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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
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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].

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