Mathematics Colloquia and Seminars

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A Riemann surface is locally modeled in the complex plainC,and transition maps are holomorphic. On the other hand, a surface S with a complex projective structure is modeled in the Riemann sphere with transition maps in PSL(2, C). Equivalently, we can define a complex projective structure on a surface as a pair of a map from the universal cover of S to the Riemann and the representation from pi_1(S) to PSL(2, C) which is equivariant. This representation is usually non-discrete. One aspect of complex projective structures is geometrization of non-discrete representations, which is a generalization of the relation between discrete representations from pi_1(S) to PSL(2,R) and 2-dimensional hyperbolic surfaces.

I will present an overview of complex projective structures including the condition for a representation to be induced from a complex projective structure on a closed surface. The main reference of this talk will be Daniel Gallo, Michael Kapovich, and Albert Marden, The monodromy groups of Schwarzian equation on closed Riemann surfaces (Ann. of Math. (2), 2000)