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Classification of closed minimal hypersurfaces with constant scalar curvature in $\mathbb{S}^5$

Chengchao He, Hongwei Xu, Entao Zhao

Abstract

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an equatorial 4-sphere, a product of spheres $\mathbb{S} ^{2}(\frac{\sqrt{2}}{2}) \times \mathbb{S} ^{2}(\frac{\sqrt{2}}{2})$ or $\mathbb{S} ^{1}(\frac{1}{2}) \times \mathbb{S} ^{3}(\frac{\sqrt{3}}{2})$, or a Cartan's minimal hypersurface. In particular, the value of the squared norm of the second fundamental form $S$ can only be 0, 4, or 12. This result strongly supports Chern's conjecture.

Classification of closed minimal hypersurfaces with constant scalar curvature in $\mathbb{S}^5$

Abstract

In this paper, we prove that any closed minimal hypersurface in the -dimensional unit sphere with constant scalar curvature and constant -th mean curvature must be isoparametric. To be precise, is either an equatorial 4-sphere, a product of spheres or , or a Cartan's minimal hypersurface. In particular, the value of the squared norm of the second fundamental form can only be 0, 4, or 12. This result strongly supports Chern's conjecture.
Paper Structure (8 sections, 9 theorems, 114 equations)

This paper contains 8 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two distinct principle curvatures at one point
  4. Proof of the main theorem
  5. Discussion on ImK
  6. Structure equations on Y
  7. The 3-form Phi
  8. Proof of Theorem \ref{['mthm']}

Key Result

Theorem 1.1

(J.Simons J.Simons1968) Let $M^n$ be an $n$-dimensional closed minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. Denote by $S$ the squared length of the second fundamental form of $M^n$. Then the following inequality holds: In particular, if $0 \leqslant S \leqslant n$, then either $S\equiv0$ and $M^n$ is the totally geodesic sphere in $\mathbb{S}^{n+1}$, or $S\equiv n$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 7 more