论文标题
最小边界G不变的最小值最小超曲面的最小值理论
Min-max theory for free boundary G-invariant minimal hypersurfaces
论文作者
论文摘要
给定一个紧凑的Riemannian歧管$ m^{n+1} $，$ 3 \ leq n+1 \ leq 7 $和$ \ \ \ \ \ partial m \ neq \ emptyset $，这是Martin Man-Chun Li和Xin li和Xin Zhou建立的自由边界最小值理论，它显示出了几乎适当地嵌入了$ hyperface $ hyperface $ hyperface $ $ $ $ $ $。在本文中，我们将它们的结构推广到模棱两可的设置中。具体而言，让$ g $是一个紧凑的谎言组，在$ m $上充当同型的同量，至少$ 3 $。然后，我们表明存在一个非平稳的平滑，几乎正确嵌入了$ g $ g $ invariant的最小超出表面，并具有自由边界。此外，如果$ m $的RICCI曲率是非负的，而$ \ partial m $则严格凸出，那么存在无限的许多适当嵌入的$ g $ g $ invariant minimal syrimal sypurfaces具有自由边界。
Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$ and $\partial M\neq\emptyset$, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in $\partial M$. In this paper, we generalize their constructions into equivariant settings. Specifically, let $G$ be a compact Lie group acting as isometries on $M$ with cohomogeneity at least $3$. Then we show that there exists a nontrivial smooth almost properly embedded $G$-invariant minimal hypersurface with free boundary. Moreover, if the Ricci curvature of $M$ is non-negative and $\partial M$ is strictly convex, then there exist infinitely many properly embedded $G$-invariant minimal hypersurfaces with free boundary.
