JournalsrmiVol. 26, No. 2pp. 651–692

Bernstein-Heinz-Chern results in calibrated manifolds

  • Guanghan Li

    Hubei University, Wuhan, China
  • Isabel M.C. Salavessa

    Instituto Superior Técnico, Lisboa, Portugal
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Given a calibrated Riemannian manifold M\overline{M} with parallel calibration Ω\Omega of rank mm and MM an orientable m-submanifold with parallel mean curvature HH, we prove that if cosθ\cos\theta is bounded away from zero, where θ\theta is the Ω\Omega-angle of MM, and if MM has zero Cheeger constant, then MM is minimal. In the particular case MM is complete with RicciM0Ricci^M\geq 0 we may replace the boundedness condition on cosθ\cos\theta by cosθCrβ\cos\theta\geq Cr^{-\beta}, when r+r\rightarrow+\infty, where 0<β<10 < \beta < 1 and C>0C > 0 are constants and rr is the distance function to a point in MM. 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\|H\| in terms of cosθ\cos\theta and an isoperimetric inequality. In a similar way, we also give some conditions to conclude MM 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. 651–692

DOI 10.4171/RMI/613