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 $Ric{c}^M\ge 0$ we may replace the boundedness condition on $\cos \theta$ by $\cos \theta \ge C{r}^{-\beta }$, when $r\to +\infty$, where $0\le \beta <1$ 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 $\Vert H\Vert$ 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.

v5: Final version, accepted for Publication in Rev. Mat. Iberoamericana. v3:We add a subsection on the foliation calibration, generalizing results of Barbosa, Kenmotsu and Oshikiri to higer codimension. We add several results and give conditions to conclude the submanifold is totally geodesic

References in corpus (1)

• Foliations and Chern-Heinz inequalities

Cited by in corpus (1)

• The geometry of closed conformal vector fields on Riemannian spaces