Revista Matemática Iberoamericana

Bernstein-Heinz-Chern results in calibrated manifolds

Guanghan Li and Isabel M. C. Salavessa

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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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Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 651-692.

First available in Project Euclid: 4 June 2010

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Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C38: Calibrations and calibrated geometries 53C40: Global submanifolds [See also 53B25] 58E35: Variational inequalities (global problems)

calibrated geometry parallel mean curvature Heinz-inequality Bernstein


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

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