LI-YAU GRADIENT BOUND FOR COLLAPSING MANIFOLDS UNDER INTEGRAL CURVATURE CONDITION
LI-YAU GRADIENT BOUND FOR COLLAPSING MANIFOLDS UNDER INTEGRAL CURVATURE CONDITION
复制标题
积分曲率条件下塌陷流形的LI-YAU梯度界限
DOI：
10.1090/proc/13418
复制
发表时间：
2017
影响因子：
1
通讯作者：
Zhu Meng
中科院分区：
文献类型：
--
作者：
Zhang Qi S.;Zhu Meng
设是一个完备的黎曼流形。对于任何常数p，\r> 0，定义，其中表示Ricci曲率张量的负部分。证明了对任意的，当足够小时，测地线球辛上热方程的正解存在一定的Li-Yau型梯度界。这里的假设是小允许的情况下，流形是崩溃的。回想一下，在Zhang和Zhu的早期论文中，作者也得到了一定的Li-Yau梯度界，假设并且流形是非塌陷的。因此，在一定程度上，本文的结果和前面的结果一样，完善了流形上热方程的Li-Yau梯度界的图象，使其在模常数的锐度下是可积的.引用
Letbe a complete Riemannian manifold. For any constants $ p,\r> 0$, define, wheredenotes the negative part of the Ricci curvature tensor. We prove that for any, whenis small enough, a certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic ballsinwith. Here the assumption thatis small allows the situation where the manifold is collapsing. Recall that in an earlier paper by Zhang and Zhu, a certain Li-Yau gradient bound was also obtained by the authors, assuming thatand the manifold is noncollapsed. Therefore, to some extent, the results in this paper, as well as the earlier one complete the picture of Li-Yau gradient bounds for the heat equation on manifolds withbeingintegrable, modulo the sharpness of constants. References