Bernstein-heinz-chern Results in Calibrated Manifolds

Abstract: Given a calibrated Riemannian manifold M with parallel calibration Ω of rank m and M an orientable m-submanifold with parallel mean curvature H , we prove that if cosθ is bounded away from zero, where θ is the Ω-angle of M, and if M has zero Cheeger constant, then M is minimal. In the particular case M is complete with RiccM ≥ 0 we may replace the boundedness condition on cosθ by cosθ ≥Cr−β , when r → +∞, where 0 < β < 1 and C > 0 are constants and r is the distance function to a point in M. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of ‖H‖ in terms of cosθ and an isoperimetric inequality. In a similar way, we also give some conditions to conclude M is totally geodesic. We study some particular cases.