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November 2014 Large time behavior of the heat kernel
Guoyi Xu
J. Differential Geom. 98(3): 467-528 (November 2014). DOI: 10.4310/jdg/1406552278

Abstract

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding’s original strategy, by blowing down the metric, using Cheeger and Colding’s theory about limit spaces of Gromov-Hausdorff convergence, combining with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure. As one application, we get the consistent large time limit of heat kernel in more general context, which generalizes the former result of P. Li. Furthermore, by choosing different sequences to blow down the suitable metric, we show the first example manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.

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Guoyi Xu. "Large time behavior of the heat kernel." J. Differential Geom. 98 (3) 467 - 528, November 2014. https://doi.org/10.4310/jdg/1406552278

Information

Published: November 2014
First available in Project Euclid: 28 July 2014

zbMATH: 1296.53080
MathSciNet: MR3263524
Digital Object Identifier: 10.4310/jdg/1406552278

Rights: Copyright © 2014 Lehigh University

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Vol.98 • No. 3 • November 2014
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