Ebook: Handbook of Teichmüller Theory, Volume VII

The present volume of the Handbook of Teichmüller theory is divided into three parts.

The first part contains surveys on various topics in Teichmüller theory, including the complex structure of Teichmüller space, the Deligne–Mumford compactification of the moduli space, holomorphic quadratic differentials, Kleinian groups, hyperbolic 3-manifolds and the ending lamination theorem, the universal Teichmüller space, barycentric extensions of maps of the circle, and the theory of Higgs bundles.

The second part consists of three historico-geometrical articles on Tissot (a precursor of the theory of quasiconfomal mappings), Grötzsch and Lavrentieff, the two main founders of the modern theory of quasiconformal mappings.

The third part comprises English translations of five papers by Grötzsch, a paper by Lavrentieff, and three papers by Teichmüller. These nine papers are foundational essays on the theories of conformal invariants and quasiconformal mappings, with applications to conformal geometry, to the type problem and to Nevanlinna's theory. The papers are followed by commentaries that highlight the relations between them and between later works on the subject. These papers are not only historical documents; they constitute an invaluable source of ideas for current research in Teichmüller theory.

Keywords: Riemann surface, Teichmüller space, Deligne–Mumford compactification, universal Teichmüller space, complex geodesic, holomorphic differential, quadratic differential, projective structure, Mostow rigidity, hyperbolic structure, Fuchsian group, quasi-Fuchsian group, Kleinian group, ending lamination, Higgs bundle, higher Teichmüller theory, Douady-Earle extension, quasisymmetric map, quadiconformal mapping, type problem, conformal invariant, extremal length, extremal domain, Tissot indicatrix, almost analytic function, measurable Riemann Mapping Theorem, value distribution, Modulsatz, reduced module, line complex, Speiser tree