Abstract
We calculate heat invariants of arbitrary Riemannian manifolds without boundary. Every heat invariant is expressed in terms of powers of the Laplacian and the distance function. Our approach is based on a multidimensional generalization of the Agmon-Kannai method. An application to computation of the Korteweg-de Vries hierarchy is also presented.
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References
[AK] S. Agmon and Y. Kannai,On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators, Israel Journal of Mathematics5 (1967), 1–30.
[Av] I. G. Avramidi,A covariant technique for the calculation of the one-loop effective action, Nuclear Physics. B355 (1991), 712–754;Erratum, Nuclear Physics. B509 (1998), 557–558.
[AvB] I. G. Avramidi and T. Branson,Heat kernel asymptotics of operators with non-Laplace principal part, math-ph/9905001, (1999), 1–43.
[AvSc] I. G. Avramidi and R. Schimming,A new explicit expression for the Korteweg-de Vries hierarchy, solv-int/9710009, (1997), 1–17.
[Be] M. Berger,Geometry of the spectrum, Proceedings of Symposia in Pure Mathematics27 (1975), 129–152.
[BGM] M. Berger, P. Gauduchon and E. Mazet,Le spectre d'une variètè riemannienne, Lecture Notes in Mathematics194, Springer-Verlag, Berlin, 1971.
[BGØ] T. Branson, P. Gilkey and B. Ørsted,Leading terms in the heat invariants, Proceedings of the American Mathematical Society109 (1990), 437–450.
[Ch] I. Chavel,Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.
[Da] E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press,
[Du] G. F. D. Duff,Partial Differential Equations, University of Toronto Press, 1956.
[Er] A. Erdélyi,Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.
[F] S. A. Fulling, ed.,Heat Kernel Techniques and Quantum Gravity, Discourses in Mathematics and its Applications, No. 4, Texas A&M University, 1995.
[GD] I. M. Gelfand and L. A. Dikii,Asymptotic behaviour of the resolvent of Sturm-Liouville equation and the algebra of the Korteweg-de Vries equations, Russian Mathematical Surveys30:5 (1975), 77–113.
[G1] P. Gilkey,The spectral geometry of a Riemannian manifold, Journal of Differential Geometry10 (1975), 601–618.
[G2] P. Gilkey,The index theorem and the heat equation, Mathematics Lecture Series, Publish or Perish, Houston, 1974.
[G3] P. Gilkey,Heat equation asymptotics, Proceedings of Symposia in Pure Mathematics54 (1993), 317–326.
[GR] I. S. Gradshtein and I. M. Ryzhik,Table of Integrals, Series and Products, Academic Press, New York, 1980.
[Gra] A. Gray,The volume of a small geodesic ball of a Riemannian manifold, The Michigan Mathematical Journal20 (1973), 329–344.
[GraV] A. Gray and L. Vanhecke,Power series expansions, differential geometry of geodesic spheres and tubes, and mean-value theorems, inGeometry and Differtial Geometry, Lecture Notes in Mathematics792, Springer-Verlag, Berlin, 1980, pp. 252–259.
[GKM] D. Gromoll, W. Klingenberg and W. Meyer,Riemannsche Geometrie im Grossen, Springer-Verlag, Berlin, 1968.
[Hö] L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, Berlin, 1969.
[MS] H. P. McKean, Jr. and I. M. Singer,Curvature and the eigenvalues of the Laplacian, Journal of Differential Geometry1 (1967), 43–69.
[MP] S. Minakshisundaram and A. Pleijel,Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian Journal of Mathematics1 (1949), 242–256.
[NMPZ] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov,Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York, 1984.
[OPS] B. Osgood, R. Phillips and P. Sarnak,Compact isospectral sets of surfaces, Journal of Functional Analysis80 (1988), 212–234.
[PWZ] M. Petkovšek, H. Wilf and D. Zeilberger,A=B, AK Peters, Wellesley, 1996.
[P1] I. Polterovich,A commutator method for computation of heat invariants, Indagationes Mathematicae, to appear.
[P2] I. Polterovich,From Agmon-Kannai expansion to Korteweg-de Vries hierarchy, Letters in Mathematical Physics, to appear.
[Rid] S. Z. Rida,Explicit formulae for the powers of a Schrödinger-like ordinary differential operator, Journal of Physics. A. Mathematical and General31 (1998), 5577–5583.
[Rio] J. Riordan,An Introduction to Combinatorial Analysis, Wiley, New York, 1958.
[Ro] S. Rosenberg,The Laplacian on a Riemannian manifold, Cambridge University Press, 1997.
[Se] R. Seeley,Complex powers of an elliptic operator, Proceedings of Symposia in Pure Mathematics10 (1967), 288–307.
[Sa] T. Sakai,On the eigenvalues of the Laplacian and curvature of Riemannian manifold, The Tôhoku Mathematical Journal23 (1971), 585–603.
[vdV] A. E. M. van de Ven,Index-free heat kernel coefficients, hep-th/9708152, (1997), 1–38.
[Wo] S. Wolfram,Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, Reading, MA, 1991.
[Xu] C. Xu,Heat kernels and geometric invariants I, Bulletin des Sciences Mathématiques117 (1993), 287–312.
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Polterovich, I. Heat invariants of Riemannian manifolds. Isr. J. Math. 119, 239–252 (2000). https://doi.org/10.1007/BF02810670
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DOI: https://doi.org/10.1007/BF02810670