A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow
A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow
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DOI：
10.1016/j.jfa.2008.05.014
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发表时间：
2006-11
影响因子：
1.7
通讯作者：
Shi-Jiang Kuang;Qi S. Zhang
中科院分区：
文献类型：
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作者：
Shi-Jiang Kuang;Qi S. Zhang
我们为共轭热方程的所有正解建立了逐点梯度估计。这与佩雷尔曼的逐点梯度估计形成对比，佩雷尔曼的逐点梯度估计主要适用于基本解而不是所有解。与佩雷尔曼的估计一样，我们梯度估计的最一般形式不需要任何曲率假设。此外，仅假设 Ricci 曲率的下界，我们还证明了类似于线性薛定谔热方程的 Li-Yau 估计的局部梯度估计。与线性情况的主要区别在于不需要对势的导数（标量曲率）进行假设。经典的哈纳克不等式如下。
We establish a point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li–Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A classical Harnack inequality follows.