Index and topology of minimal hypersurfaces in Rn

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Abstract

In this paper, we consider immersed two-sided minimal hypersurfaces in Rn with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When n= 4 , we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective Morse index bound by purely topological invariants, and is a generalization of Li and Wang (Math Res Lett 9(1):95–104, 2002). Using our index estimates and ideas from the recent work of Chodosh–Ketover–Maximo (Minimal surfaces with bounded index, 2015. arXiv:1509.06724), we prove compactness and finiteness results of minimal hypersurfaces in R4 with finite index.

Original languageEnglish (US)
Article number180
JournalCalculus of Variations and Partial Differential Equations
Volume56
Issue number6
DOIs
StatePublished - Dec 1 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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