Generalizations of Hermitian-Einstein equation of cyclic Higgs bundles, their heat equation, and inequality estimates

TL;DR

This paper extends the Hermitian-Einstein equation for cyclic Higgs bundles, introduces a heat equation approach, and generalizes inequality estimates, providing existence, uniqueness, and convergence results.

Contribution

It presents new generalizations of the Hermitian-Einstein equation and heat equations for cyclic Higgs bundles, including cases with subharmonic functions, and extends inequality estimates.

Findings

Proved existence and uniqueness of solutions to the heat equations with Dirichlet boundary conditions.

Established convergence of solutions under smooth coefficient conditions.

Generalized inequality estimates for solutions of the Hermitian-Einstein equation.

Abstract

We introduce some generalizations of the Hermitian-Einstein equation for diagonal harmonic metrics on cyclic Higgs bundles, including a generalization using subharmonic functions. When the coefficients are all smooth, we prove the existence, uniqueness, and convergence of the solution of their heat equations with Dirichlet boundary conditions. We also generalize two inequality estimates for solutions of the Hermitian-Einstein equation for cyclic Higgs bundles.