AKS]
P. Aviles, N. Korevaar and R. Schoen, The symmetry of constant scalar curvature metrics near point singularities, preprint.

[B]
K. Grosse-Brauckmann, New surfaces of constant mean curvature Math. Z. (to appear).

[CGS]
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic Symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297. MR 90c:35075

[D]
C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constant, J. Math. Pure Appl. 6 (1841), 309-320.

[Fe]
H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. MR 41:1976

[FMc]
D.L. Finn and R.C. McOwen, Singularities and asymptotics for the equation $\varDelta _{g} u -u^{q} =Su$, Indiana Univ. Math. J. 42 (1993), 1487--1523. MR 95g:58216

[FM1]
A. Fischer and J. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), 519-547. MR 52:1804

[FM2]
---, Linearization stability of nonlinear partial differential equations, Proceedings of Symposia in Pure Math., vol. 27, 1975, pp. 219-263. MR 52:4337

[Fo1]
R. H. Fowler, The form near infinity of real continuous solutions of a certain differential equation of the second order, Quart. J. Pure Appl. Math. 45 (1914), 289--349.

[Fo2]
---, Further studies of Emden's and similar differential equations, Quart. J. Math. Oxford Series 2 (1931), 259--287.

[Kap]
N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. 131 (1990), 239-330. MR 93a:53007a

[K]
T. Kato, Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976. MR 53:11389

[Ko]
O. Kobayashi, A differential equation arising from the scalar curvature function, J. Math. Soc. Japan 34 (1982), 665-675. MR 84a:53046

[KK]
N. Korevaar and R. Kusner, The global structure of constant mean curvature surfaces, Invent. Math. 114 (1993), 311-332. MR 95f:53015

[KKMS]
---,---, W. Meeks III, and B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. of Math. 114 (1992), 1-43. MR 92k:53116

[KKS]
---, --- and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geometry 30 (1989), 465-503. MR 90g:53011

[KMP]
R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. and Functional Analysis (to appear).

[L]
J. Lafontaine, Sur la géométrie d'une généralisation de l'équation différentielle d'Obata, J. Math. pures et appl. 62 (1983), 63-72. MR 84i:53047

[Li]
A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. No. 10 (1961). MR 28:967

[LN]
C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis, Academic Press, New York, 1974, pp. 245-272. MR 50:10543

[M1]
R. Mazzeo, Regularity for the singular Yamabe equation, Indiana Univ. Math. J. 40 (1991), 1277-1299. MR 92k:53071

[M2]
---, Elliptic theory of differential edge operators I, Comm. in P.D.E. 16 (1991), 1615-1664. MR 93d:58152

[MP]
--- and F. Pacard, A new construction of singular solutions for a semilinear elliptic equation, To appear, J. Differential Geometry.

[MPU]
---, D. Pollack and K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics, Preprint.

[MS]
--- and N. Smale, Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere, J. Differential Geometry 34 (1991), 581-621. MR 92i:53034

[Mc]
R. McOwen, Singularities and the conformal scalar curvature equation, Geometric Analysis and Nonlinear PDE (I. Bakelman, eds.), Marcel Decker, 1993. MR 94f:53076

[Me]
R. Melrose, The Atiyah-Patodi-Singer index theorem, AK Peters Ltd., Wellesley, MA, 1993. CMP 95:17

[O]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. MR 25:5479

[Pa]
F. Pacard, The Yamabe problem on subdomains of even dimensional spheres, preprint.

[P1]
D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. and Geom. 1 (1993), 347-414. MR 94m:58051

[P2]
---, Compactness results for complete metrics of constant positive scalar curvature on subdomains of $\mathbb{S} ^{n}$, Indiana Univ. Math. J. 42 (1993), 1441-1456. MR 95c:53052

[RS]
M. Reed and B. Simon, Methods of modern mathematical physics, vol. IV, Academic Press, 1978. MR 58:12429c

[S1]
R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479-495. MR 86i:58137

[S2]
---, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure and Appl. Math. XLI (1988), 317-392. MR 89e:58119

[S3]
---, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations; Lecture Notes in Mathematics # 1365 (M. Giaquinta, eds.), Springer-Verlag, 1989, pp. 120-154. MR 89k:58051

[S4]
---, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds.), Wiley, 1991, pp. 311-320. MR 93b:53001

[SY]
--- and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47-71. MR 89c:58139

[T]
C. Taubes, Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geometry 25 (1987), 363-430. MR 88g:58176
]]> 1996 American Mathematical Society Journal of the American Mathematical Society J. Amer. Math. Soc. jams 9 02 19960401 January 20, 1994 303 303-344 42 S0894-0347-96-00208-1 10.1090/S0894-0347-96-00208-1 0894-0347 1088-6834 English 58D27 1991 http://www.ams.org/jourcgi/jour-getitem?pii=S0894-0347-96-00208-1 1. Introduction Much has been clarified in the past ten years about the behavior of solutions of the semilinear elliptic equation relating the scalar curvature functions of two conformally related metrics. The starting point for these recent developments was R. Schoen's resolution of the Yamabe problem on compact manifolds S1 , capping the work of a number of mathematicians over many years. Soon thereafter Schoen S2 and Schoen-Yau SY made further strides in understanding weak solutions of this equation, particularly on the sphere, and its relationship with conformal geometry. Of particular interest here is the former, S2 ; in that paper, Schoen constructs metrics with constant positive scalar curvature on n , conformal to the standard round metric, and with prescribed isolated singularities (he also constructs solutions with certain, more general, singular sets). This singular Yamabe problem' is to find a metric g u 4 (n-2) g 0 on a domain n which is complete and has constant scalar curvature R(g) . This is equivalent to finding a positive function u satisfying equation gathered g 0 u - n-2 4(n-1) R(g 0)u n-2 4(n-1) R(g)u n 2 n-2 0 on n , g complete on n , R(g) constant, gathered 1.1 equation where R(g 0) n(n-1) is the scalar curvature of the round metric g 0 . The completeness of g requires that u tend to infinity, in an averaged sense, on approach to . The earliest work on this singular Yamabe problem seems to have been that of C. Loewner and L. Nirenberg LN , where metrics with constant negative scalar curvature are constructed. Later work on this negative' case was done by Aviles-McOwen AMc , cf . also Mc and Finn-McOwen FMc , where the background manifold and metric are allowed to be arbitrary. For a solution with R(g) 0 to exist, it is necessary and sufficient that ( ) (n-2) 2 . A partial converse is that if a solution with R(g) 0 exists (at least when M n ), it is necessary that k (n-2) 2 , SY . Schoen gave the first general construction of solutions with R(g) 0 S2 . These solutions have singular set which is either discrete or nonrectifiable. Many new solutions on the sphere with singular set , a smooth perturbation of an equatorial k -sphere, 1k (n-2) 2 , are constructed in MS . F. Pacard Pa has recently constructed positive complete solutions on the sphere with singular set an arbitrary smooth submanifold of dimension (n-2) 2 . Regularity of solutions near the singular set , no matter the sign of the curvature or background manifold or metric, is examined in M1 , cf . also ACF for a special case of relevance to general relativity. A more detailed account of part of this history is given in MS . Quite recently the first author and F. Pacard MP established the existence of solutions on M where M is any compact manifold with nonnegative scalar curvature and the singular set is any finite, disjoint union of submanifolds i with dimensions k i 1, , (n-2) 2 . In this paper we return to this problem in the setting studied by Schoen S2 on n , where is a finite point-set: equation p 1, , p k . 1.2 equation Hereafter, will always be taken to be this set, unless indicated specifically, and the scalar curvature R(g) attained by the conformal metric will always be n(n-1) . In this case, no solutions of (1.1) exist when k 1 . A proof of this, following from a general symmetry theorem, is indicated below. Thus we always assume that k 2 . Since this problem is conformally invariant, the set may be replaced by F() for any conformal transformation F O(n 1,1) . A simple topological argument (given in S2 ) shows that using such a transformation we can always arrange that is balanced', i .e . that the points p j sum to zero as vectors in n 1 . Henceforth this will be tacitly assumed as well. Notice that if is balanced, k 2 , and if F is a conformal transformation which preserves , then (since some power of F fixes three points on the sphere, and also the origin in n 1 ) F must be an orthogonal transformation. If k 2 then F could also be a dilation. Also, any balanced configuration is contained in a minimal equatorial subsphere k n , and an Alexandrov reflection argument similar to the one in 8 of CGS shows that if u is an arbitrary solution to (1.1), and F is a rotation preserving pointwise, then u must also be invariant under F . Observe that this implies that no complete solutions exist on p . If u were such a solution, then because any other point can be moved to be antipodal to p , this reflection argument would show that u is rotationally symmetric with respect to any other point on the sphere. Hence u would have to be constant, which contradicts completeness. Hence (except when k 2 ) we do not get trivial' families of solutions of (1.1) obtained by pulling back a fixed solution by a family of conformal transformations. Note, however, that if F preserves , but permutes the points, there is no reason to expect that most solutions of (1.1) will be fixed by F ; indeed, it is not even clear a priori that F -invariant solutions exist in this case. Our aim in this work is to consider the moduli space , which is by definition the set of all smooth positive solutions u to the problem equation gathered g 0 u - n(n-2) 4u n(n-2) 4u n 2 n-2 0 on n , g u 4 n-2 g 0 complete on n , gathered 1.3 equation where p 1, , p k is any fixed, balanced set of k points in n . We remark that the geometric condition that the metric g be complete is equivalent to the analytic one of requiring that the set consist of nonremovable singularities of u . This space might be called the PDE moduli space' PDE , to distinguish it from the geometric moduli space' geom which consists of all geometrically distinct solutions of this problem. In most instances these spaces coincide, although the second could conceivably be somewhat smaller than the first if some solutions admit nontrivial isometries (e .g . as in the case k 2 discussed in 2 below). will always denote the former of these spaces here. By Schoen's work S2 , is nonempty whenever k 1 . In fact, his construction yields families of solutions. As this was not the aim of his work, this is not made explicit there, nor are the free parameters in his construction counted. We shall examine a number of questions, both local and global, concerning the nature of this moduli space. The simplest of these is whether is a manifold, or otherwise tractable set, and if so, what is its dimension. Our main result is theorem1 is locally a real analytic variety of formal dimension k . theorem1 The formal dimension is the dimension predicted by an index theorem. As is usual in moduli space theories, obstructions may well exist to prevent from attaining this dimension. We clarify this and give a more careful statement in Theorem 5.4, Corollary 5.5 and Theorem 6.13 below. We also describe how natural parameters on may arise. On the linear level, these are given by certain scattering theoretic information for the metrics g . Another geometric description is given by the Pohozaev invariants which are defined in 3. We also obtain information on a geometrically natural compactification of which is obtained by taking the union with lower-dimensional moduli spaces corresponding to singular sets ' . We are, as yet, unable to provide a satisfactory description of the interior singularities of , or to determine whether this compactification, , is itself a compact real (semi-)analytic variety. This latter property is, by all indications, true. We hope to return to this later. There are many similarities between the theory of constant scalar curvature metrics on where is finite and that of embedded, complete, constant mean curvature (CMC) surfaces with k ends in 3 . The first examples of such CMC surfaces were given by N . Kapouleas Kap . One-parameter families of solutions with symmetry were constructed by K . Grosse-Brauckmann . Further general results on the structure of these surfaces and related problems appear in KKS , KKMS and KK . In the last of these, N . Korevaar and R . Kusner conjecture that there is a good moduli space theory for these surfaces. Our methods may be adapted immediately to this setting. If M 3 is an embedded, complete CMC surface with k3 ends which satisfies a hypothesis analogous to (5.2) below, then our results imply that the space of all nearby surfaces of this type, up to rigid motion in 3 , is a (3k-6) -dimensional real analytic orbifold. In order to understand the structure of the moduli space near surfaces where this hypothesis is not satisfied an argument different from the one employed here is needed since, for example, we do not have an analogue of the constructions in 6. Recently such an argument was derived based on the linear analysis developed in this paper. Thus we can also assert that the moduli space of such constant mean curvature surfaces is locally a real analytic variety, as was claimed in KK . This argument can be seen as an extension of Liapunov-Schmidt reduction' or the Kuranishi method' and also applies to give a new, direct proof of Theorem 1.4. These results appear in KMP . The outline of this paper is as follows. In 2 we analyze in detail the special case when k 2 . The solutions here will be called Delaunay solutions, in analogy with a similar family of complete constant mean curvature surfaces in 3 discovered in 1841 by C. Delaunay , although it was Fowler Fo1 , Fo2 who first studied the differential equation (see (2.1) below) of which these are solutions. Only in this case may the moduli space be determined completely, since, using the symmetry argument discussed above, the equation now reduces to an ODE. We also analyze the spectral theory of the linearized scalar curvature operator completely in this simple case. 3 collects a number of disparate results about solutions of (1.3) with isolated singularities which are used throughout the rest of the paper. This contains an explanation of the results of Caffarelli-Gidas-Spruck and Aviles-Korevaar-Schoen which state that the Delaunay solutions are good models for arbitrary solutions of (1.3) with isolated singularities. We also discuss here the Pohozaev invariants and compactness results for solutions g . The linearization L of the scalar curvature operator is studied in 4. We prove Fredholm results for this operator, using and expanding earlier work of C. Taubes . Other results here include more detailed information on asymptotics of solutions of Lw 0 , as well as the computation, using a relative index theorem, of the dimension of the bounded nullspace' of L . 5 uses these results to establish the structure of near its smooth points. In 6 we show that is a real analytic set by writing it as a slice of an urmoduli space' with the conformal class determined by any g . Our study of the urmoduli space draws on work of A. Fischer and J. Marsden, FM1 and FM2 . We also prove a generic slice theorem here which shows that slices of by generic nearby conformal classes are smooth, even if is not. In 7, our concluding remarks, we discuss three aspects about the nature of concerning which we have not yet obtained satisfactory results. The first of these is the nonexistence of L 2 eigenvalues for the linearization, especially for the solutions constructed in S2 . Secondly, we give a description of local coordinates on near smooth points. This is done both on the linear level and geometrically in terms of the Pohozaev invariants introduced in 3. At this point we also provide a brief discussion of the recent construction MPU of dipole solutions for the problem. Finally, we address certain natural questions concerning the boundary of the geometric compactification . The authors wish to thank Rick Schoen for his continued interest and substantial advice throughout the course of this work. The first two authors also had a number of enlightening conversations with Nick Korevaar, Rob Kusner and Tom Mrowka. 2. Delaunay solutions In this section we discuss the Delaunay family of solutions. These constitute, up to conformal equivalence, the totality of solutions when has only two elements. ODE analysis When k 2 , the condition that be balanced means that p 2 -p 1 . It is not difficult to show using the Alexandrov reflection argument CGS that any positive solution of the PDE (1.3) is only a function of the geodesic distance from either p 1 or p 2 . This equation reduces to an ODE, which takes the simplest form when written relative to the background metric dt 2 d 2 on the cylinder n-1 with coordinates (t,) , which is conformally equivalent to n . Thus, since the cylinder has scalar curvature (n-1)(n-2) , and g u 4 (n-2) (dt 2 d 2) has scalar curvature n(n-1) , u u(t) satisfies equation d 2 dt 2 u - (n-2) 2 4 u n(n-2) 4 u n 2 n-2 0. 2.1 equation This is easily transformed into a first order Hamiltonian system: setting v u' (primes denoting differentiation by t ) equation split u' v, v' (n-2) 2 4 u - n(n-2) 4 u n 2 n-2 . split 2.2 equation The corresponding Hamiltonian energy function is equation H(u,v) v 2 2 (n-2) 2 8 u 2n n-2 - (n-2) 2 8u 2. 2.3 equation The orbits of (2.2) remain within level sets of H , and since these level sets are one dimensional, this determines these orbits (but not their parameterizations) explicitly. The equilibrium points for this flow are at (0,0) and (u,0) , where equation u ( n-2 n) n-2 4 . 2.4 equation There is a special homoclinic orbit (u 0(t),v 0(t)) corresponding to the level set H 0 ; it limits on the origin as t tends to either , and encloses a bounded set in the right half-plane which is symmetric across the u -axis, given by H 0 . Somewhat fortuitously we may calculate explicitly that equation u 0(t) (t) 2-n 2 . equation Of course, H 0 decomposes into two orbits: this one and the stationary orbit (0,0) . It is simple to check that orbits not enclosed by this level set, i .e . those on which H 0 , must pass across the v -axis and into the region where u 0 . Thus, since we are only interested in solutions of (2.1) which remain positive and exist for all t , it suffices to consider only those orbits in . Notice that the second equilibrium point (u, 0) is in this region, and that all other orbits are closed curves. These correspond to periodic orbits (u (t),v (t)) , with period T() , 0 1 . The parameter may be taken as the smaller of the two u values where the orbit intersects the u -axis, so that 0 u (note that, strictly speaking, the orbit with 0 corresponds to the equilibrium point at the origin, but by convention we set it equal to the one previously defined). This ODE analysis is also described in S3 . The corresponding metrics on n-1 (or n p 1, p 2 ) have a discrete group of isometries, given on the cylinder by the translations t t T() . They interpolate between the cylindrical metric u 4 (n-2) (dt 2 d 2) (which is rescaled by the power of u so that its scalar curvature is n(n-1) ) and the solution corresponding to the conformal factor u 0(t) . This later solution is nothing other than the standard round metric on n p 1,p 2 , which is therefore incomplete and not, strictly speaking, in the moduli space . In all that follows we shall adopt the notation equation g u 4 n-2 ( dt 2 d 2) 2.5 equation when referring to these Delaunay metrics. Of greatest concern is the Laplacian for the metrics g ; in terms of the coordinates (t,) on the cylinder we may write this operator as equation split u - 2n n-2 t ( u 2 t ) u -4 n-2 u -4 n-2 t 2 2( t u)u - (n 2) n-2 t u -4 n-2 . split 2.6 equation We use t for the partial derivative with respect to t , etc . Also, is the Laplacian on the sphere n-1 of curvature 1 . We use the convention that - is a positive operator. Also, we write u u throughout this section. It is convenient to replace the variable t by a new variable r(t) , depending on , which represents geodesic distance with respect to g along the t integral curves, which are already geodesics (with varying parameterization) for each of the metrics g . r(t) is defined by setting dr dt u 2 n-2 and r 0 when t 0 . Then equation t u 2 n-2 r equation and consequently, using the first equality in (2.6), equation split u 2-2n n-2 r ( u 2n-2 n-2 r ) u -4 n-2 r 2 2n-2 n-2 r u u r u -4 n-2 . split 2.7 equation The function u is still periodic with respect to r , but the period R() now is better behaved than the period T() above. In fact, whereas equation 0 T() , and u T() 2 n-2 equation (the latter equality is proved by linearizing (2.2) at (u,0) ), we have instead equation 0 R() , and u R() 2 n , 2.8 equation so that the period stays within a compact interval in the positive real axis as varies between its limits. It is somewhat amusing that when n 4 we can solve (2.1) explicitly, once the transformation from t to r is effected; this was pointed out to us by R. Schoen. In fact, now t u r and (2.1) becomes equation u r(u r u) - u 2u 3 0. 2.9 equation Setting v u r u as the new dependent variable and u as the independent variable we get that v u v - u 2u 3 0 . Integrating to solve for v , and then integrating again leads to a general expression for u . Adjusting the constants we finally get that equation u (r) 12 (12 - 2)(2r) . 2.10 equation The parameterization has been arranged so that u() attains its minimum when r 2 , . Notice also that this shows that the period R() when n 4 . It is perhaps also instructional, although not necessary within the context of this paper, to consider the space of solutions to the problem on the real projective space with one point removed, P n p . We can identify any solution on this space with a solution on p,-p invariant under the reflection which exchanges these two antipodal points. Transforming the twice-punctured sphere to the cylinder, we are looking for solutions u such that u(t,) u(-t,-) . Any such solution is, of course, independent of and in the Delaunay family. For each value of the Delaunay parameter there are two possibilities, one taking its minimum value at t 0 and the other taking its maximum value there. These are connected via the cylindrical solution by letting increase to u . The moduli space is thus a copy of , with the cylindrical solution at the origin'. Alternatively, this moduli space is just the part of the u -axis in the set H 0 in the (u,v) -plane. As tends to zero on one end of this moduli space, the solution tends to the round spherical metric. As tends to zero at the other end, the solution tends to zero. Spectral theory for the Delaunay solutions Although the Alexandrov reflection argument shows that we have already described the full moduli space when has only two elements, we proceed further here to analyze the linearization of the scalar curvature operator around a Delaunay solution g . This case will serve as the model, and an important ingredient, for the more general linear analysis later. Thus, for v a suitably small function, let equation split N (v) (1 v) - n(n-2) 4 (1 v) n(n-2) 4 (1 v) n 2 n-2 v n v Q(v) split 2.11 equation where Q(v) is the nonlinear, quadratically vanishing term equation Q(v) n(n-2) 4( (1 v) n 2 n-2 - 1 - n 2 n-2 v). 2.12 equation Solutions of N (v) 0 correspond to other complete metrics on the cylinder with scalar curvature n(n-1) , and hence correspond to other Delaunay solutions. The linearization of N about v 0 is thus given by equation . d , d N () 0 L n. 2.13 equation Frequently we shall omit the -subscript from L when the context is clear. Using (2.6) and (2.13) and introducing an eigendecomposition j, j for on n-1 we decompose L into a direct sum of ordinary differential operators L j with periodic coefficients on . The spectral analysis of each L j , hence of L itself, is accomplished using Bloch wave theory, cf . RS . One conclusion is that spec (L) is purely absolutely continuous, with no singular continuous or point spectrum. Moreover, each L j has spectrum arranged into bands, typically separated by gaps; L itself has spectrum which is the union of all of these band structures: equation spec (L) j spec (L j). equation From (2.13) it is clear that spec (-L) is bounded below by -n . We proceed to analyze this spectrum near 0 . The first question is to understand the Jacobi fields, i .e . elements of the nullspace of L . Any solution of L 0 may be decomposed into its eigencomponents equation a j(t) , j(), equation where the a j solve L j(a j) 0 . Some of these solutions, although not necessarily all, may be obtained as derivatives of one-parameter families of solutions of N(v) 0 . It is common in geometric problems, cf . KKS , for the Jacobi fields corresponding to low values of j to have explicit geometric interpretations as derivatives of special families of solutions of the nonlinear equation. This is the case here, at least for the eigenvalues 0 0 and 1 n n-1 . For j 0 we look for families of solutions of N(v) 0 which are independent of the n-1 factor. There are two obvious examples, one corresponding to infinitesimal translations in t and the other corresponding to infinitesimal change of Delaunay parameter: equation split u (t ) u (t) -1 1(t,;) and u (t) u (t) -1 2(t,;). split 2.14 equation Now define equation j j, d . j(t,;) 0 ,j 1,2, 2.15 equation so that L j 0, j 1,2 . Of course, neither 1 nor 2 are in L 2 since L has no point spectrum. In fact, differentiating u (t T()) u (t) , first with respect to t and then with respect to , shows that equation 1(t T()) 1(t), 2(t T()) 1(t) T'() 2(t). 2.16 equation The first of these equalities states that 1 is periodic while the second shows that 2 increases linearly, at least so long as T'() 0 . Hence the j are generalized eigenfunctions for L with eigenvalue 0 ; their existence and slow growth imply that 0 is in the essential spectrum of L 0 , hence that of L . It is also not hard to see that 1 and 2 are linearly independent when u . In fact, if we translate u so that it attains a local maximum at t 0 , then 1(0) 0 , whereas 2(0) (d d) 1 , so that these functions are not multiples of one another. When u , 1 0 , but 2 does not vanish since 2(0) 1 u . A second solution of L 0 in this case is obtained by translating 2 by any non-integer multiple of its period. We shall prove below that these two functions form a basis, for each , for all temperate solutions of L 0 on the cylinder; any other solution of this equation must grow exponentially in one direction or the other. To find Jacobi fields corresponding to the first nonzero eigenvalue of , it is easiest first to transform the background cylindrical metric to the round spherical metric. The conformal invariance of the equation means that one-parameter families of solutions may be obtained from one-parameter families of conformal maps of . For example, 1 in (2.14) already corresponds to the composition of g with the family of conformal dilations fixing the two singular points. We can also consider composition with a family of parabolic conformal transformations which fixes one or the other of the singular points. Derivatives of these families lead to the new Jacobi fields. To compute them we first transform the equation yet again so that the background metric is the flat Euclidean metric on n and the singular points are at 0 and . The equation becomes equation n w n(n-2) 4 w n 2 n-2 0, 2.17 equation where the function w on n is related to u on the cylinder by the transformation equation u(t,) e (n-2)t 2 ,w(e -t ), w(,) (2-n) 2 , u(-,), 2.18 equation in terms of the polar coordinates (,) on n . The Delaunay solutions correspond to functions w w () . A parabolic transformation of fixing one of the singular points corresponds to a translation of n if that singular point corresponds to the point at infinity. Infinitesimal translation is just differentiation by one of the Euclidean coordinates x j . But equation w x j j w , equation and j x j is an eigenfunction for with eigenvalue n-1 . So w () is a solution of L 1 0 , where L 1 is written in terms of instead of t . Transforming back to the cylinder once again we arrive at the solution equation split 3(t) e t ( u'(t) u(t) n-2 2 ) e t ( 1(t) n-2 2). split 2.19 equation To obtain the other, exponentially decreasing, solution to L 1 0 , we simply observe that inversion about the unit sphere in n corresponds to reflection in t about t 0 . If u is positioned so that it assumes a maximum at t 0 , then u (-t) u (t) , and we obtain from (2.19) the solution equation 4(t) e -t ( n-2 2 - 1(t) ). 2.20 equation The solutions 1,, 4 are also recorded and employed in AKS . It is also important to have some understanding of the continuous spectrum near 0 ; we do this for each L j individually. theorem2 For every (0, u and j 1 , 0 spec (L j) . Hence the spectral analysis of L near the eigenvalue 0 reduces to that of L 0 . The only temperate solutions of L 0 0 are precisely the linear combinations of the solutions 1 and 2 of (2.15). theorem2 proof The eigenvalues of are 0 0, 1 1-n , etc. For j n (numbering the eigenvalues with multiplicity, as usual) j -n - 1 . Thus equation split L j u -4 n-2 t 2 2u - n 2 n-2 u t t u -4 n-2 j n u -4 n-2 t 2 2u - n 2 n-2 u t t n(1 - u -4 n-2 ) - u -4 n-2 . split 2.22 equation Since 0 u 1 for every and t , the term of order zero in this expression is always strictly negative. Standard ODE comparison theory now implies that an arbitrary solution of L j 0 must grow exponentially either as t or - . Clearly then 0 spec (L j) for j 1 . The same conclusion is somewhat more subtle when j 1 . First of all, observe that because 3 and 4 above span all solutions to the ODE L 1 0 , and since both of these functions grow exponentially in one direction or the other, it is evident that 0 is not in the spectrum of L 1 . We wish to know, though, that 0 is actually below all of the spectrum of -L 1 . This is no longer obvious since the term of order zero in equation L 1 u -4 n-2 t 2 2u - n 2 n-2 u t t (1-n)u -4 n-2 n 2.23 equation is no longer strictly negative. Conjugate L 1 by the function u p , where p is to be chosen. A simple calculation gives equation u -p L 1 u p u -4 n-2 t 2 (2p 2)uu t t A, equation where equation A u - 2n n-2 p(p 1)u t 2 ( p(n-2) 2 4 1-n) u 2 ( n - pn(n-2) 4 ) u 2n n-2 . equation Somewhat miraculously, upon setting p 2 (n-2) this expression reduces to equation gathered A 2n (n-2) 2 u - 2n n-2 ( u t 2 - (n-2) 2 4 u 2 (n-2) 2 4 u 2n n-2 ) 4n (n-2) 2 u - 2n n-2 H(), gathered equation where H() is the Hamiltonian energy (2.3) of the solution u (t) . This is a negative constant for every (0, u , and so once again we have obtained an ordinary differential operator with strictly negative term of order zero. u -2 (n-2) L 1 u 2 (n-2) is unitarily equivalent to L 1 , and by the previous argument, the spectrum of the first of these operators is strictly negative, and in particular does not contain 0 . Hence the same is true for L 1 and the proof is complete. proof We can say slightly more about the spectra of the L j . In fact, when u , equation L j n-2 ( t 2 j ) n, equation and equation spec (-L j) -n(1 j n-2 ), ). equation In particular, spec (-L 0) -n, ) and spec (-L 1) n (n-2), ) . The spectrum of -L u is the union of these infinite rays. As decreases, gaps appear in these rays, and bands start to form (in fact, by (2.8) the first gap appears at -3n 4 ). As decreases, the first band extends from -n to some value to the left of -3n 4 ; the second band begins somewhere to the right of this point, and always ends at 0 . It is not important for our later work that 0 is always on the end of the second band (or at least, at a point where the second Bloch band function has a turning point), but it is rather amusing that we may determine this. The explanation is that from (2.16), whenever T'() 0 (which certainly happens for almost every u ), the Jacobi field 2 grows linearly. This signifies that 0 is at the end of a band, or at the very least, at a turning point for one of the Bloch band functions for L 0 which parameterize the bands of continuous spectrum. As continues to decrease to 0 these bands shrink further and further. It is more revealing to consider the operator L written in the geodesic coordinates (r,) as in (2.7). As noted earlier, as 0 this operator remains periodic, but develops singularities every units. Geometrically, the metrics g are converging to an infinite bead of spheres of fixed scalar curvature n(n-1) . The spectrum of -L for this limiting metric is a countable union of the spectrum of -- n on one of these spheres, i .e . consists of a countable number of isolated points -n, 0, n 2, , each with infinite multiplicity. The bands of continuous spectrum have coalesced into these infinite-multiplicity eigenvalues. Note that in this limit, the infimum of spec (L 1) has increased to zero at a rate which may be estimated by a power of . 3. Generalities on solutions with isolated singularities In this section we collect various results concerning solutions of the singular Yamabe problem with isolated singularities, particularly in the context of conformally flat metrics, which will be required later. Asymptotics The Delaunay solutions discussed in the last section are interesting explicit solutions in their own right. However, their importance is due to the fact that they are asymptotic models for arbitrary solutions of the singular Yamabe problem, at least in the conformally flat setting. Before stating this result more carefully we introduce some notation. The half-cylinder (0,) t n-1 is conformally equivalent to the punctured ball n 0 ; an explicit conformal map is given by sending (t,) to the polar coordinates (,) , where e -t . A Delaunay solution u (t) may be transformed to the function equation u () 2-n 2 u (-). 3.1 equation Using the conformal invariance of the conformal Laplacian, this function solves equation u n(n-2) 4 u n 2 n-2 0. 3.2 equation theorem3 Let u ( n 0 ) be a positive solution of equation (3.2) with a nonremovable singularity at the origin. Then there exists some (0, u and A 0 so that u is asymptotic to the modified Delaunay solution u in the sense that equation u(,) (1 O( ))u (A, ), 3.4 equation for some 0 ; A corresponds to a translation of the t parameter. The analogous estimate still holds whenever u and u are replaced by any of their derivatives ( ) j . theorem3 This result was proved by Caffarelli, Gidas and Spruck CGS using a fairly general and complicated form of the Alexandrov reflection argument. They actually only give a somewhat weaker estimate where O( ) is replaced by o(1) . An alternate argument, relying on more direct geometric and barrier methods, was obtained at around the same time by Aviles, Korevaar and Schoen in the unpublished work AKS , and they obtained the stronger form. This last work is very close in method to the proof of the analogous result for complete constant mean curvature surfaces embedded in 3 presented in KKS . It is possible to calculate the decay rate in terms of spectral data of the operator L , and we shall indicate this argument in the next section. To our knowledge it is unknown whether some form of this result has been proved when the background metric is not conformally flat, but it seems likely to be true. Note that the estimate (3.4) may be restated for the transformed function u(t,) e -(n-2)t 2 u(e -t , ) on the cylinder. Now equation u(t,) (1 O(e -t )) u (t). 3.5 equation This is the form we shall use. It states that an arbitrary solution on the half-cylinder converges exponentially to a Delaunay solution. Pohozaev invariants A key ingredient in Schoen's construction S2 of solutions of (1.3) with isolated singularities is his use of balancing conditions for approximate solutions. These conditions follow from the general Pohozaev identity proved in S2 . In the present setting, when (,g) is a compact, conformally flat manifold with boundary, T is the trace-free Ricci tensor of g , X is a conformal Killing field on , and R(g) is constant, then equation T(X, ) d 0. 3.6 equation Here is the outward unit normal to , and d is surface measure along this boundary. There is a more general formula involving an integral over the interior of when the scalar curvature is not constant, or when X is not conformal Killing. When n p 1, , p k , there are many conformal Killing fields to use. An important class of these are the centered dilations'. Any such X is equal to the gradient of the restriction of a linear function (q) q, v , where v n 1 and n n 1 is the standard embedding. Explicitly, equation X q v - q, v q,q n, 3.7 equation since the unit normal to T q n is just q . Let denote the set of all such X . It is a subspace of the Lie algebra, (n 1,1) , of the conformal group O(n 1,1) . By (3.6), each X (n 1,1) determines an element of H n-1 ( n , ) , associating to a hypersurface the number equation (, X;g) T(X,) ,d. 3.8 equation The dual homology space is generated by the classes of hyperspheres i , where, for each p i , i B r(p i) for r sufficiently small, is chosen so that no other p j is in the same component of n i as p i . These classes satisfy the single homology relation 1 k 0 . The number in (3.8) will be written i(X;g) (or simply i(X) ) when i . Alternately, as suggested by this notation, we shall also regard each i as a linear functional on the X 's, i .e . as an element of (n 1,1) . These functionals will be called the Pohozaev invariants of the solution metric g . They satisfy equation 1 k 0. 3.9 equation Theorem 3.3 makes it possible to calculate at least some components of the i , by computing the Pohozaev invariants for the Delaunay solutions; we do this now. Since H n-1 ( n p 1, -p 1 ,) , it suffices to compute the single number 1(X) for each X . As described above, X may be identified with an element v n 1 . A straightforward calculation, similar to one given in P2 , shows that this invariant is, up to a constant, simply the Hamiltonian energy H() of the particular Delaunay solution: theorem4 Using the identification of X with v n 1 , the Pohozaev invariant for the Delaunay metric g on n p 1, -p 1 is given by equation 1(X) c n ,H() ,v, p 1 . equation Here c n is a non-vanishing dimensional constant (identified explicitly in P2 ). theorem4 The asymptotics in Theorem 3.3 (even with the sharp estimate of we will give later) do not give enough information for the invariant i to be computed for every X ; the point is that if X has associated vector v perpendicular to p i then Lemma 3.10 shows that there is no formal' asymptotic contribution to the invariant, but unfortunately the decay is not sufficient for there to be no perturbation' contribution. The one case where this is not an issue is when X is chosen to have associated vector exactly p i . The associated invariant i(X) in this case will be called the dilational Pohozaev invariant and denoted i (or i(g) ). theorem5 The dilational Pohozaev invariant i(g) associated to the puncture p i and the solution metric g is equal to c n ,H( i) , where i is the Delaunay parameter giving the asymptotic model for the metric g at p i , as provided by Theorem 3.3. theorem5 Local compactness properties of the moduli space It is natural to determine the possible ways that solution metrics g of our problem can degenerate. Phrased more geometrically, we wish to determine what the ends of the moduli space look like, and to determine a geometrically natural compactification. Any , when contains just two elements, may be identified with any other, and we may call this space simply 2 . In this space, all g 2 are Delaunay, and indeed 2 may be identified with the open set 2 , H 0 as described in 2. This space is two dimensional, smooth, and locally compact. A sequence of elements in 2 degenerate only when the Delaunay parameter (or neck-size' of the solutions) tends to zero. Recalling that the Hamiltonian energy H is strictly monotone decreasing for (0, u , this may be restated as saying that Delaunay solutions degenerate only when H() 0 . A similar statement is true when has more than two elements. As we have seen in Corollary 3.11, the Pohozaev invariants i , and indeed just the dilational Pohozaev invariants i , determine the Hamiltonian energies of the model Delaunay solutions for the metric g at each puncture p i . Clearly, then, if there exists some particular p i such that for a sequence of solution metrics g j the dilational Pohozaev invariant i(g j) tends to zero as j , then this sequence should be regarded as divergent in . The complementary statement, generalizing the situation for 2 , is also still valid, and was proved by Pollack P2 : theorem6 Let g j be a sequence of metrics in , such that for each i 1,,k the dilational Pohozaev invariants i(g j) are bounded away from zero. Then there is a subsequence of the g j converging to a metric g . The convergence is uniform in the topology relative to g , or indeed relative to any of the g j , on compact subsets of . theorem6 The question of what happens to divergent sequences in may also be determined. Before describing this we return to 2 . There is an obvious' compactification, 2 , which is the closure of in 2 . 2 2 decomposes into two disjoint sets: the first contains the single point 0,0 2 , which corresponds to the conformal factor 0 , while the second is the orbit u 0(t), v 0(t) passing through 1,0 , which corresponds to the incomplete spherical metric on p 1, -p 1 . These latter points on the compactification of 2 may be identified with the nonsingular round metric on itself, and these are themselves metrics of constant scalar curvature n(n-1) on . 2 is a stratified space; its principal stratum is 2 itself, the codimension one (and one dimensional) stratum consists of copies of the nonsingular round metric on , and finally the codimension two (zero dimensional) stratum consists of the single trivial solution of the PDE, which is a completely degenerate metric. There is a similar compactification of and corresponding decomposition of when the cardinality of is larger than two (and still finite). The following is a corollary of the proof of Proposition 3.12: theorem7 Let g j be a sequence such that for some nonempty subset of points p i 1 ,, p i s , the invariants i 1 (g j), , i s (g j) are uniformly bounded above by some - 0 0 for all j , and all other i(g j) tend to zero. Then this sequence has a convergent subsequence converging to a metric g , i .e . g is still singular at the points ' p j 1 , , p j s , but extends smoothly across the points in ' and R( g ) n(n-1) . If all (g i, p j) tend to zero as i then either g i tends to zero, uniformly on compact subsets of S n , or else g i converges to the round metric on S n . theorem7 This means that the compactification contains copies of ' for certain subsets ' . In addition it may contain a set whose points correspond to copies of the round metric on , and finally a set whose points correspond to the zero metric'. Later in this paper we shall give a somewhat better description of once we have determined the structure of itself better. Notice also that it must be somewhat subtle to determine precisely which subsets ' have ' occurring in . The subsets ' which can arise in the description above are determined by the Pohozaev balancing condition (3.9). For example, when has two elements, there is no piece of the boundary corresponding to ' for ' having just one element; this is because no complete solution of our problem exists on p . From the present perspective, this holds because 1(g j) - 2(g j) 0 for any j (here p 1,p 2 ) by virtue of (3.9), so that if one of these numbers tends to zero, the other must as well. (The difference in the signs here from (3.9) is because in (3.9) the same conformal Killing field X is used in each Pohozaev invariant, whereas here we use X corresponding to p 1 for one and -X corresponding to p 2 -p 1 for the other.) The subtle point is that the complete Pohozaev invariants, rather than just the dilational ones, are required when has more than two points, and these are not determined just linear algebraically by the location of the p i and the Delaunay models at these punctures. 4. Linear analysis on manifolds with asymptotically periodic ends In this section we prove various results concerning the analysis of the Laplacian and the linearization of the scalar curvature operator about solution metrics g . Many of the basic results hold more generally, e .g . for the Laplacian on manifolds with asymptotically periodic ends. The linearized operator For the remainder of this paper, L will always denote the linearization about the constant solution v 0 of the nonlinear operator equation split N(v) g(1 v) - n(n-2) 4 (1 v) n(n-2) 4(1 v) n 2 n-2 g v n v Q(v), split 4.1 equation where Q(v) is the same as in (2.12), so that, as before, equation L g n. 4.2 equation Let E j denote a neighborhood of the puncture p j which is conformally equivalent to a half-cylinder 0,) t n-1 . We fix these cylindrical coordinates around each p j . By virtue of Theorem 3.3, L can be treated on each E j as an exponentially small perturbation of the corresponding operator L j for the periodic Delaunay metric g j which is the asymptotic model for g on E j . Thus on each E j we may write equation L L j e -t F, 4.3 equation where F is a second order operator with coefficients bounded in as functions of (t,) . The linear analysis of the Laplacian on manifolds with asymptotically periodic ends is remarkably similar to that for manifolds with asymptotically cylindrical ends, as detailed for example in Me . In particular, the Fredholm theory for such an operator on exponentially weighted Sobolev (or Holder or ...) spaces has an almost identical statement in either case, although the proofs are rather different. The Fredholm theory in this asymptotically periodic setting was previously developed by Taubes in , although we proceed somewhat further into the linear analysis as we need more detailed results. It is also possible to develop a full scattering theory for the Laplacian on these manifolds. We shall actually require and develop some scattering theoretic results to clarify the nature of the moduli space . The Fourier-Laplace transform The basic tool for the parametrix construction, upon which all the linear analysis relies, is the Fourier-Laplace transform, in a form employed already by Taubes . We proceed to develop some properties of this transform, here denoted . The function spaces we shall use here are exponentially weighted Sobolev spaces based on L 2(;dV g) ; these will be written H s ( n ) , or just H s , for , s , s n 2 . The last condition ensures that the spaces behave well under nonlinear operations. To define them, decompose n into the union of the ends E 1, , E k and a compact piece K . Over K an element h H s restricts to an ordinary H s function. Over E j , h e t h , where h H s( 0,) n-1 , dt ,d) . Note that the measure here is uniformly equivalent to the one induced by g for any g . The transform is, strictly speaking, defined for functions on the whole cylinder C n-1 . It would be somewhat more natural to first develop its properties acting on e .g . the Schwartz space , but we shall specialize immediately to functions with support on just one half of C . So let h(t,)H s on E j , and assume h 0 for t 1 . Set equation h(t,,) (h) k - e -ik h(t k,). 4.4 equation Assume for the moment that h is smooth. Then, since h decays like e t , this series converges provided - . We have set i , so that . h(t,,) depends holomorphically on in the region - . When h is only assumed to be in H s , h(t,,) will still depend holomorphically on in the same region, but as a function with values in the space H s . h is continuous in - as a function with values in H s . These results follow from the Plancherel formalism. The transform is invertible, and its inverse is given by contour integration. To make the following equation clearer, assume that t , and t is its reductionmod 1 (so that 0 t 1 ). Then when t 1 so that we may write t t , equation h(t,) 1 2 0 2 e i h(t,,) d. 4.5 equation In this formula we integrate along a line 0 . By Cauchy's theorem this contour may be shifted to allow 0 to be any number less than - . If, as we are assuming, h vanishes for t 0 , then h(t,,) not only extends to the lower half plane, but decays like e there. Shifting the contour arbitrarily far down shows that the integral (4.5) vanishes for any 0 , as it should. By a similar argument, if h(t,) is defined by the integral (4.5) taken along some contour 0 , where the integrand h(t, , ) is only assumed to be defined along that line, then h H s - 0 . In particular, if h is holomorphic in some lower half plane - , and continuous with values in H s as a function of t up to this upper boundary, then h H s . Next, reindexing the sum defining h gives equation h(t 1, ,) e i h(t,, ). 4.6 equation This just means that h(t,,) is a section of the flat bundle on 1 n-1 with holonomy around the 1 loop. This bundle is isomorphic to the flat bundle with trivial holonomy; the bundle map is given by conjugating by e it . Thus the function equation h e -it h(t,,) e it 4.7 equation satisfies h(t 1,,) h(t,,) . Fredholm theory The basic Fredholm result for the linearization L may now be stated and proved. theorem8 There exists a discrete set of numbers such that the bounded operator equation L: H s 2 (M) H s (M) 4.9 equation is Fredholm for all values of the weight parameter . In particular, 0 , so the map (4.9) is not Fredholm when 0 , i .e . on the ordinary unweighted Sobolev spaces, but is Fredholm for all values of sufficiently near, but not equal to zero. theorem8 proof It suffices to construct a parametrix G for L so that LG - I and GL - I are compact operators on H s and H s 2 , respectively. As usual, G may be constructed separately on each piece of the decomposition of into a compact piece and the k ends E 1, , E k around each p j . The parametrix construction on the compact piece is the standard microlocal one since L is elliptic. We construct a parametrix on each E j using the Fourier-Laplace transform. Fix one of these ends, E j , and let g be the model asymptotic Delaunay metric for the fixed metric g there. The corresponding model operator L , as in (4.3), has periodic coefficients (for notational convenience we assume that the period is one here) so it acts on sections of the flat bundle with holonomy described above by the obvious rule (L h) L h . This induced operator L looks just like L in local coordinates (t,) . This step is the same as conjugating L by . We may proceed further and conjugate L by e it so as to act on the trivial flat bundle. This final induced operator, which depends holomorphically on , will be called L () : equation L () e -it L e it . 4.10 equation The main point of the proof is that L () has an inverse (say on L 2( 1 n-1 ) ) which depends meromorphically on . This will be a direct consequence of the analytic Fredholm theorem, which is proved for example in RS and states that a strongly holomorphic family of Fredholm operators, depending on the complex variable , either fails to be invertible for every , or else is invertible for all except for those in some discrete set in the parameter space. To check that this result is applicable, simply note that L () is elliptic for every , (as already observed) depends holomorphically on and, by the various transformations we performed, acts on a fixed function space (on a fixed bundle) over a compact manifold. Hence it forms a family of Fredholm operators; standard considerations show that the holomorphic dependence of the coefficients of this operator on ensure the strong holomorphy of the family. Our next task is to show that L () is invertible for some value of . Once this is accomplished, the analytic Fredholm theorem will imply that the set of poles j of L () -1 is discrete in . The invertibility of L () at some is equivalent to the invertibility of L acting on the flat bundle with holonomy . When is real, this operator is self-adjoint, hence invertible so long as it has no nullspace. A solution of L 0 on this bundle lifts to a function (t,) on n-1 satisfying (t 1,) e i (t,) . Since is real here, this lift is bounded and quasi-periodic.' However, we have already shown in 2 that such a function can exist only when 0 ; in fact, the arguments there show that any temperate solution must be constant on the cross-section n-1 , i .e . be a solution of the reduced operator L 0 . In addition, the functions 1 and 2 of (2.15) are independent solutions of this operator, hence span the space of all solutions since L 0 is a second order ODE. Finally, neither of these transform by a factor e it over a period except for 0 . Thus L is invertible for every real (0,2) . This proves that L () -1 exists and depends meromorphically on . We make some remarks about the set of poles j of this meromorphic family of inverses. First, if is in this set, then so is 2 for any . In fact, although L ( 2) is not equal to L () , these two operators are unitarily equivalent: the unitary operator intertwining them is multiplication by the function e 2i t . So, for some fixed , either both of these operators are invertible or neither of them is. By discreteness and this translation invariance, the set j has at most finitely many inequivalent (mod 2 ) poles in any horizontal strip a b . Note in particular that the only poles on the real line occur at the points in 2 . By discreteness again, there are no other poles in some strip - , 0 . We shall usually restrict attention then only to values of in the strip 0 2 (or equivalently, in 2 ); will denote the set of poles in this strip. Finally, note that the adjoint of L () is precisely L () , and so is invariant under conjugation. This means that we can list the elements of as follows: equation , - 2, - 1, 0 0, 1, 2, . 4.11 equation Also define equation j j: j , - 1, 0, 1, . 4.12 equation The j form a strictly increasing sequence tending to infinity, and in particular, is a discrete set in . We shall provide an interpretation of the nonzero elements of in the next subsection. Next we unwind this inverse L () -1 G () to obtain a parametrix for L on E j . The inverse for L acting on the flat bundle with holonomy is given by G e it G e -it , for every . To obtain an inverse for L we simply conjugate G with the Fourier-Laplace transform . Recall, though, that this makes sense only if the contour integral in the definition of -1 avoids . So, by conjugating, and in doing so, taking this integral along a contour - , for any , we obtain an inverse for L . Since integrating along this contour produces a function in H s ( n-1 ) , we have obtained an inverse G G , for L acting on H s functions on E j , supported in t 1 for all . Clearly this G is a parametrix with compact remainder for L L e -t F acting on H s (E j) . As a final step, couple these parametrices we have produced on each end E j to the interior parametrix to obtain a global parametrix for L on all of with compact remainder. Both right and left parametrices of this type may be obtained this way, and their existence implies immediately that L is Fredholm on H s whenever . It is a simple exercise to check that L does not even have closed range when , in particular, when 0 . proof It is obviously of interest to determine when L is actually injective or surjective. In general this is a rather complicated question, but we make note of the following result. theorem9 Suppose that L has no L 2 nullspace (i .e . is injective on H s for 0 ), then for all 0 sufficiently small (with ) align L: H s 2 H s is surjective , L: H s 2 - H s - is injective . align theorem9 This first statement follows from the second and duality, since L is self-adjoint on L 2 , i .e . when 0 . Asymptotic expansions The Fourier-Laplace transform may also be used to obtain existence of asymptotic expansions for solutions to Lw 0 on each end E j . For our purposes in this paper, it will only be important to know that any such w has a leading term in its expansion, which decays (or grows) at some specified rate, and an error which decays at a faster rate. One use of this will be to estimate the exponent appearing in (3.5). However, the full expansion is not much harder to prove, so we will do this too. The starting point is the fact, discussed above, that the Fourier-Laplace transform of a function w H s ( n-1 ) , supported in t 0 , say, is a function w(t,,) which is holomorphic in the half-plane - (and taking values in H s( 1 n-1 ) ). For a general function of this type, the most that can be concluded is that it has a limit on the line - in the appropriate L 2 -sense discussed earlier. When the solution w is transformed to a function w(r,) w(-r,) on n 0 , elliptic regularity shows that w is a conormal distribution with respect to the origin, i .e . has stable regularity when differentiated arbitrarily often with respect to the vector fields r r, (this is just the same as w itself having stable regularity with respect to t, ); M1 , for example, contains a discussion of conormal regularity. For degenerate operators of a type closely related to L , one expects solutions to be polyhomogeneous conormal, cf . M1 or Me . Polyhomogeneity is simply the property of having an asymptotic expansion in increasing (possibly complex) powers of r and integral powers of r , with coefficients smooth in (as functions of t these expansions are in analogous powers of e t and t ). Alternately, polyhomogeneous conormal distributions may be characterized as those with Mellin transforms, already defined and holomorphic in some lower half-plane in , extending meromorphically to the whole complex plane, with only finitely many poles in any lower half-plane, all of which are of finite rank. w will not be polyhomogeneous, except when u and the underlying metric is cylindrical. In fact, w (cut off so as to be supported in r 1 ) will have Mellin transform still defined and holomorphic in a half-plane and extending meromorphically to all of , but now its poles are arranged along lattices on a countable discrete set of horizontal lines. Each pole is still finite rank. This meromorphic structure of the Mellin transform of w is equivalent to the fact that the terms in the expansion for w have the form e -t w j , where w j is periodic in t . The residues at the poles along a fixed horizontal line at height will be the Fourier series coefficients of w j . This more general discussion has involved the Mellin transform as r 0 (or equivalently, the standard Fourier transform as t ). We shall revert now to the Fourier-Laplace transform; it is much less flexible than either of the other transforms, since it presupposes periodicity, but suffices here for our immediate purposes. Suppose that w H s ( n-1 ) for some is supported in t 0 and solves L w f for some compactly supported smooth function f (hence w H s for every s by elliptic regularity). For example, we could take , which solves L 0 , and let w where is a cutoff function having support in t 0 and equaling one for t 1 . Taking transforms we get Lw f ; this function on the right, f(t,,) , is obviously smooth in (t,) and entire in . Applying the inverse G from the last section gives w(t,,) G f(t,,) . The right side of this equation is meromorphic in with poles at some subset of points in , hence the same is true for the left side as well. Notice that the poles of G in - must be cancelled by zeroes of f since w is a priori known to be regular in this half-plane. The function w is recovered by inverting , integrating along the line - . However, by Cauchy's theorem this integration may be taken along any higher contour -' , so long as the interval ', does not contain any points of . Thus, for any such ' , w H s ' . If the contour is shifted even further, so as to cross a point in , i .e . a pole of G , then the resulting integral along the line -'' produces a new function v . Since (L v) L Gf f and f is entire, it follows that v also solves Lv f , and so w and v must differ by an element of the nullspace of L . By construction vH s '' , and so we have decomposed w as a sum v , where L 0 . is given by the residues of w at the points j with - (- j) - j -'' . (We use - j instead of j for notational convenience only.) If there is just one such point, say - j , then it is clear that in fact H s j for every 0 . Actually, it is not hard to show, using results from 2, that this must grow or decay exactly like a polynomial in t times e j t . This process may be continued by moving the contour past more and more poles. At each step, we have decomposed w into a sum of solutions of L 0 and a term which decays at a rate given by the height of the contour. theorem10 If w solves L w f for some compactly supported function f on (0,) n-1 , with w H s for some - , then, as t , w(t,) j(t,) , where each j solves L j 0 and j decays like a polynomial in t multiplying e - j t , where j- as in (4.12). theorem10 Note that this result shows that the poles j , or at least their imaginary parts, j , correspond to the precise growth rates of solutions of L 0 on the cylinder. A similar, though slightly more complicated, expansion holds for the elements of the nullspace of the linearization L in (4.2). theorem11 Let w solve Lw 0 , at least on some neighborhood of the puncture p j , and lie in H s for some - , where is the set of imaginary parts of poles corresponding to the inverse G for the model L for L near this puncture. Then w has an asymptotic expansion equation w(t,) j,k 0 j,k (t,) equation as t . In this sum, the leading terms' j,0 are solutions of L j,0 0 corresponding to the poles j above - . These decay, as before, like a polynomial in t multiplying e - j t . The higher terms j,k , k 0 , decay like a polynomial in t multiplying e -( j k)t . theorem11 proof As before, we may assume that w is supported in t 0 and Lw f is compactly supported and smooth. By (4.3), equation L w -e -t Fw f. 4.16 equation Now conjugate by the Fourier-Laplace transform, and apply the inverse G of L to get equation w G(-e -t Fw) Gf. 4.17 equation The term on the left is holomorphic, a priori, in the half-plane - . The second term on the right is entire, while the first term on the right is holomorphic in - and extends meromorphically to the slightly larger half-plane - . As before, w can be recovered by integrating along -- ; if this contour is moved up to - - , and if the strip - - contains no poles, then we find that w H s - . If there are poles in this strip, then, exactly as in the last proposition, w decomposes into a sum v with v H s - and L 0 . in turn decomposes into a sum of terms j,0 corresponding to the various poles in this strip. This improved information may be then fed back into (4.16) and (4.17). Now the right side of (4.17) extends meromorphically to the half-plane - 2 , and the contour may be shifted further to get more contributions to the expansion for w . Continuing this bootstrapping yields the full expansion. Details are left to the reader. proof The deficiency subspace Particularly important for us in the application of the implicit function theorem will be the pole 0 for G (for any value of the Delaunay parameter ). Specifically, we will be concerned with those solutions of L w 0 and Lw 0 which are in H s for every 0 , but not in any H s - . Here we are still only concerned with the local behavior of these solutions on each end E j . Their global nature will be discussed later. The representatives of a basis of ( Ker ,LH s) ( Ker ,LH - s) are given in (2.15) as the functions 1 1, and 2 2, . They depend on t but not on . The fact that 2 grows linearly in t indicates that G has a pole of order 2 at 0 . These are the only temperate solutions of L 0 on the whole cylinder; any other solution grows at an exponential rate (with possible exponents given by the values of ). If the solution metric g has model Delaunay parameter on the end E j , then these functions (for that value of ) may be cut off and transplanted onto this semi-cylinder. They do not decay, but it is easy to create a sequence of cut-offs i ( 1 or 2 ) with the following properties: each i is compactly supported, and has support tending to infinity as i tends to infinity. The L 2 norm of i equals one for all i , but the L 2 norm of L ( i) tends to zero ( L is the linearization, not one of the models L here). The existence of such sequence is a standard criterion for showing that L does not have closed range on L 2 . In any event, we think of 1, 2 as constituting the bounded approximate nullspace for L (here bounded' is loosely interpreted to encompass the linearly growing 2 ). Define now a linear space W generated by the functions 1, j , 2, j , cut off in a fixed way so as to be supported in t 0 and transplanted on each end E j ; the j are the Delaunay parameters for the particular model metrics g j on E j . Since there are k ends, W is 2k -dimensional. For reasons that will become clear in the next section, we call W the deficiency subspace for L . Clearly W H s for every s and every 0 . The Linear Decomposition Lemma There are two important corollaries of Proposition 4.15 and its proof which we single out in this section. The first concerns the behavior of solutions of the inhomogeneous equation Lw f on each end E j , while the second studies the exact value of the exponent appearing in Theorem 3.3 and then later in (4.3), etc. As already pointed out in Corollary 4.13, if L has no global L 2 nullspace, then we can find a solution w H s 2 to the equation Lw f for every f H s , whenever 0 . In particular, this holds whenever f H s - . Clearly, whenever f decays at some exponential rate like this, we expect the solution w to be somewhat better behaved than a general H s 2 function; of course, it is immediate that it is in this space for any 0 , but we can do even better. This is the subject of what we will call the theorem12 Suppose f H s - for some 0 sufficiently small, and w H s 2 solves Lw f . Then w H s 2 - W , i .e . w may be decomposed into a sum v with v decaying at the same rate as f and in the deficiency subspace W . theorem12 proof Clearly this question may be localized to each end E j , and in this localized decomposition will be a combination of 1 and 2 . The decomposition is achieved by exactly the same sort of shift of the contour in the integral defining -1 across a pole of G . Here the contour is being shifted from - to ; the pole crossed is the one at 0 . proof The second corollary deals with the rate at which a general solution of (1.3) on a punctured ball, singular at 0 , converges to the radial Delaunay metric. We assume the simpler statement of Theorem 3.3 that u decays to u at some exponential rate , and use the linear theory to find the optimal rate. First transform the punctured ball to the half-cylinder (0,) n-1 , and assume all functions are defined here. Write the solution u as a perturbation (1 v)u , so that N (v) 0 , where N is the nonlinear operator (2.11). This equation is the same as L v -Q(v) , where Q is the quadratically vanishing function in (2.12). We already know that v H s - (for all s ), hence Q(v) H s -2 . By the contour-shifting arguments above, this implies that v itself decays at this faster rate, at least provided 2 1 , where 1 is the first positive element in . Continue this process until this first pole at 1 has been crossed; the conclusion is that v decays exactly like e - 1 t (possibly multiplied by a polynomial in t ). We could also bootstrap further and obtain a complete asymptotic expansion for v , of the same general form as in Proposition 4.15 above, although more exponents occur because of the nonlinearity. (This is a simple form of the argument in M2 .) We summarize this discussion as theorem13 The exponent occurring in Theorem 3.3 governing the rate of decay of a general solution u to its model Delaunay solution u is equal to the first nonzero element 1 in the set corresponding to G . The function u admits a complete asymptotic expansion into terms of increasingly rapid exponential decay. theorem13 The bounded nullspace In this last subsection of 4, we finally come to the global behavior of that portion of the nullspace of L corresponding to the pole at z 0 . This space, , which we shall call the bounded nullspace' of L , is defined by equation v H s : Lv 0 wH - s:Lw 0 . 4.20 equation The full nullspace of L on H s will be the direct sum of and the L 2 nullspace (which, by Proposition 4.15, is the same as the nullspace of L in H s - ). In particular, when this latter space is trivial, is the full nullspace of L in H s . By the Linear Decomposition Lemma 4.18, is already contained in H s - W . The purpose of this subsection is to determine the dimension of . As usual, an index theorem is the principal tool for calculating () . Fortunately we require only a relative index theorem, which computes the difference between two indices in terms of asymptotic data, rather than global data. To be more explicit, for any set equation () ,ker , . L H s - ,coker , . L H s . 4.21 equation This is obviously independent of s . Since the adjoint of L on H s is L on H -s - , duality implies that equation (-) -() for every . 4.22 equation If 1 and 2 are any two allowable values (i .e . neither is in ), the relative index with respect to these two numbers is simply the difference equation ( 1, 2) ( 1) - ( 2). equation In particular, using duality again, equation split (,-) 2 ,() 2( ker . L H s - ker . L H s - ) 2(). split 4.23 equation As noted above, this relative index can be shown, on fairly general principles, to be computable in terms of asymptotic data for the operator L . Finding a specific and computable formula is another matter and, to our knowledge, there is no general result of this sort available for asymptotically periodic operators. However, such a result is available for operators associated to asymptotically cylindrical metrics, and we will use this instead. Our result is theorem14 () k . theorem14 proof By (4.23), it suffices to show that (,-) 2k . By the usual stability properties of the index (hence any relative index) under Fredholm deformations, we compute this number by choosing a one-parameter family of Fredholm operators L , 0 1 , with L 0 L , and L 1 an operator for which this relative index is computable. Since L is just g n , we shall choose a one-parameter family of metrics g and define L to be g n . The metrics g will agree with g except on each of the ends E j , where we make the following homotopy. First deform g on each end to its model Delaunay metric g , and then deform each g through Delaunay metrics to the cylindrical metric g u . The metric g 1 will agree with the original g on any large fixed compact set, and will equal g u on each end E j . This metric is now an exact b -metric , in the language of Me (more prosaically, it has asymptotically -- and in this case, exactly -- cylindrical ends), and the corresponding operator L 1 is an elliptic b -operator. Of course, we still need to prove that L is Fredholm on H s and H s - for every 0 1 , provided 0 is sufficiently small. In the part of the deformation where g is homotoped to its model Delaunay metric g on each end, this is obviously true. For the remaining part of the deformation, we need to know that the elements of j , j 1 , which are the weights for which L will not be Fredholm, remain bounded away from 0 as varies between its initial value and u . This is precisely the content of Lemma 2.21. Now, to apply Melrose's Relative Index Theorem Me , note that we may as well consider the operator L 1 n-2 n L , which on each end takes the form t 2 2 (n-2) . The set for this operator (called spec b(L 1) in Me ) is i n-2 , j, j 1 , where j are all strictly positive, tending to infinity, and obtained in a straightforward manner from spec ( ) . The prescription to calculate the relative index is to first consider the indicial family,' which in this case equals (n-2) - 2 (it is obtained by passing to the Fourier tranform with respect to t , which carries t to -i ). This is a holomorphic family of elliptic operators on n-1 , and its inverse G() has poles exactly at the points of . Since we are computing the jump in the index as the weight changes from - to , we need to compute the degree' of each pole (as defined in Me ) for each element of this set with imaginary part equal to zero. Each of the poles n-2 is simple and of rank one, and so the degree of each is also equal to one. Finally we need to sum over all poles and over all ends E j , because the preceding discussion is local on each end. For each end there are two poles of G with imaginary part zero, each contributing a degree of one to the computation, and there are k ends; the sum of all of these is 2k , and this is the relative index (,-) . proof 5. The moduli space: smooth points In this section we commence the study of the moduli space itself. Here we use the linear theory developed in the previous section in a straightforward way to study neighborhoods of the good points' g , where the associated linearization has no L 2 nullspace. In the next two sections we develop ideas to study neighborhoods of the nonsmooth points. The implicit function theorem is directly applicable only when the linearized operator L g g n is surjective on the appropriate function spaces. These function spaces should be tangent to a suitable space of metrics conformal to g and with prescribed growth conditions on each end of . Let g , and consider all nearby metrics conformal to g of the form (1 v) 4 n-2 g . The most natural class of such metrics is equation Met s 2 - (1 v) 4 n-2 g, v H s - , v -1 . 5.1 equation Clearly equation T g Met - s 2 H s 2 - . equation The operator N g(v) , as defined in (4.1), can be considered as a map from this space of metrics to H s - , and as such is obviously . However, it is never surjective, since - 0 . The obvious alternative, to consider metrics growing at the rate , is unsuitable because of the nonlinearity. The case where this difficulty is easiest to remedy is when L g has no L 2 nullspace: equation ker , (L g) L 2(,dV g) 0 . 5.2 equation This will be our standing hypothesis in the rest of this section. We can rephrase this condition by recalling, by virtue of Proposition 4.15 on asymptotics of solutions of L 0 , that if were in the L 2 nullspace of L , then it would decay at the exponential rate e - 1 t on each end, where 1 1( j) is the first nonzero element of j on the end E j , corresponding to the model Delaunay metric g j there. In particular, under the hypothesis (5.2), for (0, 1) , the operator L has no nullspace on H s - for any s . Now apply Corollary 4.13 and the Linear Decomposition Lemma 4.18 to conclude that (5.2) is equivalent to the statement that equation gathered L: H s 2 - (, dV g) W H s - (,dV g) is surjective for any 0 j 1( j) and s . gathered 5.3 equation W is the 2k -dimensional deficiency subspace introduced in the last section. Because we work with a nonlinear equation, we assume that s n 2 . theorem15 Suppose g is a metric such that the hypothesis (5.2) is satisfied. Then there exists a 2k -dimensional open manifold , the elements of which are functions v on to be described below, and with a distinguished element v 0 , such that the nonlinear map equation N g: H s 2 - (; dV g) H s - (; dV g) equation defined by N g(v,) N g(v ) where N g is the operator given by (4.1), restricted to a neighborhood of 0 in the first factor, is real analytic. Moreover, the tangent space T v 0 is identified with the deficiency subspace W , and using this identification, the linearization L g of the map N g is surjective, as in (5.3). theorem15 theorem16 Suppose g satisfies the hypothesis (5.2). Then there is an open set containing g , such that is a k -dimensional real analytic manifold. theorem16 The proof of the corollary follows directly from the surjectivity statement (5.3), by a straightforward application of the real analytic implicit function theorem. The dimension k is of course the dimension of the nullspace of L as a map (5.3). This nullspace is the bounded nullspace defined in the last section, and by Theorem 4.24 its dimension is k . So to conclude the proof of the theorem and the corollary it suffices to construct , and to show that N g is a real analytic mapping. The only difficulty in the construction of is that the elements of W are not, generally, bounded either above or below, so we could not use (1 v) 4 (n-2) as a conformal factor for most v W . However, recall from (2.15) that each element j, , j 1,2 , of the bounded nullspace for the model problem on the cylinder is the tangent to a curve j(t,,) of actual conformal factors. The actual curve v we would want to use for each such element of W on an end has the form equation v (t) cases u (t ) u (t) - 1, j 1, u (t) u (t) - 1, j 2. cases equation The manifold is constructed by gluing together these local definitions from each end E j in an essentially arbitrary, but smooth, manner. By construction, T 0 W , as desired. To check the real analyticity of N g , the main step is to write N g(v ) in such a way that it clearly lands in H s - . Hence, on each end write N g N e -t Q , and then use a common formula for the remainder in Taylor's theorem to obtain equation N g(v ) N () e -t Q(v ) 0 1 N g'( sv) ,ds v. equation Since N () 0 , every term on the right here is in H s so long as . Furthermore, every term is real analytic in (v,) . Real analyticity of N g on the interior, away from the ends, is even easier. The rest of the proof is now standard. 6. The real analytic structure of In this section we prove that is always an analytic set by representing it as the slice of an infinite dimensional real analytic manifold with the conformal class g 0 . We go on to prove a generic slice theorem, that even if itself is not smooth, generic nearby slices of by other conformal classes will be. The urmoduli space As described above, we wish to regard as the slice by the standard conformal class g 0 of the infinite dimensional set consisting of all metrics, not necessarily in the standard conformal class on n , with constant scalar curvature n(n-1) . For obvious technical reasons, we consider only those metrics which satisfy appropriate growth and asymptotics conditions near the punctures p j (this will be be elaborated on below.) Consideration of this big', or ur-', moduli space (when the underlying manifold is compact) was first undertaken by Fischer and Marsden in the early 1970's ( FM1 and FM2 ), motivated in part by concerns in general relativity. They proved that it is a smooth Banach manifold (in any one of a number of standard Banach completions), provided a certain overdetermined linear equation has only the trivial solution. Their precise set-up was to consider , the set of all metrics (of some fixed finite regularity) with scalar curvature function , a fixed function on the underlying manifold X . To show that this set is a Banach manifold, it suffices to show that the linearization of the scalar curvature map (which assigns to any metric g its scalar curvature function R(g) ) is surjective at any g . Since this map R carries metrics to functions, its linearization, which we will call g here, carries the tangent space at g of the space of all metrics, i .e . the symmetric 2 -tensors, to scalar functions. A. Lichnerowicz Li had earlier computed that, for h a symmetric 2 -tensor, equation g(h) h - g( tr , h) - r g, h . 6.1 equation Here is the divergence operator on tensors, and r g is the Ricci tensor for g . It is easy to check that g reduces to (a multiple of) the linearization operator L g we have been studying when h is a multiple of the metric g (and hence is tangent to the conformal class g ), when n(n-1) . It is straightforward to check that the symbol of g is surjective, so it follows that the symbol of its adjoint g is injective, and then that the symbol of g g is an isomorphism. Therefore, this last, fourth order operator is elliptic; at least when X is compact, it is Fredholm, and in particular has closed range of finite codimension. Since range , ( g) range , ( g g ) , the range of g itself is closed and of finite codimension. (It does, however, always have an infinite dimensional kernel.) By virtue of all this, when X is compact, g is surjective if and only if g is injective, and this ( g (f) 0 ) is precisely the overdetermined equation referred to earlier. From (6.1) it follows readily that equation g (f) Hess , g(f) - g g(f) - f ,r g. 6.2 equation If g (f) 0 , then taking the trace we also get that g(f) -( (n-1))f ; substituting this back into (6.2) we obtain finally the condition equation g (f) 0 Hess , g(f) -( r g - n-1 g ) f. 6.3 equation Fischer and Marsden were able to show that (6.3) (or equivalently, (6.2)) has only the trivial solution f 0 except possibly in the special cases where is a nonnegative constant and (n-1) is in the spectrum of - g . Thus, except for these cases, is a smooth Banach manifold. In the case of interest here, the underlying manifold is not compact, and (n-1) n is always in spec , ( g) . We address the former of these concerns first, since the Fredholm theory of 4 may be adapted easily to this problem. As before H s () will denote the Sobolev space of scalar functions decaying like e t on each end, with respect to fixed cylindrical coordinates (t,) there and with respect to a fixed metric g (any element of the moduli space can be used as a fixed background metric to define these Sobolev spaces). Define also H s (, Sym 2T ()) to be the space of symmetric 2 -tensors h e t k , with k in the unweighted Sobolev space H s(, Sym 2 T (); dV g) . theorem17 For any g , the operator (6.1) is bounded as a linear map equation g: H s 2 (, Sym 2(T ())) H s (). equation It has an infinite dimensional nullspace, and the closure of its range has finite codimension for all . There exists a discrete set ' such that for all ' , this map also has closed range. theorem17 proof The boundedness assertion is immediate. The proof of the closed range part is very close to the analogous one given in 4 for the scalar operator L g , so we just provide a sketch. As in FM2 and above, the surjectivity of the symbol of g guarantees that we merely have to show that A g g g , which is elliptic, has closed range for all ' . The operator A g is asymptotically periodic on each end E j of , and the Fourier-Laplace transform may again be employed to construct local parametrices on each of these ends. The fact needed to make this work is that the holomorphic family of elliptic operators A () , on the compact manifold 1 n-1 , and associated to the Delaunay model operator A g , is invertible at some . In this case the analytic Fredholm theory, and the rest of the construction in 4, proceeds exactly as before. Again we prove this for , since for such , A () is self-adjoint. Invertibility is equivalent to the injectivity of the induced operator () , and this in turn is implied (by taking traces) by the injectivity of the model operator L () . The invertibility of this last operator for all (0,2) (we are again assuming the period of , et al ., to be one for simplicity) was already proved in 4, and so we are done. proof To proceed with the program of FM2 , we also have to show that g is injective on for any g . If this were true, then the urmoduli space would be a real analytic manifold in a neighborhood of every g , i .e . there would exist a (local) thickening of into the real analytic moduli space such that g 0 , as desired. Unfortunately this is not quite so simple, since, as already noted, (n-1) is always in the spectrum of g ; we are thus precisely in the case not treated by FM2 . The equation (6.3) is rather similar to the well-known equation of Obata, which also appears in the study of the scalar curvature, and actually reduces to Obata's equation when X is an Einstein manifold. In analogy with the situation there -- in particular, Obata's characterization of ( n,g 0) as the only compact manifold admitting nontrivial solutions of his equation -- Fischer and Marsden conjectured that (6.3) admits a nontrivial solution, at least when X is compact, if and only if X is the standard sphere. This is now known to be false: indeed, let X be the product of the circle of length L , 1(L) , with any Einstein manifold E n-1 of positive scalar curvature, e .g . n-1 . Then, if the scalar curvature of E is normalized to be (n-1)(n-2) , (6.3) implies that f depends only on the variable t along the circle, and satisfies the ODE f'' (n-2)f 0 . Thus, (6.3) admits a nontrivial solution whenever L is an integer multiple of n-2 . There are also noncompact counterexamples to their conjecture: the simplest are just the universal covers E n-1 of the compact examples above. The characterization even of all compact manifolds admitting solutions to (6.3) is still unknown, although it is almost certain that the list of possibilities is rather small. Fortunately, though, the characterization of all complete conformally flat manifolds admitting nontrivial solutions to (6.3) was obtained about ten years ago, independently by J. Lafontaine and O . Kobayashi Ko . The simplification in this case is that the Ricci tensor of any conformally flat manifold is always harmonic. Kobayashi and Lafontaine proved that beyond the sphere and the product type examples discussed above, there is also a collection of examples which are warped products of 1 with E n-1 . We shall not write these examples explicitly here, but simply note that, provided k 2 , the solution metrics (,g) , g , are never warped products (for topological reasons alone ). We obtain then theorem18 There are no nontrivial solutions to the equation g (f) for any g , provided k 2 . theorem18 Even when k 2 , only the cylinder n-1 amongst the Delaunay metrics appears on the list of counterexamples. To check this, use the system of equations given in Lemma 1.1 of Ko . These involve both the warping function for the metric and the function f . Elementary manipulations show that the noncylindrical Delaunay metrics never satisfy this system. theorem19 is a real analytic Banach manifold in a neighborhood of . theorem19 proof It remains only to set up the precise function spaces on which the implicit function theorem (see Fe , page 239) will be applied, and make a few additional comments. We use, of course, the weighted Sobolev spaces H s of symmetric 2 -tensors and functions, as before. The crux of the argument is the fact that when g , g is surjective both on H s and on H s - , and indeed on H s for any ' This seems somewhat counterintuitive, but follows from the highly overdetermined nature of g . In fact, we already know that g has closed range on H s , ' by Proposition 6.4. Its cokernel is identified with the kernel of the adjoint, g , which is a map from H -s - to H -s-2 - . By injectivity of the symbol of g we may use elliptic regularity to replace -s here by any positive number, e .g . s . Finally we can invoke the results of Lafontaine and Kobayashi, via Proposition 6.5, to conclude that g has no nullspace, regardless of the weight - . Now consider the scalar curvature map equation g: H s 2 - ( Sym 2) H s - ( Sym 2), 6.7 equation defined in the obvious way. We are including the factor here so that points of the urmoduli space are allowed to have varying neck sizes, but all other perturbations are required to decrease exponentially on the ends. To apply the analytic implicit function theorem we need to know, first, that the linearization equation g: H s 2 - ( Sym 2) W H s - 6.8 equation is surjective, which we have just established, and second that (6.7) is a real analytic mapping of Hilbert spaces. This latter statement is also straightforward, so this completes the proof. proof One unfortunate shortfall of this theorem is that, although provides a thickening of , it is not a uniform thickening, i .e . one of fixed width' around . However, over any compact subset of we can ensure the existence of out to some fixed distance by the obvious covering argument. To find this fixed width thickening uniformly on we would need to understand more about the compactification . If the analogies of the result of Kobayashi and Lafontaine as well as the asymptotics result, Theorem 3.3, and the compactness result, Proposition 3.12, were known for non-conformally flat metrics, this would be unnecessary, and many of the results here could be given a more satisfactory form. Analyticity of and the Generic Slice Theorem In addition to its intrinsic nature and relationship to the concerns of this paper, the urmoduli space is required for two purposes: to show that itself is analytic, and to prove the Generic Slice Theorem. The proofs of these results are very closely related, so we develop the preliminaries for them simultaneously. Inside the ambient space H s 2 - ( Sym 2) there are two analytic (Hilbert) submanifolds, namely and the conformal class g 0 . (By g 0 we always mean the set of all metrics in conformal to the round metric g 0 .) Likewise we have all other conformal classes in ; a generic such class will be denoted g' . Notice that all the analytic machinery, in particular the Fredholm theory and relative index computations in 4, still holds for any metric (corresponding to an element) in . Choose any element g and let g' be the conformal class of g . We shall assume that g' is very near g 0 . The immediate aim is to prove that and g' intersect almost transversely at g ; in general these submanifolds may not be transverse at g , however they will always form a Fredholm pair there. This means that their tangent spaces, E T g g' and F T g , which are closed linear subspaces in V T g H s 2 - W , form a Fredholm pair, i .e . that E F is finite dimensional, and that E F is closed in V and has finite codimension there. To any Fredholm pair one can associate an index, which is the dimension of E F minus the dimension of V (E F) . This number is stable under perturbations of the pair. These matters are explained more thoroughly in . theorem20 and g' are a Fredholm pair at any point g in their intersection. theorem20 proof The orthogonal complements of E and F are given by equation split E h: g(h) 0 , F h: h g (f), f H s 4 - . split 6.10 equation It will suffice to show that EF and E F are finite dimensional, and that the orthogonal projection : F E has closed range. First suppose that h EF . Then h fg with f H s 2 - W and g(h) 0 . Computing g(fg) we find that L g(f) 0 . Since L is a Fredholm operator, there can be at most a finite dimensional family of such h in the intersection; the precise dimension is the same as the nullspace of L on H s 2 . On the other hand, if h E F , then h g (f) with f H s 4 - and g(h) 0 . But g( g (f)) is, up to a factor, just L g(f) , so the dimension of this intersection is the same as the dimension of the nullspace of L on H s 2 - , and is thus finite. To conclude the proof, we need to know that the projection has closed range (and hence is Fredholm, by the work above). The map is defined, for any h V , by decomposing h g k , where (k) 0 , and setting (h) k . Now suppose h j is a sequence of elements in F , h j 1 for all j , but k j (h j) 0 in norm. Since g(h j) 0 we have (n-1)L g( j) k j - r g, k j . The right hand side of this equation certainly goes to zero in norm, so by our Fredholm theory for L g , the functions j decompose as j j j with j going to zero in norm and j in the nullspace. Since j g is in the nullspace of g also, we may subtract it off from h j , and get that h j 0 , contrary to hypothesis. Hence restricted to F has closed range, and this completes the proof. proof Before stating the next result we need to introduce some notation. First, if g' is a conformal class on n (represented by elements in , and assumed to be near g 0 ), then ( g' ) will denote the moduli space of complete metrics on with constant scalar curvature n(n-1) in the conformal class g' . For 0 any sufficiently small number we also let , ( g' ) denote the subset of ( g' ) consisting of solution metrics g with the Killing norms of all Pohozaev invariants bounded below by (see 7 for a discussion of these norms). This is simply the subset of ( g' ) consisting of solutions with neck sizes uniformly bounded away from 0 by . By Proposition 3.12, each , ( g 0 ) is compact; then, provided g' is sufficiently close to g 0 , , ( g' ) will also be compact, since it is contained in a compact neighborhood of , by virtue of the implicit function theorem and the considerations above. theorem21 For any fixed 0 , the truncated moduli space , ( g' ) is a k -dimensional real analytic manifold for all conformal classes g' in a set of second category and sufficiently close to g 0 . theorem21 proof The reason for introducing the truncated moduli spaces is, of course, that we know the existence of to a fixed distance away from only over each , . We shall not comment further on this modification, but simply remark that it could be removed provided somewhat more were known about , as discussed at the end of the previous subsection. This result follows from the Sard-Smale theorem, once we have checked the hypotheses. The basic point, of course, is that ( g' ) is the same as g' . We parameterize the set of conformal classes (not modulo diffeomorphisms ) close to g 0 by the linear Hilbert space E k H s 2 - : g(k) 0 for some fixed g . Now consider the projection map equation : E . 6.12 equation is given by the preimage -1 (0) . The preimage -1 (k) in is precisely the set ( g' ) where g' g 0 k is the conformal class corresponding to k . This preimage is a smooth, in fact real analytic, manifold provided k is a regular value of . By Sard-Smale, once we know that is a Fredholm map (of index k ), then the set of regular values in E is of second category. Observing that the tangents to the orbits of the diffeomorphism group are certainly not of second category, we may conclude that the moduli spaces over generic conformal classes close to g 0 are smooth and k dimensional. However, the assertion that is Fredholm of index k is contained within the preceding lemma. Indeed, if g and F T g() , then we proved there that the projection :F E is Fredholm, and its index is the same as the relative index of L g across the weight 0 , i .e . equal to k . To finish the proof we need to eliminate the possibility that () is contained within a (finite codimensional) submanifold of E . Although this would follow from knowing that is surjective, this is a difficult issue. Instead we appeal to the existence theory for the nonlinear equation, generalizing Schoen's basic construction. In fact, the construction of solutions given in S2 may be carried out not only for the standard round metric g 0 , but also for generic metrics g' which are small perturbations of g 0 compactly supported away from . The only modification of Schoen's method needed to accomplish this is given in P1 , Proposition 1.2. The linearization of this set of compactly supported perturbations is clearly dense in E , hence there is a dense set of k E near the origin for which the preimage -1 (k) is nonempty. This is sufficient to conclude that () contains a full neighborhood of 0 , hence the condition regular value' for is not the empty one. This completes the proof. proof The reader should note that an alternate possibility for proving smoothness of generic slices would be to study the linear operator L g for any g close to but not in the standard conformal class and show that this operator generically has no L 2 kernel. The smoothness would then follow by the results of 5, combined with a compactness argument. However, proving that L g has no decaying eigenfunctions would require setting up a somewhat elaborate perturbation theory, since this is equivalent to trying to perturb point spectrum which is sitting on the end of a band of continuous spectrum. Thankfully we have been able to avoid this approach here (although we have had to appeal to the rather more difficult existence theory for the nonlinear equation instead). It should be noted that the surjectivity of (which we have established) is equivalent to this eigenvalue perturbation result. The second main result, that is (locally) a real analytic variety, also follows from Lemma 6.9. theorem22 For any g there exists a ball B in H s 2 - (; Sym 2) , a finite dimensional space M , a real analytic variety in M and a real analytic diffeomorphism equation : B equation such that (B) B' , where B' is a small ball containing (g) . theorem22 The proof reduces to the following abstract result, which is presumably well-known: theorem23 Let V be a Hilbert space, and and two real analytic submanifolds, the tangent spaces of which at any point of the intersection p form a Fredholm pair. Then for each point p there exists a neighborhood B of p , a finite dimensional subspace M V , a real analytic variety M , and a real analytic diffeomorphism equation : B V equation such that (B) B' M B' , where B' is a neighborhood of (p) in M . theorem23 proof By initially composing with a real analytic diffeomorphism we may assume that a neighborhood of p in lies in a linear subspace and that p 0 . (In our situation, is the conformal class and this may be accomplished by a translation.) Let denote a defining function' for in V . By this we mean that is a real analytic map from V into another Hilbert space U such that -1 (0) . By the stability of Fredholm pairs, any level set -1 (w) forms a Fredholm pair with , for all sufficiently small w U , and for all points of intersection near the origin in . It is evident that this statement is equivalent to the assertion that . : U is a Fredholm map. The intersection is the same as -1 (0) . Thus the lemma may be rephrased as saying that if is a real analytic Fredholm map between two Hilbert spaces and U , then the level set -1 (0) is locally equivalent, by an analytic diffeomorphism, to a finite dimensional analytic variety. A standard implicit function theorem argument proves this assertion. Let A . 0 , and set M (A) , L W range (A) . Then both L and M are finite dimensional. Define a new map : L U by (q,f) (q) f . Then equation . (0,0) (h,) A(h) , equation so this differential is obviously surjective. Note that equation (. (0,0) ) (h,): A(h) 0 . equation Since A(h) and lie in orthogonal spaces, this nullspace is just M (A) . The analytic implicit function theorem gives the existence of two analytic maps equation k: M M , : M L equation and a ball B around (0,0) L such that all zeroes of in this ball lie in the graph of the map (k,): M M L : equation (q,f): (q) f 0 B (m k(m),(m)): m M B . equation Thus, for m in this ball, (m k(m)) (m) vanishes identically. The zeroes of in B are then just m k(m): m B M: (m) 0 , for these points are all zeroes of , and are the only such zeroes at which also vanishes. If is an analytic diffeomorphism of carrying the graph of k into M , then B -1 (0) equals -1 (M B) m M B: (m) 0 as desired. This completes the proof of Lemma 6.14. proof It is also possible to establish the real analytic structure of by using a modification of Liapunov-Schmidt reduction' (also referred to as the Kuranishi method'). The closely related and analogous result concerning the real analyticity of the moduli space of complete, embedded, constant mean curvature surfaces in 3 is proved by this method in KMP . This technique is more direct than the one used above, however in using the approach provided by the urmoduli space, , we also obtain the Generic Slice Theorem rather easily. 7. Concluding remarks and informed speculation L 2 -nullspace and singularities of We have not yet discussed whether it is possible to give conditions ensuring that a given g is a smooth point', as defined in 5. It is possible for to be both smooth and of the correct dimension near g even if L g has L 2 -nullspace; however, absence of this nullspace is our only criterion for guaranteeing this smoothness. We expect that it should be difficult to establish such a criterion in general. It would also be quite interesting to understand when is not smooth in the neighborhood of some element g . In particular, constructing solutions near which is singular seems like another very challenging and important problem. On the other hand, the two known constructions for producing points in , those of Schoen S2 and MPU , yield solutions with explicit geometries. The dipole solutions of MPU , described below, are manifestly smooth points. These solutions exist only for certain configurations , whereas the ones in S2 exist for any . Unfortunately, the existence of L 2 -nullspace for these latter solutions is less evident, although we expect this to be true; outlined below is a strategy to prove this. Note that once the existence of one smooth point in any component of is established, the real analyticity of the moduli space then shows that almost every element in that component is a smooth point. We now give a brief description of Schoen's solutions and their construction. These solutions are uniformly small 0 perturbations of explicit approximate solutions. Each of these approximate solutions is constructed from an admissible conformal structure' (,) ; is an infinite tree with a labeling, , of strong dilations G ij for each directed edge e ij . We assume for simplicity that has one vertex of order k and all other vertices of order 2. The admissibility of the labeling refers to certain compatibility conditions that the dilations must satisfy. The strength, ij , of these dilations can be related to a measurement of the neck sizes' ij of the approximate solution g constructed from the data (,) . (, g ) consists of almost spherical regions, corresponding to the vertices V of , joined by small necks, corresponding to the edges, the sizes of which are dictated by the ij . There are infinitely many parameters in the construction of these approximate solutions since the admissible conformal structures can be varied, even with fixed. In particular, one can begin with an initial conformal structure 1 , so that the corresponding approximate solution g 1 consists of one central spherical region 0 (the vertex of order k ) and k periodic, spherically symmetric ends. But in order to find an exact solution, it is necessary to deform the conformal structure, as dictated by the Pohozaev balancing condition, and break this symmetry. The conformal structure (,) decomposes into a union of almost spherical regions, i , i V . Each i is the pullback by a conformal diffeomorphism F i of of a large region of . The metric g on i is constructed so that F i: ( i,g ) (,g 0) is an isometry off a small neighborhood of i . This decomposition gives rise to a basic analytic property of the approximate solution metric g . This is the existence of a basis of functions ij , i V and j 1, ,n 1 , for an infinite dimensional space K (the small eigenspace') corresponding to all the spectrum of L g g n in a small interval around 0 . The functions ij have explicit geometric descriptions. Each ij has support concentrated on i and corresponds to the linear function j on F i( i) . Linear functions are of course eigenfunctions for g 0 with eigenvalue n . On the orthogonal complement K , L g is invertible, uniformly in . As the ij 0 , these approximations improve and L g converges, as an operator on L 2 , to the operator L g 0 on the disjoint union of spheres indexed by V . Writing the solution g as a perturbation, g (1 ) 4 n- 2 g and using the conformal invariance of the conformal Laplacian, we find that equation split L g , g n ,(1 ) -4 n-2 g 2(1 ) - n 2 n-2 , , n. split 7.1 equation Now, is small in 0 (by which we mean that 0 can be made arbitrarily small by taking all ij sufficiently small), and estimates from S2 can be used to show that is small in L 2 over any fixed compact set, but these facts do not immediately imply the existence of an analogous good basis for the small eigenspace' K for L g . Nonetheless, we expect this basis to exist, namely that there exist an 0 such that if ij for all ij , then there exists a set of smooth functions ij , i V , j 1, , n 1 , which satisfy the estimates in Lemma 3.6 of S2 . The span of these functions is a subspace KL 2() such that equation split L g L 2() c() L 2() for K, L 2() c L g L 2() for K , split equation where c is a constant independent of , and c() tends to 0 as 0 . Note that from (7.1) this conjecture would be immediate if H 1 were finite and small. Unfortunately, this is never the case, since the in Schoen's construction is not even in L 2 . The likeliest method to establish the existence of this space K with its explicit basis is to write the solution along each end as a conformal perturbation of the Delaunay metric g on the half-cylinder to which it converges at infinity. There is good evidence that perturbation determining the solution is uniformly small and even exponentially decaying, again provided the neck sizes are sufficiently small. With this information in hand, a not very complicated transference procedure would produce the basis for K with all necessary estimates. If these could be settled, it would then be possible to resolve some basic questions about the existence of an L 2 -nullspace of L g , for Schoen's solutions. theorem24 Suppose g is a solution similar to that constructed in S2 , so that g (1 ) 4 n-2 g as above. Assume that does not lie in any round hypersphere. Then there exists an () 0 , such that if satisfies ij for each edge (i .e . all neck sizes' are less than ) , then L g has no L 2 -nullspace, and hence g is a smooth point of . If does lie in a hyperplane, then any element in the L 2 -nullspace of L g is not integrable, i .e . it is not tangent to a path in . theorem24 We provide a brief sketch of our plan for proving this. If g is a solution constructed from some (,) , with all neck sizes sufficiently small, and if L 2 , L g 0 , then admits two different decompositions, one into the almost linear functions' ij concentrated on the almost spherical regions i , and the other (which is only local along each end E ) into eigenfunctions for the Laplacian along the cross-sectional sphere, as in 2. For this latter decomposition we need the (putative) fact that the solution stays uniformly close to the Delaunay solution to which it asymptotically converges. The former expansion would show that is very close to a linear function on each of the almost spherical regions. A very important point here is that the linear function determined by on each of the spherical regions can never vanish. In terms of the eigenfunctions of the second decomposition, this first decomposition implies that along each end, has most of its mass concentrated in the eigenspaces corresponding to the zeroth and first nonzero eigenvalues of the cross-section. Since g is only approximately Delaunay, the zeroth Fourier coefficient 0 of is not an exact solution of L 0 0 0 , however, the error terms are sufficiently controllable to show that 0 must be quite small (again, depending on the size of ). Hence, in fact, most of the mass of is contained in the coefficients 1, , n , corresponding the the first nonzero eigenvalue of the Laplacian on n-1 (which has multiplicity n ). We call these transverse' linear functions, since each of the spherical regions where we consider them has a natural axis picked out. Thus these arguments show that an L 2 solution of L 0 restricts on each spherical region to be almost linear. More specifically, on the innermost sphere along each end (the one adjoining the central spherical region), actually restricts to be approximately transverse linear. But now, on the central sphere, is linear and near each of the neighborhoods where the various ends are attached, must restrict to a transverse linear function relative to the axis determined by that end. This is clearly impossible if the points do not lie in any subsphere. For the second assertion, concerning the situation when n-1 , we only need to use that is sufficiently close to a linear function on the central sphere. Of necessity, this linear function is one which vanishes on the equator determined by . Hence, if were an integrable Jacobi field, we would obtain a family of solutions which were not symmetric about this equator, contradicting the reflectional symmetry guaranteed by the Alexandrov reflection argument. We hope to be able to settle these issues in the near future. There are, of course, a number of more refined questions about the singular structure of , beginning with the problem, already noted, of constructing solutions near which is singular. Coordinates on In this section we discuss two ways in which coordinates for may be given. We begin with the one arising from the linear analysis. If g is a point for which L g has no L 2 -nullspace, then a neighborhood of g in is parameterized by a small neighborhood of the origin in the bounded nullspace . Thus, linear coordinates on yield local coordinates on near g . The problem then is to get precise information about these linear coordinates on . Unfortunately, this seems difficult, in general. We now discuss how one would go about this, and at least set the problem up in scattering-theoretic terms. By definition, any element in the deficiency subspace W can be expanded on each end as a combination of the model solutions 1, 2 plus an exponentially decreasing error. For clarity, we label these model solutions on E j with a superscript (j) . Thus equation W a j 1 (j) b j 2 (j) on E j. 7.3 equation So any W determines a map equation gathered S: W 2k , (a 1, b 1, , a k, b k). gathered 7.4 equation By the definition of W , S is an isomorphism. Now suppose is in the bounded nullspace . theorem25 Under the hypothesis (5.2), the linear map equation .C S : 2k equation is injective. Its image S is a k -dimensional subspace which is Lagrangian with respect to the natural symplectic form j 1 k da j db j on 2k . theorem25 proof C is obviously linear, and it is injective, because otherwise there would exist an element with all its asymptotic coefficients a j, b j vanishing. By the results of the last section, such a would lie in L 2 , which contradicts the hypothesis (5.2). That S is Lagrangian is a consequence of Green's theorem. There is an analogous result for manifolds with asymptotically cylindrical ends proved in Me . Since we already know that S is k -dimensional, it suffices to prove that it is isotropic. For this, let , and set C() (a 1, b 1, , a k, b k), C() ( 1, 1, , k, k) . Using the coordinate t j along the end E j , let M A denote the region in where each t j A . Then, since L L 0 , equation 0 M A ((L) - (L)) ,dV g M A ( - ) ,d g, equation where is the unit vector-field normal to M A . Substitute the expansions for and at each end, and drop all terms which decrease exponentially as A in this expression. This means that we can replace the metric g on E j by the Delaunay metric g j there, and also replace by u j -2 (n-2) t j and d g by u j (2n-2) (n-2) . Taking the limit as A we get equation 0 j 1 k ( j b j - a j j) n-1 ( 1 (j) t j 2 (j) - 1 (j) 2 (j) t j ) ,u j 2 ,d. 7.6 equation Here d is the standard volume measure on n-1 . If we set A j equal to the integrand for the j th end, then it suffices to show that these constants A j are nonzero and independent of j . However, 1 (j) and 2 (j) are independent solutions of the ODE L 0 0 , and the expression in parentheses in the integrand in (7.6) is just the Wronskian for this ODE. This Wronskian may be written explicitly as some nonzero multiple of equation (-2 u j ' u j ) u j -2 . equation This shows that the integrand, hence the integral itself, is a non-vanishing constant, independent of j . proof The relevance of S to the description of coordinates on near g is explained as follows. Suppose first, for simplicity, that there exists an element for which all a j, ,b j 0, j 1 , but a 1 1, ,b 1 0 . If g(s) is a curve in moduli space tangent to this , then the Delaunay parameters j on each of the ends would remain fixed, at least infinitesimally, along this curve. In fact, the metrics g(s) (or the underlying curve of solutions u s of the PDE) would be strongly asymptotic to the initial metric g on E j , j 1 . On E 1 however, g(s) would be strongly asymptotic to a translation of g , or alternately, of its model Delaunay metric there. If, on the other hand, a 1 0, b 1 1 and all other a j, ,b j 0 , then again each g(s) would be strongly asymptotic to g on all ends except E 1 ; on that end the Delaunay parameter for the model metric would be changing, but the model Delaunay metric would not be translating. Of course, neither of these types of elements of need exist. If is a general element of , and g(s) is a curve in through g tangent to , then describes infinitesimal changes in translation and Delaunay parameters along each end. The most precise description of coordinates on would relate the proportions of these various changes on each end to one another for all directions T g . This is equivalent to describing the coefficients a j,b j for each , and this, in turn, is equivalent to describing the Lagrangian subspace S . An explicit description of S requires a better understanding of the Pohozaev balancing condition (3.9). For the sake of illustration, let us examine all this for 2 , the moduli space when k 2 . Of course, we have already given a complete description of this space, but it provides a concrete example of this description. We let the background metric be any Delaunay metric g with u . Now is 2 dimensional, consisting of the elements 1 and 2 . Hence equation S a 1 -a 2, b 1 b 2 4, equation which is Lagrangian, as expected. The two natural curves g(s) emanating from any g 2 are simply the families 1 and 2 defined in (2.14). It may be possible to give a geometric description of parameters on by considering the Killing norms of the Pohozaev invariants. Recall that these invariants are elements of (n 1,1) , the dual of the Lie algebra of the conformal group. Any element X(n 1,1) may be uniquely decomposed as X X 0 w , where X 0(n 1) and w n 1 . The Killing form equation B:(n 1,1)(n 1,1) equation is the nondegenerate symmetric quadratic form given by equation B(X,X) 12 Tr(X 0X 0) w w . equation Thus, B(,) is positive definite on n 1 and negative definite on (n 1) (with respect to this decomposition). Moreover, the adjoint representation preserves B in the sense that B(Ad(F)(X),Ad(F)(X)) B(X,X) , for all FO(n 1,1) and X,X(n 1,1) . Since B is nondegenerate it provides an identification between (n 1,1) and its dual space. Thus we may use the Killing form to identify, for the Pohozaev invariants 1,, k of a metric g , corresponding elements 1,, k(n 1,1) . The squared Killing norm of the Pohozaev invariant i is then defined to be equation i 2 B( i, i). equation These k numbers are conformal invariants. By this we mean the following. If FO(n 1,1) is a conformal diffeomorphism of and g , then F (g) F -1 () . The Pohozaev invariants of g and F (g) do not coincide but transform via the co-adjoint representation (see S2 and KKMS ). Since B is invariant under this representation this implies that the squared Killing norms of the Pohozaev invariants of g and F (g) are the same. Schoen has suggested the following way to obtain parameters on . Let p 1,,p k be a balanced singular set. One can then try to produce, for some 0 , a 1-parameter family of solutions g t , t(0,) , such that the asymptotic neck sizes 1,, k for g t are all equal to t . To realize the other (k-1) -parameters, choose any k -tuple of numbers, (a 1,,a k) , close to 1 and normalized so that a 1 1 . Then there is a conformal diffeomorphism F taking to p 1,,p k so that j 1 ka jp j 0 . As before, there is now a 1-parameter family of solutions g t , on with asymptotic neck sizes given by (a 1t,,a kt) . Thus F (g t ) is another 1-parameter family of solutions. This exhibits the k -parameters as (t, a 2,,a k) . It should be possible to phrase this in terms of the squared Killing norms of the Pohozaev invariants described above. Dipole solutions We briefly describe a new construction of solutions MPU for certain special configurations . In general terms, this construction shows that any two nondegenerate' complete manifolds M 1 and M 2 with constant positive scalar curvature may be grafted together to obtain a one parameter family of complete conformal metrics of constant positive scalar curvature on the connected sum M 1 M 2 ; furthermore, these solution metrics are nondegenerate as well. Here nondegenerate means simply that the linearized scalar curvature operator L has no L 2 -nullspace, and that it is surjective on a suitable extension of L 2 (this is explained more carefully in MPU ). This is proved by a fairly standard gluing procedure, using the implicit function theorem. Since, in particular, Delaunay metrics on n-1 are nondegenerate, we obtain nondegenerate solutions on p 1, p 2, q 1, q 2 , where p 1, p 2 and q 1, q 2 are two pairs of points, with each pair clustered tightly near two antipodal points P, Q . The conformal factor u on which is singular at these four points is very small on the complement of small balls around P and Q containing the p i , q i , and is highly concentrated in these balls. For this reason, we call these dipole solutions'. The large region where u is small corresponds to the small neck joining the two Delaunay solutions. As the neck gets smaller, the p i converge to P and analogously for the q i . By the final assertion of the result, this new solution (or rather, family of solutions) is nondegenerate, so the process may be iterated. We obtain theorem26 Let P 1, , P be a balanced set of points on . Then for sufficiently small, there exist points p i (1) and p i (2) in each ball B (P i) and a moduli space of dimension 2 of nondegenerate solutions on where p i (j) , 1 i , j 1,2 . Moreover the map from this moduli space to 2 , sending a solution to the translation and neck parameters at p i (1) , for i 1,, , is an isomorphism onto an open set. theorem26 There are several important consequences of this result. The first is that since we can fix the neck parameters of the initial Delaunay solutions arbitrarily before the gluing, and since we may choose the gluing to alter these parameters only on one end of each Delaunay solution, we obtain solutions on the complement of 2 points, where the Delaunay parameters are prescribed at of the points. In particular, we obtain solutions with 2 ends, of which are asymptotically cylindrical. This is important, in part, because it produces solutions with the simplest type of asymptotic behavior. Another consequence concerns nondegeneracy and follows from an elaboration of our perturbation techniques. We can consider the moduli space of pairs equation (, ) . equation A development of the ideas here shows that itself is a real analytic set. Its expected dimension is k(1 n) , where k is the cardinality of . If there is a nondegenerate solution for a given configuration , then the whole component of containing that solution has this dimension; by considering as a fibration over the kn -dimensional space of configurations, we see that generic fibers also have the correct dimension k , and hence that generic points in this fiber are nondegenerate. Since we have found nondegenerate solutions for certain configurations when k 2 , we conclude that generic points in the same component of (presumably including those where the points are not so tightly clustered in pairs) are also nondegenerate. Unfortunately, no uniqueness theorem has been proved, even in the restricted setting of Theorem 7.7. This means that we have no information on the number of connected components other than to assert, as a consequence of the compactness theorem P2 , that there are finitely many. The structure of the boundary There are a number of interesting questions about the structure of the compactification . Recall from 3 that this compactification is obtained by adding certain solutions g ' to , for certain subsets ' . There are two basic problems. The first is to determine which subsets ' have elements occurring in . This is again intimately connected to the balancing condition (3.9). There are few examples where we can say anything explicit about this. The case k 2 was already discussed in 3. The existence of dipole solutions MPU may give additional information concerning this problem when k is even. The other fundamental problem is to determine how much of ' is contained in when at least one point g ' lies in . More specifically, we propose the following: theorem27 Suppose that g ' . Then the entire component of ' containing g also lies in . theorem27 It is easy to show, using the compactness theorem of P2 , that the set of points g' , in the same component of ' as g and which also lie in , is closed. One method to prove that this set is open would be to generalize the general grafting procedure of MPU . This would prove the conjecture. amsalpha