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Results in Mathematics

, 74:68 | Cite as

Rigidity and Gap Results for the Morse Index of Self-Shrinkers with any Codimension

  • Xu-Yong Jiang
  • He-Jun SunEmail author
  • Peibiao Zhao
Article
  • 80 Downloads

Abstract

In this paper, we investigate an n-dimensional complete properly immersed self-shrinker M in the \((n+p)\)-dimensional Euclidean space \(\mathbb {R}^{n+p}\). We prove that the Morse index of M is great than or equal to p, with the equality holds if and only if M is an n-plane. Moreover, we prove that if the self-shrinker is non-totally geodesic, then its index has to be at least \(n+p+1\). We also show that the index of the cylinder \(\mathbb {S}^k(\sqrt{2k})\times \mathbb {R}^{n-k}\) (for some \(1\le k \le n\)) in \(\mathbb {R}^{n+p}\) is \(n+p+1\).

Keywords

Self-shrinker Morse index eigenvalue Jacobi operator 

Mathematics Subject Classification

53C21 53C42 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of ScienceNanjing University of Science and TechnologyNanjingPeople’s Republic of China

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