Stability of submanifolds with parallel mean curvature in calibrated manifolds

On a Riemannian manifold $ \bar M^{m + n} $ \bar 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 ℝH ⋇ 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 $ \bar M $ \bar 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 ℝ ^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.