Closed minimal hypersurfaces in $\mathbb S^5$ with constant $S$ and $A_3$

TL;DR

The paper proves that a closed minimally immersed hypersurface $M^4 o S^5$ with constant $S= frac{}{} extstyle\sum_{i=1}^4 \lambda_i^2$ and $A_3= frac{}{} extstyle\sum_{i=1}^4 \lambda_i^3$, under nonnegative scalar curvature $R_M$, must be isoparametric with $S\

Abstract

In this paper, we prove that a closed minimally immersed hypersurface $M^4\subset\mathbb S^5$ with constant $S:=\sum\limits_{i=1}^4λ_i^2$ and $A_3:=\sum\limits_{i=1}^4λ_i^3$ whose scalar curvature $R_M$ is nonnegative must be isoparametric. Moreover, $S$ can only be $0, 4,$ and $12.$ That is $M^4$ is either an equatorial $4$-sphere, a clifford torus, or a Cartan's minimal hypersurface.

Theorems & Definitions (19)

• Conjecture 1.1: Chern's Conjecture 1
• Conjecture 1.2: Chern's Conjecture 2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Theorem \ref{['thm2']}
• Theorem 3.1
• Lemma 3.2
• Remark 3.3