论文标题
在$ \ mathbb {r}^3 $中存在具有恒定平均曲率的自由边界磁盘的存在
Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$
论文作者
论文摘要
Given a surface $Σ$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $Σ$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $Σ$ orthogonally.我们使用不同的方法重现了此结果，并在$σ$上的其他凸度假设下改进了结果。具体而言，当$σ$本身为凸并且具有平均曲率以下方为$ H_0 $时，我们获得了（0，H_0）$的所有$ h \ in的存在。我们使用Sacks-uhlenbeck类型的扰动，而不是Struwe使用的热流。就像以前与周（Arxiv：2012.13379）的联合合作一样，在$ h $的零零集中扩展存在的关键要素是Morse指数上限。
Given a surface $Σ$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $Σ$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $Σ$ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on $Σ$. Specifically, when $Σ$ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of $H$'s is a Morse index upper bound.
