Teichmüller Spaces of Riemann Surfaces

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 9 in Quasiconformal Mappings in the Plane and Complex Dynamics, 2025, pp 315-345 from Springer

Abstract: Abstract Extension of the study of Teichmüller spaces initiated in Chapter 4 and continued in Chapter 8. Quasiconformal mappings between Riemann surfaces. Riemann and Teichmüller spaces of a Riemann surface, the latter as by Bers in 1965. Modular group of a Riemann surface. Differential, higher order differential and Beltrami differential in a Riemann surface. Conformal structures of a Riemann surface as given by its Teichmüller space. Universal Teichmüller space, that of the open upper complex half plane, which includes the Teichmüller spaces of Riemann surfaces covered by that half plane, and six different descriptions of the space. Geodesics in Teichmüller space of a Riemann surface covered by the open upper complex halfplane. Generalization of the Bers embedding to Teichmüller spaces of such Riemann surfaces. Foliation of one-dimensional complex manifold. Teichmüller theorem on existence of unique extremal quasiconformal mapping in each homotopy class of orientation preserving homeomorphisms of a finite type Riemann surface. Kobayashi metric (that of the largest distance with the norm of the derivative at 0 of holomorphic functions from the unit disk centered at 0 to the manifold upper bounded by 1) and Royden theorem of 1971 establishing that Kobayashi and Teichmüller metrics are equal.

Date: 2025
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DOI: 10.1007/978-3-031-80115-0_9

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