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

TL;DR

This paper investigates the ACS criterion on minimal isoparametric hypersurfaces with positive Ricci curvature in spheres, establishing conditions for the inequality to hold and deriving implications for the Morse index of minimal hypersurfaces.

Contribution

It provides a precise characterization of the ACS inequality for hypersurfaces with specific principal curvature multiplicities and links the index to topological invariants.

Findings

ACS inequality holds iff minimum multiplicity ≥ 4 for g=4 cases.

Verified ACS condition for g=3 with specific multiplicities.

Derived lower bounds on Morse index based on first Betti number.

Abstract

We study the Ambrozio--Carlotto--Sharp (ACS) criterion on minimal isoparametric hypersurfaces $N^{n+1}\subset S^{n+2}$ with positive Ricci curvature, motivated by the Schoen--Marques--Neves conjecture on Morse index.For $g=4$ distinct principal curvatures with multiplicities $m_1,m_2$, we prove that the pointwise ACS inequality holds if and only if $\min\{m_1,m_2\}\ge 4$. Sufficiency is obtained via a moment-relaxation technique yielding the sharp bound $4a^2$ on the quadratic part of the integrand; necessity follows from an explicit extremal configuration in the top eigenspace of the shape operator. We also verify the ACS condition for $g=3$ with $m_1=m_2\in\{4,8\}$.As a consequence, for any closed embedded minimal hypersurface $M^n$ in such an ambient manifold, $\operatorname{index}(M)\ge \tfrac{2}{d(d-1)}\, b_1(M)$ with $d=n+3$.