The local isometric embedding in $\mathbb{R}^3$ of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in \mathbb{R}^3. We prove a general local existence result for a large class of 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 to arbitrary finite order on a single smooth curve.