Prescribing Gaussian curvature on $R^2$

Abstract

We derive a sufficient condition for a radially symmetric function K(x) which is positive somewhere to be a conformal curvature on R2. In particular, we show that every nonnegative radially symmetric continuous function K(x) on R2 is a conformal curvature. In this paper, we consider the prescribing Gaussian curvature problem. Let (M, g) be a Riemannian manifold of dimension 2 with Gaussian curvature k. Given a function K on M , one may ask the following question: Can we find a new conformal metric g1 on M (i.e., there exists u on M such that g1 = e g) such that K is the Gaussian curvature of g1? This is equivalent to the problem of solving the elliptic equation ∆u − k +Ke = 0 (0) on M , where ∆ is the Laplacian of (M, g). This problem has been considered by many authors. In case M is compact, we refer to [6] for details and references. In case M = R, equation (0) becomes ∆u+K(x)e = 0 (1) and this problem is well understood if K(x) is nonpositive; in particular, if |K(x)| decays slower than |X |−2 at infinity, then equation (1) has no solution (see [11], [13]). However, if K(x) is positive at some point, the situation is totally different. If K(x0) > 0 for some x0 ∈ R, R. C. McOwen [10] proved that, for K(x) = O(r−l) as r → ∞, equation (1) has a C solution, where l is a positive constant. Also, it is not difficult to see that equation (1) has solutions for every positive constant K(x) = C. Since there is no known nonexistence result for K ≥ 0 on R, one may propose the following Problem 1. Is it true that every nonnegative function (smooth enough) on R is a conformal Gaussian curvature function? Received by the editors May 10, 1996. 1991 Mathematics Subject Classification. Primary 58G30; Secondary 53C21.