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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The asymptotic expansion for the trace of the heat kernel on a generalized surface of revolution


Author: Ping Charng Lue
Journal: Trans. Amer. Math. Soc. 273 (1982), 93-110
MSC: Primary 58G11; Secondary 35K05
MathSciNet review: 664031
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Abstract: Let $ M$ be a smooth compact Riemannian manifold without boundary. Let $ I$ be an open interval. Let $ h(r)$ be a smooth positive function. Let $ g$ be the metric on $ M$. Consider the fundamental solution $ E(x,y,{r_1},{r_2};t)$ of the heat equation on $ M \times I$ with metric $ {h^2}(r)g + dr \otimes dr$ (when $ E$ exists globally we call it the heat kernel on $ M \times I$). The coefficients of the asymptotic expansion of the trace $ E$ are studied and expressed in terms of corresponding coefficients on the basis $ M$. It is fulfilled by means of constructing a parametrix for $ E$ which is different from a parametrix in the standard form. One important result is that each of the former coefficients is a linear combination of the latter coefficients.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0664031-2
PII: S 0002-9947(1982)0664031-2
Article copyright: © Copyright 1982 American Mathematical Society