Toeplitz operators on Bergman spaces with locally integrable symbols

Abstract

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 $Ricc{i}^M\ge 0$ we may replace the boundedness condition on $\cos \theta$ by $\cos \theta \ge C{r}^{-\beta }$, when $r\to +\infty$, 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 $\Vert H\Vert$ 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.