On the $\Theta$-function of a Riemannian manifold with boundary
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- by Pei Hsu PDF
- Trans. Amer. Math. Soc. 333 (1992), 643-671 Request permission
Abstract:
Let $\Omega$ be a compact Riemannian manifold of dimension $n$ with smooth boundary. Let ${\lambda _1} < {\lambda _2} \leq \cdots$ be the eigenvalues of the Laplace-Beltrami operator with the boundary condition $[\partial /\partial n + \gamma ]\phi = 0$ . The associated $\Theta$-function ${\Theta _\gamma }(t) = \sum \nolimits _{n = 1}^\infty {\exp [ - {\lambda _n}t]}$ has an asymptotic expansion of the form \[ {(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .\] The values of ${a_0}$ , ${a_1}$ are well known. We compute the coefficients ${a_2}$ and ${a_3}$ in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel $p(t,x,y)$ near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].References
-
M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1974.
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
- Thomas P. Branson and Peter B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), no. 2, 245–272. MR 1032631, DOI 10.1080/03605309908820686
- Peter Greiner, An asymptotic expansion for the heat equation, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–135. MR 0265784
- Pei Hsu, Sur la fonction $\Theta$ d’une variété riemanniene à bord, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 7, 507–510. MR 1055469
- Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI 10.2307/2313748
- Gerard Kennedy, Raymond Critchley, and J. S. Dowker, Finite temperature field theory with boundaries: stress tensor and surface action renormalisation, Ann. Physics 125 (1980), no. 2, 346–400. MR 567353, DOI 10.1016/0003-4916(80)90138-4
- G. Louchard, Mouvement Brownien et valeurs propres du Laplacien, Ann. Inst. H. Poincaré Sect. B (N.S.) 4 (1968), 331–342 (French, with English summary). MR 0261695
- S. A. Molčanov, Diffusion processes, and Riemannian geometry, Uspehi Mat. Nauk 30 (1975), no. 1(181), 3–59 (Russian). MR 0413289
- 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
- Åke Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954), 553–569. MR 61257, DOI 10.1007/BF02591229
- R. T. Waechter, On hearing the shape of a drum: An extension to higher dimensions, Proc. Cambridge Philos. Soc. 72 (1972), 439–447. MR 304887, DOI 10.1017/s0305004100047277
- E. M. E. Zayed, Eigenvalues of the Laplacian: an extension to higher dimensions, IMA J. Appl. Math. 33 (1984), no. 1, 83–99. MR 763405, DOI 10.1093/imamat/33.1.83
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 333 (1992), 643-671
- MSC: Primary 58G18; Secondary 58G20
- DOI: https://doi.org/10.1090/S0002-9947-1992-1055808-9
- MathSciNet review: 1055808