Local solvability of degenerate Monge-Ampère equations and applications to geometry

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in ℝ³, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes and possesses a nonvanishing Hessian matrix at a critical point.