Bernstein-Heinz-Chern results in calibrated manifolds

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Given (M,Ω) a calibrated Riemannian manifold with a parallel calibration of rank m, and Mᵐ an immersed orientable submanifold with parallel mean curvature H we prove that if 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 Riccᵐ≥ 0 we may replace the boundedness condition on θ by 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 θ and an isoperimetric inequality. We also give some conditions to conclude M is totally geodesic. We study some particular cases.