Leading terms in the heat invariants
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- by Thomas P. Branson, Peter B. Gilkey and Bent Ørsted PDF
- Proc. Amer. Math. Soc. 109 (1990), 437-450 Request permission
Abstract:
Let $D$ be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on $D$ are given by homogeneous, invariant, local formulas. Within the set of allowable expressions of a given homogeneity there is a filtration by degree, in which elements of the smallest class have the highest degree. Modulo quadratic terms, the linear terms integrate to zero, and thus do not contribute to the asymptotics of the ${L^2}$ trace of the heat operator; that is, to the asymptotics of the spectrum. We give relations between the linear and quadratic terms, and use these to compute the heat invariants modulo cubic terms. In the case of the scalar Laplacian, qualitative aspects of this formula have been crucial in the work of Osgood, Phillips, and Sarnak and of Brooks, Chang, Perry, and Yang on compactness problems for isospectral sets of metrics modulo gauge equivalence in dimensions 2 and 3.References
- Thomas P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), no. 2, 293–345. MR 832360, DOI 10.7146/math.scand.a-12120
- Thomas P. Branson and Bent Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), no. 3, 261–293. MR 869104
- Thomas P. Branson and Bent Ørsted, Conformal deformation and the heat operator, Indiana Univ. Math. J. 37 (1988), no. 1, 83–110. MR 942096, DOI 10.1512/iumj.1988.37.37004 —, Conformal geometry and global invariants, preprint.
- Robert Brooks, Peter Perry, and Paul Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), no. 1, 131–150. MR 1016417, DOI 10.1215/S0012-7094-89-05808-0
- Sun-Yung A. Chang and Paul C.-P. Yang, Isospectral conformal metrics on $3$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 1, 117–145. MR 1015647, DOI 10.1090/S0894-0347-1990-1015647-3
- Sun-Yung A. Chang and Paul C. Yang, Compactness of isospectral conformal metrics on $S^3$, Comment. Math. Helv. 64 (1989), no. 3, 363–374. MR 998854, DOI 10.1007/BF02564682
- Sun-Yung A. Chang and Paul C. Yang, The conformal deformation equation and isospectral set of conformal metrics, Recent developments in geometry (Los Angeles, CA, 1987) Contemp. Math., vol. 101, Amer. Math. Soc., Providence, RI, 1989, pp. 165–178. MR 1034980, DOI 10.1090/conm/101/1034980
- J. S. Dowker and Gerard Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11 (1978), no. 5, 895–920. MR 479266
- Peter B. Gilkey, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38 (1979), no. 2, 201–240. MR 528840
- Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
- Peter B. Gilkey, Leading terms in the asymptotics of the heat equation, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 79–85. MR 954631, DOI 10.1090/conm/073/954631 J. Hadamard, Le problème de Cauchy et les èquations aux dérivées partielles linéaires hyperboliques, Hermann et Cie, Paris, 1932.
- H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739 R. Melrose, Isospectral sets of drumheads are compact in ${C^\infty }$, preprint.
- S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242–256. MR 31145, DOI 10.4153/cjm-1949-021-5
- B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), no. 1, 212–234. MR 960229, DOI 10.1016/0022-1236(88)90071-7
- Thomas Parker and Steven Rosenberg, Invariants of conformal Laplacians, J. Differential Geom. 25 (1987), no. 2, 199–222. MR 880183
- D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
- Rainer Schimming, Lineare Differentialoperatoren zweiter Ordnung mit metrischem Hauptteil und die Methode der Koinzidenzwerte in der Riemannschen Geometrie, Beiträge Anal. 15 (1980), 77–91 (1981) (German). MR 614779
- Volkmar Wünsch, Konforminvariante Variationsprobleme und Huygenssches Prinzip, Math. Nachr. 120 (1985), 175–193 (German). MR 808340, DOI 10.1002/mana.19851200115
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 437-450
- MSC: Primary 58G25; Secondary 58G11
- DOI: https://doi.org/10.1090/S0002-9939-1990-1014642-X
- MathSciNet review: 1014642