Prescribed Gaussian Curvature On Surfaces

In this thesis,we consider the prescribed Gaussian curvature problem on 2-dimensional compact surfaces without boundary.Specifically speaking,the problem of prescribing Gauss curve is the following:Let(M,g0)be a compact surface without boundary.Assume that f is a smooth function defined on M.We,then,ask if one can find a metric g,conformal related to g0(i.e.g=e2u·g0 for some smooth function u)such that the Gaussian curvature of the metric g,defined by Kg,satisfies Kg=f?In the present thesis,we will use the method of conformal Gauss curvature flow to study this problem.Consider a family of time-dependent metrics g(t)satisfying the evolution equation(?)=-2(K-?(t)·f)·g.where ?(t)is a function that depends only on time parameter t.Under a proper assumption on/,we are able to prove the short-time existence,long-time existence and convergence of the flow.Furthermore,as time goes to infinity,the limit metric g? will be the solution of the prescribed Gaussian curvature problem.