Traces of heat operators on Riemannian foliations

Richardson, Ken

doi:10.1090/S0002-9947-09-05069-7

Traces of heat operators on Riemannian foliations
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by Ken Richardson

Trans. Amer. Math. Soc. 362 (2010), 2301-2337

DOI: https://doi.org/10.1090/S0002-9947-09-05069-7

Published electronically: December 8, 2009

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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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