无穷厚流形中最小超曲面的二分法

IF 1.3 1区 数学 Q1 MATHEMATICS
Antoine SONG
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引用次数: 17

摘要

本文章由计算机程序翻译,如有差异,请以英文原文为准。
A dichotomy for minimal hypersurfaces in manifolds thick at infinity
Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that $(M,g)$ is thick at infinity, i.e. any connected finite volume complete minimal hypersurface is closed. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface if $(M,g)$ has finite volume, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result of Irie-Marques-Neves when $(M,g)$ is shrinking to zero at infinity.
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来源期刊
CiteScore
3.00
自引率
5.30%
发文量
25
审稿时长
>12 weeks
期刊介绍: The Annales scientifiques de l''École normale supérieure were founded in 1864 by Louis Pasteur. The journal dealt with subjects touching on Physics, Chemistry and Natural Sciences. Around the turn of the century, it was decided that the journal should be devoted to Mathematics. Today, the Annales are open to all fields of mathematics. The Editorial Board, with the help of referees, selects articles which are mathematically very substantial. The Journal insists on maintaining a tradition of clarity and rigour in the exposition. The Annales scientifiques de l''École normale supérieures have been published by Gauthier-Villars unto 1997, then by Elsevier from 1999 to 2007. Since January 2008, they are published by the Société Mathématique de France.
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