Stability of submanifolds with parallel mean curvature in calibrated manifolds

TL;DR

This paper investigates the stability of certain submanifolds with constant mean curvature in calibrated manifolds, introducing an $ ext{Omega}$-Jacobi operator and demonstrating Euclidean spheres' uniqueness as stable solutions.

Contribution

It extends stability analysis to submanifolds with parallel mean curvature in calibrated manifolds, defining $ ext{Omega}$-stability and proving Euclidean spheres are uniquely stable under natural conditions.

Findings

Euclidean $m$-spheres are the only $ ext{Omega}$-stable submanifolds in $ ext{R}^{m+n}$.

Introduces an $ ext{Omega}$-Jacobi operator for second variation analysis.

Studies $ ext{Omega}$-stability of geodesic spheres in fibred space forms.

Abstract

On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $\Omega$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $\Omega$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $\Omega$ is the volume element of $\bar{M}$. To the second variation we associate an $\Omega$-Jacobi operator and define $\Omega$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $\Omega$-stable submanifolds of $\mathbb{R}^{m+n}$. We study the $\Omega$-stability of geodesic $m$-spheres of a fibred space form $M^{m+n}$ with totally geodesic $(m+1)$-dimensional fibres.