Locally univalent functions, VMOA and the Dirichlet space

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmüller space of such Riemann surfaces and demonstrate that in the case that 2g + 2 − n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic.We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of nonoverlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.2010 Mathematics Subject Classification. Primary 30F60 ; Secondary 30C55, 30C62, 32G15, 46E20, 81T40. Key words and phrases. Refined Teichmüller space, Hilbert manifold, quasiconformal maps, moduli space of rigged Riemann surfaces, conformal field theory.Eric Schippers is partially supported by the National Sciences and Engineering Research Council. He would like to thank Nina Zorboska for several helpful conversations.

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“…However, we need a uniform estimate in t. Although this does not follow from the theorem as stated in[12, Theorem 1 …”

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confidence: 99%

A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

Radnell

Schippers

Staubach

2015

Commun. Contemp. Math.

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“…The following lemma will be needed to prove Theorem 1. We refer the reader to [5, Lemmas 2.2 and 3.3] (see also [8]) for the details of the proof. (1 − |z| 2 ) < α .…”

Section: A Criterion For Bounded Valence Of Harmonic Mappings In Termmentioning

confidence: 99%

Criteria for Bounded Valence of Harmonic Mappings

Huusko

Martín

2017

Comput. Methods Funct. Theory

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“…In some aspects, the present paper is a continuation of the paper [4]. Section 3 revolves around the well-known expression What can we derive from the knowledge of bounds for this expression?…”

Section: Introductionmentioning

confidence: 97%