On the index of minimal hypersurfaces in $\mathbb{S}^{n+1}$ with $\lambda_1<n$

TL;DR

This paper establishes a lower bound on the Morse index of certain minimal hypersurfaces in spheres with specific eigenvalue conditions, advancing understanding of their stability and contributing to longstanding conjectures.

Contribution

It proves a new lower bound on the Morse index for minimal hypersurfaces with eigenvalue constraints, and offers simplified proofs of related index estimates.

Findings

Minimal hypersurfaces with $\lambda_1<n$ have Morse index at least $n+4$.

Re-proves partial Urbano Theorem for minimal tori in $\mathbb{S}^3$ with Morse index at least 5.

Introduces a comparison theorem between eigenvalues of elliptic operators that simplifies existing proofs.

Abstract

In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $\lambda_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in $\mathbb{S}^3$: a minimal torus in $\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and $r$-minimal hypersurfaces in a sphere.