Surfaces of Prescribed Mean Curvature in Quasi-Fuchsian Manifolds

TL;DR

This paper constructs closed incompressible surfaces with prescribed mean curvature in quasi-Fuchsian 3-manifolds, extending the understanding of geometric structures and curvature properties in these complex spaces.

Contribution

It introduces a method to produce surfaces with specific mean curvature functions in quasi-Fuchsian manifolds, including constant mean curvature surfaces within a certain range.

Findings

Existence of embedded surfaces with prescribed mean curvature in quasi-Fuchsian manifolds.

Construction of constant mean curvature surfaces with values in (-2,2).

Extension of geometric analysis in 3-manifold theory.

Abstract

Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed incompressible surface with mean curvature $H$ . A direct application is the existence of embedded surfaces of prescribed constant mean curvatures with constants in $(-2,2)$.