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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a smooth compact Riemannian manifold without boundary. Let be an open interval. Let be a smooth positive function. Let be the metric on . Consider the fundamental solution of the heat equation on with metric (when exists globally we call it the heat kernel on ). The coefficients of the asymptotic expansion of the trace are studied and expressed in terms of corresponding coefficients on the basis . It is fulfilled by means of constructing a parametrix for 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.

**[1]**Marcel Berger, Paul Gauduchon, and Edmond Mazet,*Le spectre d’une variété riemannienne*, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR**0282313****[2]**Robert S. Cahn and Joseph A. Wolf,*Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one*, Comment. Math. Helv.**51**(1976), no. 1, 1–21. MR**0397801****[3]**Jeff Cheeger and Shing Tung Yau,*A lower bound for the heat kernel*, Comm. Pure Appl. Math.**34**(1981), no. 4, 465–480. MR**615626**, 10.1002/cpa.3160340404**[4]**I. M. Gel′fand and L. A. Dikiĭ,*Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations*, Uspehi Mat. Nauk**30**(1975), no. 5(185), 67–100 (Russian). MR**0508337****[5]**Peter B. Gilkey,*The index theorem and the heat equation*, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks; Mathematics Lecture Series, No. 4. MR**0458504****[6]**H. A. Lauwerier,*Asymptotic analysis*, Mathematisch Centrum, Amsterdam, 1974. Mathematical Centre Tracts, No. 54. MR**0467123****[7]**S. Minakshisundaram and Å. Pleijel,*Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds*, Canadian J. Math.**1**(1949), 242–256. MR**0031145****[8]**S. Minakshisundaram,*Eigenfunctions on Riemannian manifolds*, J. Indian Math. Soc. (N.S.)**17**(1953), 159–165 (1954). MR**0061750****[9]**A. H. Zemanian,*Generalized integral transformations*, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1968. Pure and Applied Mathematics, Vol. XVIII. MR**0423007**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58G11,
35K05

Retrieve articles in all journals with MSC: 58G11, 35K05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0664031-2

Article copyright:
© Copyright 1982
American Mathematical Society