# Bernstein-Heinz-Chern results in calibrated manifolds

### Guanghan Li

Hubei University, Wuhan, China### Isabel M.C. Salavessa

Instituto Superior Técnico, Lisboa, Portugal

## 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 $Ricci_{M}≥0$ 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.

## Cite this article

Guanghan Li, Isabel M.C. Salavessa, Bernstein-Heinz-Chern results in calibrated manifolds. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 651–692

DOI 10.4171/RMI/613