The objective of this class is to develop the tools to study geometric properties of the Weil–Petersson metric in Universal Teichmüller space T(1) [1]. Universal Teichmüller space is a Banch complex manifold that naturally contains any Teichmüller space given by a group action, as well as having interpretations in terms of univalent maps and diffeomorphisms of the circle. The Weil-Petersson metric is a natural Hermitian metric defined on a Hilbert structure of T(1). With this metric the space becomes Kähler–Einstein of negative curvature. Previous background in differential geometry, complex analysis and differential topology is recommended.

Referencias:

[1] Teo, Lee-Peng and Takhtajan, Leon A., Weil-Petersson Metric on the Universal Teichmüller Space Mem. Amer. Math. Soc. 183 (2006), no. 861, viii+119 pp.
[2] Ahlfors, Lars V., Lectures on quasiconformal mappings. Univ. Lecture Ser., 38 American Mathematical Society, Providence, RI, 2006. viii+162 pp.
[3] Lehto, Olli, Univalent functions and Teichmüller spaces. Grad. Texts in Math., 109 Springer-Verlag, New York, 1987. xii+257 pp.
[4] Wolpert, Scott A., Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85 (1986), no. 1, 119–145.
[5] Hubbard, John Hamal, Teichmüller theory and applications to geometry, topology, and dy-namics. Vol. 1.