Free boundary stable hypersurfaces in manifolds with density and rigidity results
Free boundary stable hypersurfaces in manifolds with density and rigidity results
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DOI：
10.1016/j.geomphys.2014.01.013
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发表时间：
2013-11
影响因子：
1.5
通讯作者：
Katherine Castro;César Rosales
中科院分区：
文献类型：
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作者：
Katherine Castro;César Rosales
设M是一个具有边界的加权流形，即一个黎曼流形，其中一个密度函数用于加权黎曼Hausdorff测度。本文给出了带边界超曲面形变的内权面积的第一、第二变分公式。作为结果，我们得到变分特征的临界点和二阶极小的加权面积有或没有体积约束。此外，在紧情形下，在一定的曲率和边界假设下，我们得到了自由边界稳定的面积极小化超曲面的拓扑估计和刚性性质。我们的结果和证明推广了以前的黎曼流形（常密度）和加权流形中的空边界超曲面。
Let M be a weighted manifold with boundary∂ M, ie, a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in∂ M. As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on M. Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.