On a convexity property of the space of almost fuchsian immersions

TL;DR

This paper investigates the convexity properties of the space of Hopf differentials associated with almost Fuchsian minimal immersions, revealing that the set of these differentials forms a convex subset and establishing bounds in the non-equivariant case.

Contribution

It demonstrates the convexity of the set of Hopf differentials for almost Fuchsian immersions and provides bounds in the non-equivariant scenario.

Findings

The extrinsic curvature is a concave function of the Hopf differential.

The set of Hopf differentials is convex within the space of holomorphic quadratic differentials.

Bounds are established for the size of this set in the non-equivariant case.

Abstract

We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set.