On the stability of minimal submanifolds in conformal spheres

TL;DR

This paper proves that certain conformally round, $ ext{pinched}$ spheres do not contain stable minimal submanifolds of specific dimensions, highlighting stability constraints in conformal geometry.

Contribution

It establishes non-existence results for stable minimal submanifolds in $ ext{pinched}$ conformal spheres, extending understanding of stability in conformal geometry.

Findings

No closed stable minimal submanifolds of certain dimensions exist in $ ext{pinched}$ conformal spheres.

The result applies to spheres conformal to the round sphere with specific curvature bounds.

Provides new insights into stability conditions in conformal Riemannian geometry.

Abstract

Given an $n$-dimensional Riemannian sphere conformal to the round one and $\delta$-pinched, we show that it does not contain any closed stable minimal submanifold of dimension $2\le k\le n-\delta^{-1}$.