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Leading terms in the heat invariants

Authors: Thomas P. Branson, Peter B. Gilkey and Bent Ørsted
Journal: Proc. Amer. Math. Soc. 109 (1990), 437-450
MSC: Primary 58G25; Secondary 58G11
MathSciNet review: 1014642
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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.

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  • [B] T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345. MR 832360 (88a:58212)
  • [BØ1] T. Branson and B. Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261-293. MR 869104 (88b:58131)
  • [BØ2] -, Conformal deformation and the heat operator, Indiana Univ. Math. J. 37 (1988), 83-110. MR 942096 (89g:58214)
  • [BØ3] -, Conformal geometry and global invariants, preprint.
  • [BPY] R. Brooks, P. Perry, and P. Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), 131-150. MR 1016417 (90i:58193)
  • [CY1] S.-Y. A. Chang and P. C. Yang, Isospectral conformal metrics on $ 3$-manifolds, J. Amer. Math. Soc. 3 (1990), 117-145. MR 1015647 (91c:58140)
  • [CY2] -, Compactness of isospectral conformal metrics on $ {S^3}$, Comment. Math. Helv. 64 (1989), 363-374. MR 998854 (90c:58181)
  • [CY3] -, The conformal deformation equation and isospectral sets of conformal metrics, Contemp. Math. (to appear). MR 1034980 (90k:53072)
  • [DK] J. Dowker and G. Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11 (1978), 895-920. MR 0479266 (57:18709)
  • [G1] P. Gilkey, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38 (1979), 201-240. MR 528840 (80i:53020)
  • [G2] -, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish, Wilmington, Delaware, 1984. MR 783634 (86j:58144)
  • [G3] -, Leading terms in the asymptotics of the heat equation, Contemp. Math. 73 (1988), 79-85. MR 954631 (89h:58199)
  • [H] J. Hadamard, Le problème de Cauchy et les èquations aux dérivées partielles linéaires hyperboliques, Hermann et Cie, Paris, 1932.
  • [MS] H. P. McKean, Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43-69. MR 0217739 (36:828)
  • [M] R. Melrose, Isospectral sets of drumheads are compact in $ {C^\infty }$, preprint.
  • [MP] 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 0031145 (11:108b)
  • [OPS] B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), 212-234. MR 960229 (90d:58160)
  • [PR] T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom. 25 (1987), 199-222. MR 880183 (89e:58118)
  • [RS] D. B. Ray and I. M. Singer, $ R$-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. MR 0295381 (45:4447)
  • [S] R. Schimming, Lineare Differentialoperatoren zweiter Ordnung mit metrischem Hauptteil und die Methode der Koinzidenzwerte in der Riemannschen Geometrie, Beitr. z. Analysis 15 (1981), 77-91. MR 614779 (82h:58051)
  • [W] V. Wünsch, Konforminvariante Variationsprobleme und Huygenssches Prinzip, Math. Nachr. 120 (1985), 175-193. MR 808340 (87f:53049)

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