Minimizing immersions of a hyperbolic surface in a hyperbolic $3$-manifold

TL;DR

This paper studies energy-minimizing maps from hyperbolic surfaces to hyperbolic 3-manifolds, establishing their uniqueness, geometric characterization via holomorphic data, and the complex structure of the moduli space of such maps.

Contribution

It introduces a new class of energy-minimizing maps, proves their uniqueness, and describes their structure using holomorphic data, connecting geometric analysis with complex geometry.

Findings

Uniqueness of smooth minimizing maps in each homotopy class.

Characterization of maps via holomorphic Codazzi tensors.

The moduli space of data has a natural complex structure.

Abstract

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps from $(S,h)$ are described by a simple holomorphic data on $S$: a complex self-adjoint Codazzi tensor of determinant $1$. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the holonomy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of $F$ analoguous to the complex length.