稳定最小超曲面的两个刚性结果

IF 2.4 1区数学 Q1 MATHEMATICS

Geometric and Functional Analysis Pub Date : 2024-02-01 DOI:10.1007/s00039-024-00662-1

Giovanni Catino, Paolo Mastrolia, Alberto Roncoroni

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引用次数: 0

摘要

本文的目的是证明两个关于完整的、浸没的、可定向的、稳定的最小超曲面的刚度的结果：我们证明它们在 R4 中是超平面，而当 n≤5 时，它们不存在于正曲封闭的黎曼（n+1）-manifold 中；特别是，当 n≤5 时，在 Sn+1 中不存在稳定的最小超曲面。第一个结果最近也由 Chodosh 和 Li 证明了，第二个结果是关于有限指数极小曲面的一个更普遍结果的结果。这两个定理都依赖于保角方法，其灵感来自费舍尔-科尔布里（Fischer-Colbrie）的经典著作。

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Two Rigidity Results for Stable Minimal Hypersurfaces

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in R⁴, while they do not exist in positively curved closed Riemannian (n+1)-manifold when n≤5; in particular, there are no stable minimal hypersurfaces in Sⁿ⁺¹ when n≤5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.

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来源期刊

Geometric and Functional Analysis 数学-数学

CiteScore

3.70

自引率

4.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.