  We consider two natural problems arising in geometry which are equivalent to
the local solvability of specific equations of Monge-Ampere type. These are:
the problem of locally prescribed Gaussian curvature for surfaces in R^3, and
the local isometric embedding problem for two-dimensional Riemannian manifolds.
We prove a general local existence result for a large class of Monge-Ampere
equations in the plane, and obtain as corollaries the existence of regular
solutions to both problems, in the case that the Gaussian curvature possesses a
nondegenerate critical point.
