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ACS Condition on Minimal Isoparametric Hypersurfaces of Positive Ricci Curvature in Unit Spheres

Niang Chen

Abstract

Motivated by the Schoen--Marques--Neves conjecture relating the Morse index of a minimal hypersurface to its first Betti number in ambient manifolds with positive Ricci curvature, we study the Ambrozio--Carlotto--Sharp (ACS) criterion for index growth. We verify a sufficient pointwise ACS inequality for several families of minimal isoparametric hypersurfaces with positive Ricci curvature in the unit sphere. As a consequence, for any closed embedded minimal hypersurface $M^n$ in such an ambient manifold $\mathcal{N}^{n+1}\subset \mathbb{S}^{n+2}\subset\mathbb{R}^{n+3}$, one has $\operatorname{index}(M)\ge \frac{2}{d(d-1)}\, b_1(M), d=n+3,$ where $b_1(M)$ is the first Betti number of $M$.

ACS Condition on Minimal Isoparametric Hypersurfaces of Positive Ricci Curvature in Unit Spheres

Abstract

Motivated by the Schoen--Marques--Neves conjecture relating the Morse index of a minimal hypersurface to its first Betti number in ambient manifolds with positive Ricci curvature, we study the Ambrozio--Carlotto--Sharp (ACS) criterion for index growth. We verify a sufficient pointwise ACS inequality for several families of minimal isoparametric hypersurfaces with positive Ricci curvature in the unit sphere. As a consequence, for any closed embedded minimal hypersurface in such an ambient manifold , one has where is the first Betti number of .
Paper Structure (3 sections, 4 theorems, 44 equations)

This paper contains 3 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The computations

Key Result

Theorem 1.1

Let $(N^{n+1},g)$ be a Riemannian manifold isometrically embedded in some Euclidean space $\mathbb{R}^d$. Let $M^n$ be a closed embedded minimal hypersurface of $N^{n+1}$. Assume that for every nonzero vector field $X$ on $M$, where $R^N$ is the curvature tensor of $N$, $\Pi$ is the second fundamental form of the embedding $N\hookrightarrow \mathbb{R}^d$, $\nu$ is a local unit normal vector field

Theorems & Definitions (8)

  • Conjecture : Schoen-Marques-Neves, see MarquesNeves
  • Theorem 1.1: Ambrozio-Carlotto-Sharp ACS1
  • Theorem 1.2
  • Theorem 2.1: Münzner, see ChiMuenznerIMuenznerII
  • Proposition 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 3.2