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

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## 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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