High energy harmonic maps and degeneration of minimal surfaces

Abstract

Let $S$ be a closed surface of genus $g\ge 2$ 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 $\tilde{\Sigma }$ 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 $\tilde{\Sigma }$ 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 $\mathrm{PSL}(2,\mathbb{R})\times \mathrm{PSL}(2,\mathbb{R})$.

38 pages