A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary
A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary
复制标题
具有自由边界的嵌入极小曲面的一般存在性定理
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
M. Li
中科院分区：
文献类型：
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作者：
M. Li
本文证明了任意边界为∂M的紧黎曼3 -流形M中具有自由边界的适当嵌入极小曲面的一般存在性定理。这些最小曲面要么与∂M不相交，要么与∂M正交。我们的结果的主要特征是没有关于M的曲率或∂M的凸性的假设。证明了极小曲面在自由边界处的边界规则性。此外，我们定义了具有边界的紧化3 -流形的拓扑不变量填充格，并证明了我们可以用环境流形m的填充格来约束上述构造的最小曲面的格。我们的证明采用了Colding和De Lellis在封闭嵌入最小曲面上使用的最小-最大构造的一种变体，该构造最初由Almgren和Pitts提出。©2014 Wiley期刊公司
In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂M. These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.
由最小-最大构造产生的最小曲面的属界
DOI：
10.1515/crelle.2010.052
发表时间：
2010
期刊：
影响因子：
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作者：
De Lellis;Pellandini
通讯作者：
Pellandini