On the Singular Set of Harmonic Maps into DM-Complexes
type: Item
instance of: scholarly article
title: On the Singular Set of Harmonic Maps into DM-Complexes
title: Rigidity of Teichmüller space
zbMATH Open document ID: 1331.53093
zbMATH Open document ID: 1470.32032
author: Chikako Mese
author: Georgios D. Daskalopoulos
published in: Inventiones Mathematicae
published in: Memoirs of the American Mathematical Society
publication date: 2015-12-04T00:00:00Z
publication date: 2021-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1502.03367
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 32A50
zbMATH DE Number: 6516128
zbMATH DE Number: 7355944
zbMATH Keywords: harmonic maps
zbMATH Keywords: NPC spaces
zbMATH Keywords: Riemannian DM-complexes
zbMATH Keywords: singular sets
zbMATH Keywords: Hausdorff codimension
zbMATH Keywords: monotonicity formulas
zbMATH Keywords: Teichmüller space
zbMATH Keywords: holomorphic rigidity
MaRDI profile type: Publication
OpenAlex ID: W2284582699
OpenAlex ID: W3110099823
arXiv ID: 1502.03367
review text: Le résultat central de cet article est la preuve de la rigidité holomorphique de l'espace de Teichmüller. Plus précisément, soit \(\Gamma\) le groupe modulaire d'une surface orientée de genre \(g\geq2\). Si \(\Gamma\) agit sur une variété de Kähler contractile \(\tilde{M}\) tel qu'il existe un sous-groupe \(\Gamma' \subset \Gamma\) d'indice fini vérifiant: \begin{itemize} \item[(i)] \(M:=\tilde{M}/\Gamma' \) est une variété quasiprojective lisse, \item[(ii)]\(M\) admet une compactification \(\bar M\) comme variété algébrique telle que la dimension de \(\bar M \setminus M\) est supérieure ou égale à \(3\), \end{itemize} alors \(\tilde M\) est biholomorphiquement équivariante ou conjuguée biholomorphiquement à l'espace de Teichmüller \(\mathcal{T}_{g}\) de \(S\) où \(\Gamma\) agit sur \(T\) comme le groupe modulaire. Ce résultat est montré en utilisant la théorie des fonctions harmoniques sur la complétion métrique l'espace de Teichmüller avec la métrique de Weil-Petersson. Deux difficultés principales doivent être résolues afin d'obtenir le résultat. La première difficulté est que la métrique de Weil-Petersson n'est pas un produit mais seulement asymptotiquement un produit. La seconde difficulté est le fait que la complétion est non localement compacte au bord. Les techniques sophistiquées utilisées pour surmonter ces difficultés sont excellemment exposées dans une longue introduction.
reviewed by: Quentin Gendron
cites work: The real analytic theory of Teichmüller space
cites work: Some remarks on Teichmüller's space of Riemann surfaces
cites work: 𝑄 valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two
cites work: Compactification of arithmetic quotients of bounded symmetric domains
cites work: Some metric properties of arithmetic quotients o f symmetric spaces and an extension theorem
cites work: On compact, locally symmetric Kähler manifolds
cites work: Archimedean superrigidity and hyperbolic geometry
cites work: Analytic cycles and vector bundles on non-compact algebraic varieties
cites work: Essential regularity of the model space for the Weil-Petersson metric
cites work: $C^1$ estimates for the Weil-Petersson metric
cites work: Superrigidity of hyperbolic buildings
cites work: Classification of Weil-Petersson isometries
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Superrigidity and mapping class groups
cites work: Q5565773
cites work: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension
cites work: Q3834387
cites work: Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one
cites work: Asymptotic flatness of the Weil-Petersson metric on Teichmüller space
cites work: Q4380377
cites work: Universal moduli spaces of Riemann surfaces
cites work: The strong rigidity of locally symmetric complex manifolds of rank one and finite volume
cites work: On the rigidity of certain discrete groups and algebraic varieties
cites work: Sobolev spaces and harmonic maps for metric space targets
cites work: Global existence theorems for harmonic maps to nonlocally compact spaces
cites work: Q5521862
cites work: Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena
cites work: Canonical metrics on the moduli space of Riemann surfaces. I
cites work: Canonical metrics on the moduli space of Riemann surfaces. II
cites work: Good geometry on the curve moduli
cites work: The extension of the Weil-Petersson metric to the boundary of Teichmüller space
cites work: Dynamics on Thurston's sphere of projective measured foliations
cites work: Geometric superrigidity
cites work: Three circle theorems in partial differential equations and applications to improperly posed problems
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Strong rigidity of Q-rank 1 lattices
cites work: The Ahlfors-Schwarz lemma in several complex variables
cites work: A regularity theory for harmonic maps
cites work: On the removal of singularities of analytic sets
cites work: Q3134943
cites work: Boundary regularity of harmonic maps to nonpositively curved metric spaces
cites work: Q4896042
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
cites work: Q3791828

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: On the Singular Set of Harmonic Maps into DM-Complexes

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: Chern forms and the Riemann tensor for the moduli space of curves

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: Noncompleteness of the Weil-Petersson metric for Teichmüller space

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: Geodesic length functions and the Nielsen problem

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: Extension of the Weil-Petersson connection

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: The Riemannian sectional curvature operator of the Weil-Petersson metric and its application

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: On the geometry of Weil-Petersson completion of Teichmüller spaces

On the Singular Set of Harmonic Maps into DM-Complexes (cites work)
cites work: Q4435760

On the Singular Set of Harmonic Maps into DM-Complexes (DOI)
DOI: 10.1090/MEMO/1129

On the Singular Set of Harmonic Maps into DM-Complexes (DOI)
DOI: 10.1007/S00222-020-01020-2