JournalsrmiVol. 26, No. 2pp. 693–706

Toeplitz operators on Bergman spaces with locally integrable symbols

  • Jari Taskinen

    University of Helsinki, Finland
  • Jani Virtanen

    University of Helsinki, Finland
Toeplitz operators on Bergman spaces with locally integrable symbols cover
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Abstract

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

Jari Taskinen, Jani Virtanen, Toeplitz operators on Bergman spaces with locally integrable symbols. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 693–706

DOI 10.4171/RMI/614