Large time behavior of the heat kernel. (English) Zbl 1296.53080

Xu, Guoyi

J. Differ. Geom. 98, No. 3, 467-528 (2014).

Summary: 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, combined with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on the manifold to the values of heat kernels on tangent cones at infinity of manifolds with renormalized measure. As one application, we get the consistent large time limit of the 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 of a manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

volume growth; blowing down; Gromov-Hausdorff convergence