Abstract
We consider a singular holomorphic vector field in a neighborhood of 0 ∈ C 3 and suppose that there is a singular holomorphic foliation of codimension one (outside its singular set, given by a holomorphic decomposition of this neighborhood into complex surfaces, called leaves) to which it is tangent. This means that, when both objects are non-singular, the orbits of the vector field are contained in the leaves of the foliation. First we consider the desingularizations of both objects, trying to relate their final models. Then we analyse the situation where the vector field is tangent to three independent foliations.
Highlights
The last decades have seen a steadily growing interest in the study of a priori estimates for the height function of constant mean curvature compact graphs or, more generally, compact hypersurfaces with boundary having some constant higher order mean curvature
More than twenty years after that, Korevaar et al 1992 obtained a sharp bound for compact graphs and for compact embedded hypersurfaces in the hyperbolic space Hn+1 with nonzero constant mean curvature and boundary in a horosphere
Given an arbitrary Riemannian manifold Mn, height estimates in the product space R × Mn for constant mean curvature compact embedded hypersurfaces with boundary in a slice were exhibited by Hoffman et al 2006 and Aledo et al 2008, for n = 2, and by Alías & Dajczer 2007, for an arbitrary dimension n
Summary
The last decades have seen a steadily growing interest in the study of a priori estimates for the height function of constant mean curvature compact graphs or, more generally, compact hypersurfaces with boundary having some constant higher order mean curvature. Given an arbitrary Riemannian manifold Mn, height estimates in the product space R × Mn for constant mean curvature compact embedded hypersurfaces with boundary in a slice were exhibited by Hoffman et al 2006 and Aledo et al 2008, for n = 2, and by Alías & Dajczer 2007, for an arbitrary dimension n. Cheng & Rosenberg 2005 were able to generalize these estimates for compact graphs with some constant higher order mean curvature in the product manifold R × Mn, with boundary in a slice.
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