Handbook of Teichmuller Theory, Volume I (Irma Lectures in by Athanase Papadopoulos

By Athanase Papadopoulos

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34 37 37 38 39 40 42 44 46 47 49 51 53 54 54 57 59 63 63 63 66 69 69 72 74 77 77 78 81 83 85 85 34 Georgios D. Daskalopoulos and Richard A. 2 Properness of the energy . . . . . . 3 Convexity of energy and length functionals . 4 Further applications . . . . . . . . 5 Harmonic maps to Teichmüller space . . .

86 87 88 90 90 90 92 93 95 95 96 99 1 Introduction Teichmüller theory is rich in applications to topology and physics. By way of the mapping class group the subject is closely related to knot theory and three-manifolds. From the uniformization theorem, Teichmüller theory is part of the more general study of Kleinian groups and character varieties. Conformal field theory and quantum cohomology make use of the algebraic and geometric properties of the Riemann moduli space.

Georgios D. Daskalopoulos was partially supported by NSF Grant DMS-0204191. Richard A. Wentworth was partially supported by NSF Grants DMS-0204496 and DMS-0505512. Notation For simplicity, we shall deal with connected compact oriented surfaces without boundary and of genus p ≥ 2. The notation we shall use is the following: S will denote the underlying smooth surface, and j will denote a complex structure on S. Hence, a Riemann surface is a pair (S, j ). The hyperbolic metric on S will be denoted by σ .

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