  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.
