Large time behavior of the heat kernel

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.

Weyl’s law on $RCD^{\ast} (K, N)$ metric measure spaces

“…As a corollary, we have the following convergence for the solutions of Poisson equations living on varying spaces, which is due essentially to [19,29,51,23].…”

“…However, the construction of the parametrix on smooth manifolds does not work on singular metric measure spaces. To deal with this lack of the parametrix, we shall get the small time asymptotic behavior via the (locally) uniform convergce of Dirichlet heat kernels living on a converging sequence of metric measure spaces in the sense of pointed measured Gromov-Hausdorff topology, as in [19,51,23].…”

Rigidity of vector valued harmonic maps of linear growth

“…Notice that when M has vanishing asymptotic volume ratio at infinity, (1.9) is not necessarily true for any bounded function, see [23]. However it is not clear if (1.10) only holds for sub-harmonic functions.…”

Local spectral convergence inRCD∗(K,N)spaces

“…In [KS03], the spectral convergence as well as the Mosco convergence of the local Cheeger energies with Dirichlet boundary conditions are claimed. Finally, the stability of the Dirichlet boundary condition seems to play a role in Proposition 7.5 of [Xu14]. One of the main purposes of the paper is to give an example such that (1.4) is not satisfied in general, providing at the same time positive results and, in particular, convergence results for generic balls.…”

“…In several papers (e.g. [D02], [KS03], [Xu14], [ZZ17]), the local spectral convergence is investigated, i.e. lim where Γ is the carré du champ operator associated to the metric measure structure.…”