# Adiabatic limits and Kazdan-Warner equations

## TL;DR

This paper investigates the behavior of solutions to abelian vortex equations on Riemann surfaces as their volume increases, establishing convergence properties and connecting to related geometric PDEs.

## Contribution

It provides new a priori estimates for generalized Kazdan-Warner equations and links the asymptotic vortex behavior to existing results in gauge theory.

## Key findings

- Solutions converge smoothly away from finitely many points as volume grows
- Established a priori estimates for generalized Kazdan-Warner equations
- Connected vortex solutions to Seiberg-Witten equations with multiple spinors

## Abstract

We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan-Warner equations. We relate our results to the work of Hong, Jost, and Struwe on classical vortices, and that of Haydys and Walpuski on the Seiberg-Witten equations with multiple spinors.