High-energy harmonic maps and degeneration of minimal surfaces

Ouyang, Charles

doi:10.2140/gt.2023.27.1691

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High-energy harmonic maps and degeneration of minimal surfaces

Charles Ouyang

Geometry & Topology 27 (2023) 1691–1746

DOI: 10.2140/gt.2023.27.1691

Abstract

Let $S$ be a closed surface of genus $g \geq 2$ and $ρ$ a maximal $PSL (2, ℝ) \times PSL (2, ℝ)$ surface group representation. By a result of Schoen, there is a unique $ρ$ –equivariant minimal surface $\tilde{Σ}$ 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 ${AdS}^{3}$ . In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $\tilde{Σ}$ 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 $π_{1} (S)$ into $PSL (2, ℝ) \times PSL (2, ℝ)$ .

Keywords

minimal lagrangian, harmonic maps, mixed structures, higher Teichmüller space

Mathematical Subject Classification 2010

Primary: 49Q05, 53C43

References

Publication

Received: 6 January 2020

Revised: 25 July 2021

Accepted: 22 December 2021

Published: 27 July 2023

Proposed: Anna Wienhard

Seconded: Tobias H Colding, Mladen Bestvina

Authors

Charles Ouyang
Department of Mathematics Washington University St Louis, MO United States

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Volume 27, issue 5 (2023)