High energy harmonic maps and degeneration of minimal surfaces

TL;DR

This paper investigates the degeneration of maximal surface group representations into 4444, showing how associated minimal surfaces degenerate to mixed structures and 4-trees, leading to a compactification of the representation space.

Contribution

It establishes the limits of 4444-maximal surfaces as mixed structures and interprets their degeneration to 4-trees, providing a new compactification of the representation space.

Findings

Limits of minimal surfaces are mixed structures.

Degeneration to the core of 4-trees.

Compactification of the space of maximal representations.

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 $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$.