Higher Bers maps

TL;DR

This paper introduces higher Bers maps, generalizations of classical Schwarzian operators, which induce holomorphic mappings on Teichmüller spaces, expanding the understanding of their structure and potential applications.

Contribution

It proves that certain differential operators induce holomorphic maps on Teichmüller spaces and develops the theory of higher Bers maps based on higher Schwarzians.

Findings

Higher Bers maps are holomorphic functions on Teichmüller spaces.

Derived a formula for the differential of these maps at the origin.

Discussed potential applications and open questions in the field.

Abstract

The Bers embebbing realizes the Teichm\"uller 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 Teichm\"uller 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 Teichm\"uller 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.