ON NONPOSITIVE CURVATURE FUNCTIONS ON NONCOMPACT SURFACES OF FINITE TOPOLOGICAL TYPE

We study the conformal deformation equation Delta u - 2k(g) + 2Ke(u) = 0 on noncompact surfaces of finite topological type with K nonpositiv e. Sharp existence and nonexistence results are obtained. There are tw o kinds of nonexistence results for this equation with K nonpositive. The first is of a global geometric nature and therefore it should be d etermined by the topology of the surface. We show that it is the Euler characteristic of the surface that determines this kind of nonexisten ce result for the class of noncompact conformal surfaces of finite top ological type with no hyperbolic ends. This also reveals the topologic al significance of the results previously obtained by D. H. Sattinger and W. M. Ni on the Euclidean plane R(2). Its most general form is sta ted in Theorem 0.9. The second kind of nonexistence result is a purely geometric property of the ends. Typical examples are theorems 0.5 and 0.6. Thus if a conformal surface contains a hyperbolic end, topology plays no role in the study of the equation. In other words, if K is a nonpositive function on M, then the above equation has a solution on M if and only if restricted to each end, this equation has a solution.