A note On the existence of solutions to Hitchin's self-duality equations

TL;DR

This paper clarifies and completes Hitchin's original proof of the existence of solutions to the self-duality equations on rank-2 bundles over Riemann surfaces, by providing detailed steps involving gauge transformations and energy minimization.

Contribution

It offers detailed technical steps to establish the existence of solutions to Hitchin's self-duality equations, filling gaps in the original proof.

Findings

Existence of $L_1^2$ solutions via energy minimization.

Construction of smooth solutions using Coulomb gauge and gauge transformations.

Technical clarification of Hitchin's original proof.

Abstract

In 1987, Hitchin introduced the self-duality equations on rank-2 complex vector bundles over compact Riemann surfaces with genus greater than one as a reduction of the Yang-Mills equation and established the existence of solutions to these equations starting from a Higgs stable bundle. In this paper, we fill in some technical details in Hitchin's original proof by the following three steps. First, we reduce the existence of a solution of class $L_1^2$ to minimizing the energy functional within a Higgs stable orbit of the $L_2^2$ complex gauge group action. Second, using this transformation, we obtain a solution of class $L_1^2$ in this orbit. These two steps primarily follow Hitchin's original approach. Finally, using the Coulomb gauge, we construct a smooth solution by applying an $L_2^2$ unitary gauge transformation to the $L_1^2$ solution constructed previously. This last step provides additional technical details to Hitchin's original proof.