Curvature and Uniformization

Details Curvature and Uniformization Curvature and Uniformization

2001-05-02

arxiv.org/abs/math/0105016v1

Mathematics

Differential Geometry

26 pages

Scientific paper

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincar\'e metrics (i.e., complete metrics of constant negative curvature) by solving the equation $\Delta u - e^{2u} = K_0(z)$ on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factor $e^{2u}$ giving the Poincar\'e metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.

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