Given a calibrated Riemannian manifold with parallel calibration of rank and an orientable m-submanifold with parallel mean curvature , we prove that if is bounded away from zero, where is the -angle of , and if has zero Cheeger constant, then is minimal. In the particular case is complete with we may replace the boundedness condition on by , when , where and are constants and is the distance function to a point in . 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 in terms of and an isoperimetric inequality. In a similar way, we also give some conditions to conclude 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. 651–692DOI 10.4171/RMI/613