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High-energy harmonic maps and degeneration of minimal surfaces

Charles Ouyang

Geometry & Topology 27 (2023) 1691–1746
Abstract

Let S be a closed surface of genus g 2 and ρ a maximal PSL (2, ) × PSL (2, ) surface group representation. By a result of Schoen, there is a unique ρ–equivariant minimal surface Σ~ in 2 × 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 Σ~ degenerate to the core of a product of two –trees. As a consequence, we obtain a compactification of the space of maximal representations of π1(S) into PSL (2, ) × PSL (2, ).

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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