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ISSN 1088-6850(e) ISSN 0002-9947(p)

Traces of heat operators on Riemannian foliations

Author(s): Ken Richardson
Journal: Trans. Amer. Math. Soc. 362 (2010), 2301-2337.
MSC (2010): Primary 53C12, 58J37, 58J35, 58J50
Posted: December 8, 2009
MathSciNet review: 2584602
Retrieve article in: PDF

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Abstract: We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace $K_{B}(t)$ of this operator has a particular asymptotic expansion as $t\to 0$ . The coefficients of $t^{\alpha}$ and of $t^{\alpha}(\log t)^{\beta}$ in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure.

References:

1.: J. A. Álvarez López, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), 179-194. MR 1175918 (93h:53027)
2.: M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and index theorem, Invent. Math. 19(1973), 279-330. MR 0650828 (58:31287)
3.: M. Berger, Sur les spectre d'une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 163(1963), 13-16. MR 0158332 (28:1557)
4.: M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d'Une Variété Riemannienne, Springer Verlag, Berlin, 1971. MR 0282313 (43:8025)
5.: N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer Verlag, Berlin, 1991. MR 1215720 (94e:58130)
6.: D. Bleecker, The supertrace of the steady asymptotic of the spinorial heat kernel, J. Math. Phys. 33(1992), no. 6, 2053-2070. MR 1164316 (93d:58153)
7.: G. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
8.: T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York, 1985. MR 781344 (86i:22023)
9.: J. Brüning and E. Heintze, Representations of compact Lie groups and elliptic operators, Inventiones Math. 50 (1979), 169-203. MR 517776 (81b:58039)
10.: J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case, Duke Math. J. 51(1984), 959-979. MR 771390 (86b:58124)
11.: J. Brüning and F. W. Kamber, Vanishing theorems and index formulas for transversal Dirac operators, A.M.S. Meeting 845, Special Session on Operator Theory and Applications to Geometry, Lawrence, Kansas, A.M.S. Abstracts, October 1988.
12.: R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Comm. Math. Phys. 148(1992), no. 2, 283-308. MR 1178146 (93g:58140)
13.: R. A. Carmona and W. A. Zheng, Reflecting Brownian motions and comparison theorems for Neumann heat kernels, J. Funct. Anal. 123(1994), no. 1, 109-128. MR 1279298 (95k:60202)
14.: I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, New York, 1984. MR 768584 (86g:58140)
15.: J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109(1979), 259-322. MR 528965 (80j:58065a)
16.: M. Craioveanu and M. Puta, Asymptotic properties of eigenvalues of the basic Laplacian associated to certain Riemannian foliations, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 35(83) (1991), no. 1-2 , 61-65. MR 1307911 (95j:58164)
17.: H. Donnelly, Spectrum and the fixed point sets of isometries I, Math. Ann. 224(1976), 161-170. MR 0420743 (54:8755)
18.: H. Donnelly, Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23(1979), 485-496. MR 537804 (80h:58049)
19.: E. B. Dryden, C. S. Gordon, S. J. Greenwald, and D. L. Webb, Asymptotic expansion of the heat kernel for orbifolds, Mich. Math. J., 56 (2008), no. 1, 205-238. MR 2433665 (2009h:58057)
20.: A. El Kacimi-Alaoui, Équation de la chaleur sur les espaces singulièrs, C. R. Acad. Sci. Paris 303(1986) no. 6, 243-246. MR 860827 (87k:58258)
21.: A. El Kacimi-Alaoui and G. Hector, Décomposition de Hodge basique pour un feuilletage Riemannien, Ann. Inst. Fourier, Grenoble 36(1986), no. 3 , 207-227. MR 865667 (87m:57029)
22.: G. Esposito, G. Fucci, A. Y. Kamenshchik, and K. Kirsten, Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions, Classical Quantum Gravity 22 (2005), no. 6, 957-974. MR 2131582 (2005m:83044)
23.: H. D. Fegan, The heat equation and modular forms, J. Diff. Geo. 13(1978), 589-602. MR 570220 (81k:22006)
24.: W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York, 1966. MR 0210154 (35:1048)
25.: P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Wilmington, DE, 1984. MR 783634 (86j:58144)
26.: J. F. Glazebrook and F. W. Kamber, Transversal Dirac families in Riemannian foliations, Comm. Math. Phys. 140(1991), 217-240. MR 1124268 (92j:58103)
27.: P. Greiner, An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41(1971), 163-218. MR 0331441 (48:9774)
28.: F. B. Hildebrand, Advanced Calculus for Applications, second ed., Prentice Hall, Inc., Englewood Cliffs, N.J., 1976.
29.: F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann. 277(1987), 415-431. MR 891583 (89d:53070)
30.: K. Kawakubo, The Theory of Transformation Groups, Oxford University Press, 1991. MR 1150492 (93g:57044)
31.: J. Lee and K. Richardson, Riemannian foliations and eigenvalue comparison, Ann. Global Anal. Geom. 16(1998), no. 6, 497-525. MR 1651376 (99k:53047)
32.: P. Malliavin and D. Stroock, Short time behavior of the heat kernel and its logarithmic derivatives, J. Diff. Geom. 44(1996), 550-570. MR 1431005 (98c:58164)
33.: H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1(1967), 43-69. MR 0217739 (36:828)
34.: J. Milnor, Morse Theory, Ann. Math. Stud., no. 51, Princeton University Press, 1963. MR 0163331 (29:634)
35.: S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1(1949), 242-256. MR 0031145 (11:108b)
36.: P. Molino, Riemannian foliations, Progress in Mathematics, Birkhauser, Boston, 1988. MR 932463 (89b:53054)
37.: S. Nishikawa, M. Ramachandran, and Ph. Tondeur, The heat equation for Riemannian foliations, Trans. Amer. Math. Soc. 319 (1990), 619-630. MR 987165 (90j:58145)
38.: S. Nishikawa, Ph. Tondeur, and L. Vanhecke, Spectral geometry for Riemannian foliations, Ann. Global Anal. Geom. 10(1992), 291-304. MR 1186017 (93h:58160)
39.: B. Osgood, R. Phillips, and P. Sarnak, Extremals of Determinants of Laplacians, J. Funct. Anal. 80(1988), no. 1, 148-211. MR 960228 (90d:58159)
40.: B. Osgood, R. Phillips, and P. Sarnak, Compact Isospectral Sets of Surfaces, J. Funct. Anal. 80(1988), no. 1, 212-234. MR 960229 (90d:58160)
41.: E. Park, Toeplitz algebras associated to isometric flows, Illinois J. Math. 41(1997), no. 1, 93-102. MR 1433188 (98g:47023)
42.: E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118(1996), 1249-1275. MR 1420923 (97i:58165)
43.: A. S. Petrow, Einstein-Räume, Akademie-Verlag, Berlin, 1964. MR 0162594 (28:5792)
44.: D.B. Ray and I.M. Singer, -Torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7(1971), 145-210. MR 0295381 (45:4447)
45.: B. Reinhart, Differential Geometry of Foliations, Springer-Verlag, Berlin, 1983. MR 705126 (85i:53038)
46.: K. Richardson, Critical points of the determinant of the Laplace operator, J. Funct. Anal. 122(1994), no. 1, 52-83. MR 1274583 (95j:58176)
47.: K. Richardson, The asymptotics of heat kernels on Riemannian foliations, Geom. Funct. Anal. 8(1998), 1-46. MR 1616151 (99e:58188)
48.: K. Richardson, The transverse geometry of G-manifolds and Riemannian foliations, Illinois J. Math. 45(2001), 517-535. MR 1878616 (2002k:53041)
49.: J. Roe, Elliptic operators, topology, and asymptotic methods, Pitman Research Notes in Math., no. 179, Longman Scientific and Technical, Harlow, and Wiley, New York, 1988. MR 960889 (89j:58126)
50.: Ph. Tondeur, Foliations on Riemannian manifolds, Springer Verlag, New York, 1988. MR 934020 (89e:53052)
51.: H. Weyl, Der Asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71(1912), 441-469. MR 1511670
52.: H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1(1983), no. 3, 51-75. MR 739904 (85j:57043)

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