An algebraic construction of a solution to the mean field equations on hyperelliptic Curves and its diabatic limit
In this paper, we give an algebraic construction of the solution to the following mean field equation $$ Δψ+e^ψ=4π\sum_{i=1}^{2g+2}δ_{P_{i}}, $$ 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 $$ Δψ+γe^ψ=4πγ\sum_{i=1}^{2g+2}δ_{P_{i}} $$ and its adiabatic limit at $γ=0$.
math.DG↗