High energy harmonic maps and degeneration of minimal surfaces

Abstract

Let $S$ be a closed surface of genus $g \geq 2$ and let $\rho$ be a maximal $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $\rho$-equivariant minimal surface $\widetilde{\Sigma}$ in $\mathbb{H}^{2} \times \mathbb{H}^{2}$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $\widetilde{\Sigma}$ degenerate to the core of a product of two $\mathbb{R}$-trees. As a consequence, we obtain a compactification of the space of maximal representations of $\pi_{1}(S)$ into $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$.