Stability of submanifolds with parallel mean curvature in calibrated manifolds

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.

arXiv commentFinal version (Bull. Braz. Math. Soc. NS 41(4)(2010), 495-530. v.4 We add a criteria on the calibration for spheres in Euclidean spaces to be $\Omega$-stable. We give some natural conditions that enable us to conclude that the m-spheres are the unique $\Omega$-stable submanifolds in Euclidean spaces. We correct a misprint (missing a term in $C_{\Omega }$)