Min-max minimal disks with free boundary in Riemannian manifolds
Min-max minimal disks with free boundary in Riemannian manifolds
复制标题
DOI：
10.2140/gt.2020.24.471
复制
发表时间：
2018-06
影响因子：
2
通讯作者：
Longzhi Lin;Ao Sun;Xin Zhou
中科院分区：
文献类型：
--
作者：
Longzhi Lin;Ao Sun;Xin Zhou
在本文中，我们建立了最小-最大理论，用于在任何封闭黎曼流形中构造具有自由边界的最小圆盘。主要结果是弗雷泽建立的具有自由边界的最小圆盘的部分莫尔斯理论的有效版本。我们的理论还包括最小盘 Plateau 问题的最小-最大理论作为特例，该理论可用于将 Morse-Thompkins 和 Shiffman 在 $\mathbf{R}^n$ 中的最小曲面上的著名工作推广到黎曼设置。更准确地说，我们使用 Colding 和 Minicozzi 引入的调和替换将最小曲面的最小-最大构造推广到自由边界设置。作为这种构造的关键要素，我们展示了弱调和映射的能量凸性，其中混合狄利克雷和自由边界从 $\mathbf{R}^2$ 中的半单位 $2$-disk 到任何封闭的黎曼流形，这特别产生了这种弱调和映射的唯一性。这是由 Colding 和 Minicozzi 证明的单位 $2$ 盘上具有狄利克雷边界的弱调和映射的能量凸性和唯一性的自由边界模拟。
In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit $2$-disk in $\mathbf{R}^2$ into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved by Colding and Minicozzi.