INFINITE ENERGY EQUIVARIANT HARMONIC MAPS, DOMINATION, AND ANTI-DE SITTER 3-MANIFOLDS

SAGMAN, Nathaniel

doi:10.4310/jdg/1689262064

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INFINITE ENERGY EQUIVARIANT HARMONIC MAPS, DOMINATION, AND ANTI-DE SITTER 3-MANIFOLDS

SAGMAN, Nathaniel

2023 • In Journal of Differential Geometry, 124 (3), p. 553 - 598

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https://hdl.handle.net/10993/58168

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10.4310/jdg/1689262064

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Keywords :

Analysis; Algebra and Number Theory; Geometry and Topology

Abstract :

[en] We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT(−1) Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichmüller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter 3-manifolds. We also present an application to the theory of maximal immersions into the Grassmanian of timelike planes in R2,2

Disciplines :

Mathematics

Author, co-author :

SAGMAN, Nathaniel ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Mathematics Dept., Pasadena, United States

External co-authors :

Language :

English

Title :

INFINITE ENERGY EQUIVARIANT HARMONIC MAPS, DOMINATION, AND ANTI-DE SITTER 3-MANIFOLDS

Publication date :

July 2023

Journal title :

Journal of Differential Geometry

ISSN :

0022-040X

eISSN :

1945-743X

Publisher :

International Press, Inc.

Volume :

124

Issue :

Pages :

553 - 598

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBilu :

since 29 November 2023

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