Asian Journal of Mathematics

Higher Bers maps

Guy Buss

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The Bers embebbing realizes the Teichmüller space of a Fuchsian group G as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for G. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator.

In this paper we prove that a certain class of differential operators acting on functions of the disc induce holomorphic mappings of Teichmüller spaces, and we also obtain a general formula for the differential of the induced mappings at the origin. The main focus of this work, however, is on two particular series of such mappings, dubbed higher Bers maps, because they are induced by so-called higher Schwarzians – generalizations of the classical Schwarzian operator. For these maps, we prove several further results.

The last section contains a discussion of possible applications, open questions and speculations.

Article information

Asian J. Math., Volume 16, Number 1 (2012), 103-140.

First available in Project Euclid: 13 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15] 30C55: General theory of univalent and multivalent functions 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

Teichmüller spaces quasiconformal mappings higher Schwarzians univalent functions


Buss, Guy. Higher Bers maps. Asian J. Math. 16 (2012), no. 1, 103--140.

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