Large time behavior of the heat kernel
Large time behavior of the heat kernel
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DOI：
10.4310/jdg/1406552278
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发表时间：
2013-10
期刊：
影响因子：
--
通讯作者：
Guoyi Xu
中科院分区：
文献类型：
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作者：
Guoyi Xu
本文研究了P. Li在附加最大体积增长假设下研究的具有非负Ricci曲率的完全黎曼流形上的热核的大时间行为。遵循丁氏的原始策略，通过扬弃度量，利用Cheeger和Colding关于Gromov-Hausdorff收敛的极限空间理论，结合极限空间上的热核的高斯上界，成功地将流形上的热核的极限行为化约为流形在无穷远处的切锥上的热核的值。作为一个应用，我们在更一般的情况下得到了热核的一致大时限，推广了Li P.之前的结果。在此基础上，通过选择不同的序列来降低合适的度量，给出了热核具有不一致极限行为的第一个流形实例，从而否定地回答了P. Li提出的一个开放性问题。
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.