Rigidity and gap results for low index properly immersed self--shrinkers in ℝ^m+1

2014/08/15 by Debora Impera, Impera, Debora · 1 citation
Mathematics · #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DG

paper · pdf · doi:10.48550/arxiv.1408.3479

13 pages; fixed the last part of the proof of Theorem 1.1

openalex publication_date 2014/08/15 · arxiv created 2015/01/11 · arxiv updated 2015/01/13 · openalex created_date 2022/08/27 · openalex updated_date 2026/07/28

Abstract

In this paper we show that the only properly immersed self--shrinkers Σ in ℝ^m+1 with Morse index 1 are the hyperplanes through the origin. Moreover, we prove that if Σ is not a hyperplane through the origin then the index jumps and it is at least m+2, with equality if and only if Σ is a cylinder ℝ^m-k× \mathbbS^k(√(k)) for some 1≤ k≤ m-1.

Citations

• Local Rigidity Theorems for Minimal Hypersurfaces

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