Explicit Solutions to the mean field equations on hyperelliptic curves of genus two

TL;DR

This paper derives explicit solutions to mean field equations on genus two hyperelliptic curves by linking the Gaussian curvature function of the canonical metric to the solutions.

Contribution

It provides a novel explicit solution method for mean field equations on complex hyperelliptic curves of genus two based on curvature functions.

Findings

Explicit solutions to mean field equations are obtained.

The Gaussian curvature function uniquely determines solutions.

The approach links geometric properties to differential equations.

Abstract

Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit solution to a mean field equation.