On the asymptotics for minimizers of Donaldson functional in Teichmüller theory

Abstract

We discuss the asymptotic behavior of minimizers for a Donaldson functional of interest in Teichmüller theory. For example, such minimizers allow one to parametrize the moduli space of constant mean curvature immersions of a closed surface $S$ of genus $\mathfrak{g}\ge 2$ into a $3$-manifold with sectional curvature $-1$, by elements of the tangent bundle of the Teichmüller space of $S$. The minimizers are governed by a system of PDEs which include a Gauss equation of Liouville type and a holomorphic $\kappa$-differential.

In our asymptotic analysis, we face the difficulty to describe the possible blow-up behavior of minimizers, especially when it occurs at a point where different zeroes of the holomorphic $\kappa$-differential coalesce. Therefore, we need to pursue accurate estimates of the blow-up profile of solutions for Liouville type equations, in the “collapsing” case.