Complete self-shrinkers with bounded the second fundamental form in ℝⁿ⁺¹

2023/06/17 by Yayun Chen, Chen, Yayun, Tongzhu Li +1
Computer Science · Mathematics · #53C24 #53C40 #Advanced Graph Theory Research #Cellular Automata and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2306.10343

openalex publication_date 2023/06/17 · openalex created_date 2023/06/22 · openalex updated_date 2026/07/28

Abstract

Let X:Mⁿ→ ℝⁿ⁺¹ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form S satisfies 1≤ S< C for some constant C, then S=1. Further we classify the n-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in ℝⁿ⁺¹, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.

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