Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the $ \Theta$-function of a Riemannian manifold with boundary

Author: Pei Hsu
Journal: Trans. Amer. Math. Soc. 333 (1992), 643-671
MSC: Primary 58G18; Secondary 58G20
MathSciNet review: 1055808
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle {(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .$

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 [Enhancements On Off] (What's this?)

  • [BGM] 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.
  • [CE] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam, 1975. MR 0458335 (56:16538)
  • [GB] P. B. Gilkey and T. P. Branson, The asymptotics of the Laplacian on a manifold with boundary, preprint (1989). MR 1032631 (90m:58201)
  • [G] P. Greiner, An asymptotic expansion for the heat equations, Proc. Sympos. Pure Math., vol. 16, Amer. Math. Soc., Providence, R.I., 1970, pp. 133-135. MR 0265784 (42:693)
  • [H] P. Hsu, Sur la théta fonction d'une variété riemannienne á bord, C.R. Acad. Sci. Sér. I Math. 309 (1989), 507-510. MR 1055469 (91h:58117)
  • [K] M. Kac, Can one hear the shape of a drum? Amer. Math. Soc. Monthly 73 (1966), 1-23. MR 0201237 (34:1121)
  • [KCD] G. Kennedy, R. Critchley, and J. S. Dowker, Finite temperature field theory with boundaries: stress tensor and surface action renormalization, Ann. Phys. 125 (1980), 346-400. MR 567353 (81c:81040)
  • [L] G. Louchard, Mouvement Brownien et valueurs propres du Laplacien, Ann. Inst. Henri Poincaré 4 (1968), 331-342. MR 0261695 (41:6308)
  • [M] S. A. Molchanov, Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975), 1-63. MR 0413289 (54:1404)
  • [MS] H. P. McKean, Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1969), 43-69. MR 0217739 (36:828)
  • [P] A. Pleijel, A study of certain Green's functions with applications in the theory of vibrating membranes, Ark. Math. 2 (1954), 533-569. MR 0061257 (15:798g)
  • [W] R. T. Waechter, On hearing the shape of a drum: an extension to higher dimensions, Proc. Cambridge Philos. Soc. 72 (1972), 439-447. MR 0304887 (46:4019)
  • [Z] E. M. E. Zayed, Eigenvalues of the Laplacian: an extension to higher dimensions, IMA J. Appl. Math. 33 (1984), 83-99. MR 763405 (86c:58153)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G18, 58G20

Retrieve articles in all journals with MSC: 58G18, 58G20

Additional Information

Keywords: Laplace-Beltrami operator, eigenvalue, heat kernel, asymptotic expansion, $ \Theta $-function, parametric method
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society