Li-Yau gradient bounds on compact manifolds under nearly optimal curvature conditions
Li-Yau gradient bounds on compact manifolds under nearly optimal curvature conditions
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近乎最优曲率条件下紧流形上的Li-Yau梯度界
DOI：
10.1016/j.jfa.2018.02.001
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发表时间：
2018
影响因子：
1.7
通讯作者：
Meng Zhu
中科院分区：
文献类型：
--
作者：
Qi S. Zhang;Meng Zhu
本文证明了热方程在固定度量流形上或在Ricci流下的Li-Yau型梯度界。在前一种情况下，曲率条件为|R i c−| ∈ L p，对于某些p> n/2，或sup M ∈ M| R i c−| 2（y）d 2− n（x，y）dy <∞，其中n是流形的维数。在后一种情况下，只需要标量曲率有界。我们将解释为什么条件是接近最优的，并给出一个应用。具有固定度量的流形上的热方程的Li-Yau界似乎是第一个允许Ricci曲率从下到上没有界的界。
Abstract We prove Li–Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is| R i c−|∈ L p for some p> n/2, or sup M∫ M| R i c−| 2 (y) d 2− n (x, y) d y<∞, where n is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li–Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below.