A Bernstein-chern-heinz Type Result in Calibrated Manifolds

Given (M,Ω) a calibrated Riemannian manifold with a parallel calibration of rank m, and M an immersed orientable submanifold with parallel mean curvature H we prove that if cos θ 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 Ricc ≥ 0 we may replace the boundedness condition on cos θ by cos θ ≥ 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 a estimation of ‖H‖ in terms of cos θ and an isoperimetric constant. We also give one application in quaternionic geometry.