Stability of submanifolds with parallel mean curvature in calibrated manifolds

摘要

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