## Moduli Spaces of Singular Yamabe Metrics

HTML articles powered by AMS MathViewer

- by Rafe Mazzeo, Daniel Pollack and Karen Uhlenbeck PDF
- J. Amer. Math. Soc.
**9**(1996), 303-344 Request permission

## Abstract:

Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ 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.## References

- Lars Andersson, Piotr T. Chruściel, and Helmut Friedrich,
*On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations*, Comm. Math. Phys.**149**(1992), no. 3, 587–612. MR**1186044**, DOI 10.1007/BF02096944 - Patricio Aviles and Robert C. McOwen,
*Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds*, Duke Math. J.**56**(1988), no. 2, 395–398. MR**932852**, DOI 10.1215/S0012-7094-88-05616-5 - P. Aviles, N. Korevaar and R. Schoen,
*The symmetry of constant scalar curvature metrics near point singularities*, preprint. - K. Grosse-Brauckmann,
*New surfaces of constant mean curvature Math. Z.*(to appear). - Luis A. Caffarelli, Basilis Gidas, and Joel Spruck,
*Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math.**42**(1989), no. 3, 271–297. MR**982351**, DOI 10.1002/cpa.3160420304 - C. Delaunay,
*Sur la surface de revolution dont la courbure moyenne est constant*, J. Math. Pure Appl.**6**(1841), 309-320. - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - David L. Finn and Robert C. McOwen,
*Singularities and asymptotics for the equation $\Delta _gu-u^q=Su$*, Indiana Univ. Math. J.**42**(1993), no. 4, 1487–1523. MR**1266103**, DOI 10.1512/iumj.1993.42.42068 - Arthur E. Fischer and Jerrold E. Marsden,
*Deformations of the scalar curvature*, Duke Math. J.**42**(1975), no. 3, 519–547. MR**380907** - Arthur E. Fischer and Jerrold E. Marsden,
*Linearization stability of nonlinear partial differential equations*, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 219–263. MR**0383456** - 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. - —,
*Further studies of Emden’s and similar differential equations*, Quart. J. Math. Oxford Series**2**(1931), 259–287. - Nicolaos Kapouleas,
*Complete constant mean curvature surfaces in Euclidean three-space*, Ann. of Math. (2)**131**(1990), no. 2, 239–330. MR**1043269**, DOI 10.2307/1971494 - Tosio Kato,
*Perturbation theory for linear operators*, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR**0407617** - Osamu Kobayashi,
*A differential equation arising from scalar curvature function*, J. Math. Soc. Japan**34**(1982), no. 4, 665–675. MR**669275**, DOI 10.2969/jmsj/03440665 - Nick Korevaar and Rob Kusner,
*The global structure of constant mean curvature surfaces*, Invent. Math.**114**(1993), no. 2, 311–332. MR**1240641**, DOI 10.1007/BF01232673 - Nicholas J. Korevaar, Rob Kusner, William H. Meeks III, and Bruce Solomon,
*Constant mean curvature surfaces in hyperbolic space*, Amer. J. Math.**114**(1992), no. 1, 1–43. MR**1147718**, DOI 10.2307/2374738 - Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon,
*The structure of complete embedded surfaces with constant mean curvature*, J. Differential Geom.**30**(1989), no. 2, 465–503. MR**1010168** - R. Kusner, R. Mazzeo and D. Pollack,
*The moduli space of complete embedded constant mean curvature surfaces*, Geom. and Functional Analysis (to appear). - Jacques Lafontaine,
*Sur la géométrie d’une généralisation de l’équation différentielle d’Obata*, J. Math. Pures Appl. (9)**62**(1983), no. 1, 63–72 (French). MR**700048** - André Lichnerowicz,
*Propagateurs et commutateurs en relativité générale*, Inst. Hautes Études Sci. Publ. Math.**10**(1961), 56 (French). MR**157736** - Charles Loewner and Louis Nirenberg,
*Partial differential equations invariant under conformal or projective transformations*, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245–272. MR**0358078** - Rafe Mazzeo,
*Regularity for the singular Yamabe problem*, Indiana Univ. Math. J.**40**(1991), no. 4, 1277–1299. MR**1142715**, DOI 10.1512/iumj.1991.40.40057 - Rafe Mazzeo,
*Elliptic theory of differential edge operators. I*, Comm. Partial Differential Equations**16**(1991), no. 10, 1615–1664. MR**1133743**, DOI 10.1080/03605309108820815 - — and F. Pacard,
*A new construction of singular solutions for a semilinear elliptic equation*, To appear, J. Differential Geometry. - —, D. Pollack and K. Uhlenbeck,
*Connected sum constructions for constant scalar curvature metrics*, Preprint. - Rafe Mazzeo and Nathan Smale,
*Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere*, J. Differential Geom.**34**(1991), no. 3, 581–621. MR**1139641** - Robert C. McOwen,
*Prescribed curvature and singularities of conformal metrics on Riemann surfaces*, J. Math. Anal. Appl.**177**(1993), no. 1, 287–298. MR**1224820**, DOI 10.1006/jmaa.1993.1258 - R. Melrose,
*The Atiyah-Patodi-Singer index theorem*, AK Peters Ltd., Wellesley, MA, 1993. - Morio Obata,
*Certain conditions for a Riemannian manifold to be isometric with a sphere*, J. Math. Soc. Japan**14**(1962), 333–340. MR**142086**, DOI 10.2969/jmsj/01430333 - F. Pacard,
*The Yamabe problem on subdomains of even dimensional spheres*, preprint. - Daniel Pollack,
*Nonuniqueness and high energy solutions for a conformally invariant scalar equation*, Comm. Anal. Geom.**1**(1993), no. 3-4, 347–414. MR**1266473**, DOI 10.4310/CAG.1993.v1.n3.a2 - Daniel Pollack,
*Compactness results for complete metrics of constant positive scalar curvature on subdomains of $S^n$*, Indiana Univ. Math. J.**42**(1993), no. 4, 1441–1456. MR**1266101**, DOI 10.1512/iumj.1993.42.42066 - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419** - Richard Schoen,
*Conformal deformation of a Riemannian metric to constant scalar curvature*, J. Differential Geom.**20**(1984), no. 2, 479–495. MR**788292** - Richard M. Schoen,
*The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation*, Comm. Pure Appl. Math.**41**(1988), no. 3, 317–392. MR**929283**, DOI 10.1002/cpa.3160410305 - M. Giaquinta (ed.),
*Topics in calculus of variations*, Lecture Notes in Mathematics, vol. 1365, Springer-Verlag, Berlin, 1989. Lectures given at the Second 1987 C.I.M.E. Session held in Montecatini Terme, July 20–28, 1987. MR**994016**, DOI 10.1007/BFb0089175 - Blaine Lawson and Keti Tenenblat (eds.),
*Differential geometry*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. A Symposium in Honor of Manfredo do Carmo. MR**1173028** - R. Schoen and S.-T. Yau,
*Conformally flat manifolds, Kleinian groups and scalar curvature*, Invent. Math.**92**(1988), no. 1, 47–71. MR**931204**, DOI 10.1007/BF01393992 - Clifford Henry Taubes,
*Gauge theory on asymptotically periodic $4$-manifolds*, J. Differential Geom.**25**(1987), no. 3, 363–430. MR**882829**

## Additional Information

**Rafe Mazzeo**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- Email: mazzeo@math.stanford.edu
**Daniel Pollack**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- Email: pollack@math.uchicago.edu
**Karen Uhlenbeck**- Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
- Email: uhlen@math.utexas.edu
- Received by editor(s): January 20, 1994
- Additional Notes: The first author’s research was supported in part by NSF Young Investigator Award, the Sloan Foundation, NSF grant # DMS9001702, the second author’s research was supported by NSF grant # DMS9022140, and the third author’s research was supported by the Sid Richardson and O’Donnell foundations.
- © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**9**(1996), 303-344 - MSC (1991): Primary 58D27
- DOI: https://doi.org/10.1090/S0894-0347-96-00208-1
- MathSciNet review: 1356375