On the local isometric embedding in R of surfaces with Gaussian curvature of mixed sign

Does every smooth two-dimensional Riemannian manifold admit a smooth local isometric embedding into R3, or heuristically, can every abstract surface be visualized at least locally? This natural question was first posed in 1873 by Schlaefli [16], and remarkably has remained to a large extent unanswered. It is the purpose of this paper to provide a general sufficient condition under which local embeddings exist. The local isometric embedding problem for surfaces is equivalent to finding local solutions of a particular Monge–Ampère equation, usually referred to as the Darboux equation. The primary difficulty in analyzing this equation arises from the fact that it changes from elliptic to hyperbolic type, whenever the Gaussian curvature of the given metric passes from positive to negative curvature. Consequently, the hypotheses of any result must take into account the manner in which the Gaussian curvature, K, vanishes. The classical results deal with the cases in which the curvature does not vanish, or the metric is analytic. It was not until 1985/1986 that the first degenerate cases (whenK vanishes) were treated, by Lin. He showed the existence of sufficiently smooth embeddings if the metric is sufficiently smooth and K ≥ 0 [11], or K(0) = 0, |∇K(0)| = 0 [12]. Smooth embeddings of smooth surfaces were obtained by Han et al. [5] when K ≤ 0 and ∇K possesses a certain nondegeneracy, and by Han [3] when K vanishes across a single smooth curve (see also [1, 2, 8] for related results). Lastly if K = |∇K(0)| = 0, |∇2K(0)| = 0 then Khuri [9] has proven the existence of sufficiently smooth