Abstract

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

中文翻译：

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Dirichlet和Neumann特征函数的梯度和Hessian估计

摘要

我们建立了带有边界的黎曼流形上的Ricci曲率，平均曲率和第二基本形式的积分公式和尖锐的两面界。作为应用，可以为Dirichlet和Neumann本征函数导出陡峭的梯度和Hessian估计。