Cyclic Higgs bundles, subharmonic functions, and the Dirichlet problem

TL;DR

This paper proves the existence and uniqueness of solutions to a generalized Dirichlet problem for cyclic Higgs bundles, using subharmonic functions to handle lower regularity coefficients.

Contribution

It introduces a new formulation of Hitchin's equations with less regular coefficients, extending the theory to more general cyclic Higgs bundles.

Findings

Existence of solutions established

Uniqueness of solutions proved

Generalization to less regular coefficients achieved

Abstract

We demonstrate the existence and uniqueness of the solution to the Dirichlet problem for a generalization of Hitchin's equation for diagonal harmonic metrics on cyclic Higgs bundles. The generalized equations are formulated using subharmonic functions. In this generalization, the coefficient exhibits worse regularity than that in the original equation.