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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.
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참고문헌 (7)
참고문헌 신청
J. Choe / 1993 / On the analytic reflection of a minimal surface / Pacific J. Math 157(1):29~36
Y. Fang / 1996 / Lectures on minimal surfaces in R3 / Proceedings of the Centre for Mathematics and its Applications, Australian National University 35
D. Gilbarg / 2001 / Classics in Mathematics / Springer-Verlag
John McCuan / 1997 / Symmetry via spherical reflection and spanning drops in a wedge / Pacific Journal of Mathematics 180(2):291~323
R. Osserman / 1986 / A Survey of Minimal Surfaces / Dover Publications, Inc
D. Gilbarg / 2001 / Classics in Mathematics / Springer-Verlag
John McCuan / 1997 / Symmetry via spherical reflection and spanning drops in a wedge / Pacific Journal of Mathematics 180(2):291~323
R. Osserman / 1986 / A Survey of Minimal Surfaces / Dover Publications, Inc
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