CONFORMAL METRICS WITH PRESCRIBED GAUSSIAN CURVATURE ON S2

We consider on S2 the problem of which functions K can be the scalar curvature of a metric conformal to the standard metric on S2. We assume that K is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on k (which are easy to check), in order that the problem be solvable.