论文标题
用于Varifolds和应用的移动平面方法
Moving plane method for varifolds and applications
论文作者
论文摘要
在本文中，我们介绍了一种移动平面方法的版本，该方法适用于潜在的相当单一的超曲面，从而推广了经典的移动平面方法以进行平滑的曲面。宽松地说，我们的varifolds版本表明，无穷大的平滑度和对称性（分别在边界处）可以促进到内部的平滑度和对称性。与移动平面原理的经典公式相反，关键特征是平滑度是一个结论，而不是假设。 我们在具有光滑边界的紧凑型Varifolds和完整的Varifolds的设置中实现了我们的移动平面方法。关键成分是用于固定和cmc-varifolds的HOPF引理。我们的HOPF引理提供了一种新工具来建立Varifolds的平滑度，并在任意维度上起作用，而没有任何稳定性假设。作为我们新移动平面方法的应用，我们证明了由Schoen，Alexandrov，Meeks和Korevaar-Kusner-Solomon启发的catenoid，球形帽和Delaunay表面的Varifold唯一性结果。我们还证明了Alexandrov定理的varifold版本，用于双曲线空间中的紧凑型cmc-varifolds。
In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compact varifolds with smooth boundary and in the setting of complete varifolds. A key ingredient is a Hopf lemma for stationary and CMC-varifolds. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compact CMC-varifolds in hyperbolic space.
