Symmetry and uniqueness of embedded minimal hypersurfaces in $\mathbb R^{n+1}$

Abstract

In this paper, we prove some rigidity results about embedded minimal hypersurface $M\subset \mathbb R^{n+1}$ with compact $\partial M$ that has one end which is regular at infinity. We first show that if $M \subset \mathbb R^{n+1}$ meets a hyperplane in a constant angle $\ge \pi/2$, then $M$ is part of an $n$-dimensional catenoid. We show that if $M$ meets a sphere in a constant angle and $\partial M$ lies in a hemisphere determined by the hyperplane through the center of the sphere and perpendicular to the limit normal vector $n_M$ of the end, then $M$ is part of either a hyperplane or an $n$-dimensional catenoid. We also show that if $M$ is tangent to a $C^2$ convex hypersurface $S$, which is symmetric about a hyperplane $P$ and $n_M$ is parallel to $P$, then $M$ is also symmetric about $P$. In special, if $S$ is rotationally symmetric about the $x_{n+1}$-axis and $n_M=e_{n+1}$, then $M$ is also rotationally symmetric about the $x_{n+1}$-axis.