Bernstein-Heinz-Chern results in calibrated manifolds

Abstract

Given $(\bar{M},\Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos \theta$ is bounded away from zero, where $\theta$ is the $\Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\geq 0$ we may replace the boundedness condition on $\cos \theta$ by $\cos \theta\geq Cr^{-\beta}$, when $r\to +\infty$, where $ 0\leq\beta <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 $\cos\theta$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.