On the local isometric embedding in \(\mathbb R^3\) of surfaces with Gaussian curvature of mixed sign (Q641521)

The local isometric embedding of a two-dimensional Riemannian metric with positive Gaussian curvature \(K\) into Euclidean 3-space was often treated. The present authors include a case where \(K\) changes sigm. They assume that the regions of positive or negative \(K\) are separated by two curves intersecting transversally. A version of the Nash-Moser iteration is used to establish the existence of local solutions for the elliptic and hyperbolic regions separately, and it is shown how these solutions can be patched together.