A Bernstein Type Theorem For Self-similar Shrinkers

2009/12/09 by Lu Wang, Wang, Lu · 1 citation
Computer Science · Engineering · Mathematics · #53C42 #53C44 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:53C42 #msc:53C44

paper · pdf · doi:10.48550/arxiv.0912.1809

7 pages

arxiv created 2009/12/09 · openalex publication_date 2009/12/09 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we prove that smooth self-shrinkers in \Realⁿ⁺¹, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.

Citations

• Curvature estimates for minimal hypersurfaces
• Minimal Varieties in Riemannian Manifolds
• Generic mean curvature flow I; generic singularities

Cited by

• Rigidity Results for Spacelike Self-Shrinkers via Different Maximum Principles

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