Donaldson Functional in Teichm\"uller Theory

TL;DR

This paper introduces a Donaldson-type functional related to Hitchin's self-duality equations on Riemann surfaces, overcoming analytical challenges to establish existence and uniqueness of solutions, with applications to geometric parametrizations.

Contribution

It defines a new Donaldson functional with a variational approach, proving existence and uniqueness of solutions to Hitchin's equations on closed Riemann surfaces.

Findings

Unique critical point corresponding to the global minimum

Existence and uniqueness of solutions to Hitchin's self-duality equations

Parametrization of constant mean curvature immersions in hyperbolic manifolds

Abstract

In this paper we define a Donaldson type functional whose Euler-Lagrange equations are a system of differential equations which corresponds to Hitchin's self-duality equations for a suitable choice of Higgs bundle on closed Riemann surfaces. The main challenge of this functional is its lack of regularity and lack of compactness when defined in its natural domain of definition. Though a standard variational approach cannot directly be applied, we provide the appropriate analytical tools that make Donaldson functional treatable by a variational viewpoint. We prove that this functional admits a unique critical point corresponding to its global minimum. As an immediate consequence, we find that this system of self-duality equations admits a unique solution. Among the applications in geometry of this fact, we obtain a parametrization of closed constant mean curvature immersions in hyperbolic manifolds (possibly incomplete), and their moduli spaces.