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


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 \(n+p+1\).


Self-shrinker Morse index eigenvalue Jacobi operator 

Mathematics Subject Classification

53C21 53C42 



  1. 1.
    Alías, L.J.: On the stability index of minimal and constant mean curvature hypersurfaces in spheres. Rev. De La Unión Mat. Argent. 47, 39–61 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrozio, L., Carlotto, A., Sharp, B.: Comparing the Morse index and the first Betti number of minimal hypersurfaces. J. Differ. Geom. 108, 379–410 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, B., Li, H., Wei, Y.: F-stability for self-shrinking solutions to mean curvature flow. Asian J. Math. 18, 757–778 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, H.D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879–889 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheng, Q.-M., Ogata, S.: 2-Dimensional complete self-shrinkers in \(R^3\). Math. Z. 284, 537–542 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheng, Q.-M., Peng, Y.J.: Complete self-shrinkers of the mean curvature flow. Calc. Var. 52, 497–506 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. Math. 175, 755–833 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Colding, T.H., Minicozzi II, W.P.: Smooth compactness of self-shrinkers. Comment. Math. Helv. 87, 463–475 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cao, H., Shen, Y., Zhu, S.: The structure of stable minimal hypersurfaces in \(\mathbb{R}^{n+1}\). Math. Res. Lett. 4, 637–644 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ding, Q., Xin, Y.L.: The rigidity theorems of self-shrinkers. Trans. Am. Math. Soc. 366, 5067–5085 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in three manifolds. Invent. Math. 82, 121–132 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990). In: Proceedings of Symposia in Pure Mathematics, vol. 54, Part 1. American Mathematical Society, Providence, pp. 175–191 (1993)Google Scholar
  13. 13.
    Hussey, C.: Classification and analysis of mean curvature flow self-shrinkers. arXiv:1303.0354v1
  14. 14.
    Impera, D.: Rigidity and gap results for low index properly immersed self-shrinkers in \({\mathbb{R}}^{m+1}\). arXiv:1408.3479
  15. 15.
    Lee, Y.I., Lue, Y.K.: The stability of self-shrinkers of mean curvature flow in higher co-dimension. Trans. Am. Math. Soc. 367, 2411–2435 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, P., Wang, J.P.: Minimal hypersurfaces with finite index. Math. Res. Lett. 9, 95–103 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, H., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. J. Math. Soc. Jpn. 66, 709–734 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Savo, A.: Index bounds for minimal hypersurfaces of the sphere. Indiana Univ. Math. J. 59, 823–838 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Urbano, F.: Minimal surfaces with low index in the three-dimensional sphere. Proc. Am. Math. Soc. 108, 989–992 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xin, Y.: Minimal Submanifolds and Related Topics. World Scientific, Hackensack (2003)CrossRefGoogle Scholar

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© 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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