Vortex-type equations on compact Riemann surfaces

TL;DR

This paper establishes a priori estimates for vortex-type equations on compact Riemann surfaces, leading to existence, uniqueness results, and stability correspondences in complex geometry.

Contribution

It introduces new a priori estimates for vortex equations, proves existence and uniqueness theorems, and links stability conditions with Einstein metrics on vortex bundles.

Findings

Recovered existing estimates for vortex bundle Monge-Ampère equations

Proved an existence and uniqueness theorem for Calabi-Yang-Mills equations

Established a Kobayashi-Hitchin type correspondence for Gieseker stability

Abstract

In this paper, we prove \emph{a priori} estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Amp\`ere equation, prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles, and get estimates for $J-$vortex equation. We prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove K\"ahlerness of the negative of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations in \cite{Vamsi3}