Higher Bers maps

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.

Keywords

Teichmüller spaces, quasiconformal mappings, higher Schwarzians, univalent functions

2010 Mathematics Subject Classification

Primary 30C55, 30F60, 32H02. Secondary 46G20.

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Published 8 March 2012