# Solutions to Donaldson's hyperk\"ahler reduction on a curve

## TL;DR

This paper extends the understanding of Donaldson's hyperk"ahler reduction on a Riemann surface by establishing broader existence results, thereby enlarging the associated hyperk"ahler moduli space.

## Contribution

It provides a more general existence theorem for solutions to the hyperk"ahler reduction equations, expanding the known moduli space beyond special solutions.

## Key findings

- Existence of more general solutions to the hyperk"ahler equations.
- Construction of a larger hyperk"ahler moduli space.
- Extension of solutions beyond those derived from holomorphic quadratic differentials.

## Abstract

We study an infinite-dimensional hyperk\"ahler reduction introduced by Donaldson and associated with the constant scalar curvature equation on a Riemann surface. It is known that the corresponding moment map equations admit special solutions constructed from holomorphic quadratic differentials. Here we obtain a more general existence result and so a larger hyperk\"ahler moduli space.