Isometric submersions of Teichmüller spaces are forgetful (original) (raw)
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Teichmüller spaces and holomorphic dynamics
Handbook of Teichmüller Theory, Volume IV, 2014
One fundamental theorem in the theory of holomorphic dynamics is Thurston's topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston's theorem (the marked Thurston's theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.
Isomorphisms between generalized Teichmüller spaces
Contemporary mathematics, 1999
Let X and Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of X onto the Teichmüller space of Y is induced by a quasiconformal homeomorphism of X onto Y. These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.
Geometric isomorphisms between infinite dimensional Teichmüller spaces
TRANSACTIONS-AMERICAN MATHEMATICAL …, 1996
Let X and Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of X onto the Teichmüller space of Y is induced by a quasiconformal homeomorphism of X onto Y . These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.
Liftings of holomorphic maps into Teichmüller spaces
Kodai Mathematical Journal, 2009
We study liftings of holomorphic maps into some Teichmü ller spaces. We also study the relationship between universal holomorphic motions and holomorphic lifts into Teichmü ller spaces of closed sets in the Riemann sphere.
Holomorphic Motions and Teichmuller Spaces
Transactions of the American Mathematical Society, 1994
We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane C extends to a holomorphic motion of C. As a consequence we prove that every holomorphic map of the unit disc into Teichmüller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmüller metric.
On dynamical Teichmüller spaces
Conformal Geometry and Dynamics of the American Mathematical Society, 2010
Following ideas from a preprint of the second author, see [2], we investigate relations of dynamical Teichmüller spaces with dynamical objects. We also establish some connections with the theory of deformations of inverse limits and laminations in holomorphic dynamics, see .
TEICHMULLER SPACES AND HOLOMORPHIC MOTIONS
The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmiiller space of a closed subset of the Riemann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the "Harmonic A-Lemma" of Bers and Royden. We also study some other applications.
Inventiones mathematicae
We prove the holomorphic rigidity conjecture of Teichmüller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmüller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth n-dimensional Riemannian domain to the Weil-Petersson completion T of Teichmüller space has Hausdorff dimension at most n − 2, and moreover, u has certain decay near the singular set. Combining this with the earlier work of Schumacher, Siu and Jost-Yau, we provide a proof of the holomorphic rigidity of Teichmüller space. In addition, our results provide as a byproduct a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by Farb-Masur and Yeung. We will derive Theorem 1.1 as a consequence of the following more general holomorphic rigidity Theorem and its Corollary. Theorem 1.2. Let M be a complete, finite volume Kähler manifold with universal coverM and π 1 (M) finitely generated. Let Γ be the mapping class group of an oriented surface S of genus g and p marked points such that k = 3g − 3 + p > 0, T the Weil-Petersson completion of the Teichmüller space T of S and ρ : π 1 (M) → Γ a homomorphism. If there exists a finite energy ρ-equivariant harmonic map u :M → 1
Harmonic Maps to Teichmüller Space
Mathematical Research Letters, 2000
We give sufficient conditions for the existence of equivariant harmonic maps from the universal cover of a Riemann surface B to the Teichmüller space of a genus g ≥ 2 surface Σ. The condition is in terms of the representation of the fundamental group of B to the mapping class group of Σ. The metric on Teichmüller space is chosen to be the Kähler hyperbolic metric. Examples of such representations arise from symplectic Lefschetz fibrations.