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