An algebraic construction of a solution to the mean field equations on hyperelliptic Curves and its diabatic limit

Abstract

In this paper, we give an algebraic construction of the solution to the following mean field equation $$ \Delta \psi+e^{\psi}=4\pi\sum_{i=1}^{2g+2}\delta_{P_{i}}, $$ on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$ Furthermore, we study the rescaled equation $$ \Delta \psi+\gamma e^{\psi}=4\pi\gamma \sum_{i=1}^{2g+2}\delta_{P_{i}} $$ and its adiabatic limit at $\gamma=0$.