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 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.