Book Conformal Deformation Of The Gaussian Curvature And Is The Whole Curvature Of The Hyperbolic Plane H 2 (-1)

Given a hyperbolic plane H2(-1) = (B,g),whose sectional curvature k is function K is the Gaussian curvature of the conformal metric g = e2ug ,then K satisfies the conformal Gauss curvature equation . On the other hand,the conformal deformation's problem is to find a metric on H2(-1), conformal to g,with the given function K as its Gaussian curvature ,that is,it is important for us to study the solvability of the conformal Gauss curvature equation in geometry analysis.The problem that the conformal Gauss curvature equation may have a solution for every nonegative Holder continuous function K(x) is also an open problem.In this paper,we introduce the weighted Sobolev space on H2(-1) where we study the solvability of the conformal Gauss curvature equation, we get the following main conclusion:Theorem: If K is a Holder continuous function which is positive somewhere on H2(-1) and for some constants E > 0, C > 0,such that,where r(x) = dist(O,x) is a distance function, then for every we can find a C2-function u which satisfies that the Gaussian curvature of conformal metric g = e2ug is K(x) and that the total curvature of conformal metric .