Transactions of the American Mathematical Society

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



Traces of heat operators on Riemannian foliations

Author: Ken Richardson
Journal: Trans. Amer. Math. Soc. 362 (2010), 2301-2337
MSC (2010): Primary 53C12, 58J37, 58J35, 58J50
Published electronically: December 8, 2009
MathSciNet review: 2584602
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Abstract: We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace $ K_{B}(t) $ of this operator has a particular asymptotic expansion as $ t\to 0$. The coefficients of $ t^{\alpha}$ and of $ t^{\alpha}(\log t)^{\beta}$ in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure.

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Ken Richardson
Affiliation: Department of Mathematics, Texas Christian University, TCU Box 298900, Fort Worth, Texas 76129

Keywords: Foliation, heat equation, asymptotics, basic, Laplacian
Received by editor(s): October 8, 2007
Published electronically: December 8, 2009
Additional Notes: The author’s research at MSRI was supported in part by NSF grant DMS-9701755.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.