Boundary value problems for surfaces of constant Gauss Curvature

The compact smooth surfaces in 3 with constant positive Gauss Curvature (Ksurfaces) form a natural class. A K-surface without boundary is itself the boundary of a convex body, so it must be embedded. The surfaces of interest to us have non-empty boundary and so are not necessarily embedded. A fundamental question is this: given a collection γ = {C1 . . . , Cn} of Jordan curves in , what are the K-surfaces with boundary γ? For example, if γ is a single Jordan curve with no inflection points, does γ bound a K-surface? When γ is a single planar Jordan curve, the simplest case, a great deal can be said. We begin our discussion of this case by recalling several elementary facts. Let P be a plane and S a smooth surface in . Let γ be a component of S ∩ P that is not a point. The normal curvature (up to sign) of S ∩ P is the projection of the curvature vector of S ∩P , considered as a curve in P , onto the normal line of S at P . This means that an inflection point of γ ⊂ P ∩ S will have an asymptotic direction Research supported in part by NSF grants DMS-900083, DMS-8802858 and DOE grant DEFG02-86ER250125.