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June, 2010 Bernstein-Heinz-Chern results in calibrated manifolds
Guanghan Li , Isabel M. C. Salavessa
Rev. Mat. Iberoamericana 26(2): 651-692 (June, 2010).


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.


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Guanghan Li . Isabel M. C. Salavessa . "Bernstein-Heinz-Chern results in calibrated manifolds." Rev. Mat. Iberoamericana 26 (2) 651 - 692, June, 2010.


Published: June, 2010
First available in Project Euclid: 4 June 2010

zbMATH: 1197.53077
MathSciNet: MR2677011

Primary: 53C38 , 53C40 , 53C42 , 58E35

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

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 2 • June, 2010
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