High-energy harmonic maps and degeneration of minimal surfaces

Ouyang, Charles¹

¹ Department of Mathematics, Washington University, St Louis, MO, United States

Geometry & topology, Tome 27 (2023) no. 5, pp. 1691-1746.

Résumé

Let $S$ be a closed surface of genus $g\ge 2$ and $\rho$ a maximal $\mathrm{PSL}(2,ℝ)\times \mathrm{PSL}(2,ℝ)$ surface group representation. By a result of Schoen, there is a unique $\rho$–equivariant minimal surface $\tilde{\mathrm{\Sigma }}$ in ${ℍ}^2\times {ℍ}^2$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. We prove a similar result for maximal surfaces in ${\mathrm{AdS}}^3$. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $\tilde{\mathrm{\Sigma }}$ degenerate to the core of a product of two $ℝ$–trees. As a consequence, we obtain a compactification of the space of maximal representations of ${\pi }_1(S)$ into $\mathrm{PSL}(2,ℝ)\times \mathrm{PSL}(2,ℝ)$.

Détail

DOI : 10.2140/gt.2023.27.1691

Classification : 49Q05, 53C43
Keywords: minimal lagrangian, harmonic maps, mixed structures, higher Teichmüller space

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