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 \(n+p+1\).
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The first author was supported by the Scientific Research Innovation Project of Jiangsu Province (Grant No. KYCX17_0327). The second author was supported by the Fundamental Research Funds for the Central Universities (Grant No. 30917011335). The third author was supported by the National Natural Science Foundation of China (Grant Nos. 11371194, 11871275).
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Jiang, XY., Sun, HJ. & Zhao, P. Rigidity and Gap Results for the Morse Index of Self-Shrinkers with any Codimension. Results Math 74, 68 (2019). https://doi.org/10.1007/s00025-019-0993-z
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DOI: https://doi.org/10.1007/s00025-019-0993-z