Bernstein-Heinz-Chern results in calibrated manifolds

TL;DR

This paper proves that under certain geometric conditions, submanifolds in calibrated manifolds with bounded or decaying cosine of the calibration angle must be minimal, extending classical results on mean curvature estimates.

Contribution

It introduces a simple, unified proof extending Heinz and Chern's estimates to a broad class of submanifolds in calibrated manifolds, with new conditions for minimality and total geodesicity.

Findings

Submanifolds with zero Cheeger constant are minimal under bounded cosine conditions.

Decay conditions on cosine of the angle imply minimality for complete manifolds with non-negative Ricci curvature.

The proof generalizes classical Euclidean results to calibrated manifolds with a unified approach.

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.