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

Niang Chen

paper

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

arXiv:2603.09705

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 $min {m_{1}, m_{2}} \geq 4$ . Sufficiency is obtained via a moment-relaxation technique yielding the sharp bound $4 a^{2}$ 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, $index (M) \geq \frac{2}{d ( d - 1 )} b_{1} (M)$ with $d = n + 3$ .