Revista Matemática Iberoamericana

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Volume 26, Issue 2, 2010, pp. 651–692
DOI: 10.4171/RMI/613

Published online: 2010-08-31

Bernstein-Heinz-Chern results in calibrated manifolds

Guanghan Li[1] and Isabel M.C. Salavessa[2]

(1) Hubei University, Wuhan, China
(2) Instituto Superior Técnico, Lisboa, Portugal

Given a calibrated Riemannian manifold $\overline{M}$ with parallel calibration $\Omega$ of rank $m$ and $M$ an orientable m-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 $Ricci^M\geq 0$ we may replace the boundedness condition on $\cos\theta$ by $\cos\theta\geq Cr^{-\beta}$, when $r\rightarrow+\infty$, where $0 < \beta < 1$ 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 an estimation of $\|H\|$ in terms of $\cos\theta$ and an isoperimetric inequality. In a similar way, we also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.

Keywords: Calibrated geometry, parallel mean curvature, Heinz-inequality, Bernstein

Li Guanghan, Salavessa Isabel: Bernstein-Heinz-Chern results in calibrated manifolds. Rev. Mat. Iberoamericana 26 (2010), 651-692. doi: 10.4171/RMI/613