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Oct 12, 2010 at 2:55 comment added Will Jagy Thanks, Deane. I thought $C^\omega$ would be known but I was not sure. For Han and Khuri, it would appear that the next case would be a monkey saddle, where the curvature function behaves as the imaginary part of $(x + i y)^3$ near the origin.
Oct 12, 2010 at 0:59 comment added Deane Yang On the other hand, in the $C^\infty$ category, the isometric embedding equation is the nastiest naturally occurring system of PDE's that I know. Bryant, Griffiths, and I showed that in some sense the linearized system is a generic first order $n$-by-$n$ system of PDE's. So in general it is not elliptic, hyperbolic, or parabolic. Either very clever ad hoc techniques specific to the system need to be developed, or microlocal analysis is used to analyze the linearized operator.
Oct 12, 2010 at 0:55 comment added Deane Yang The local existence of an isometric embedding in $R^3$ of a real analytic $2$-dimensional Riemannian metric is classical and easy to prove by applying Cauchy-Kovalevski to the Monge-Ampere equation that arises. The generalization to higher dimensions is known as the Cartan-Janet theorem and is a bit trickier, because the system of PDE's is highly degenerate. However, it can be proved by induction of the dimension of the domain, where you use Cauchy-Kovalevski at each step. Cartan formulated the question as an exterior differential system and used the Cartan-Kahler theorem.
Oct 12, 2010 at 0:23 history answered Will Jagy CC BY-SA 2.5