Numerical Solutions of Boundary Value Problems for K-Surfaces in R

A K-surface is a surface whose Gauss curvature K is equal to a positive constant. In this paper, we will consider K-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing K-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, if an approximate numerical subsolution can be determined, then the corresponding K-surface can be computed. We will consider two boundary value problems. In the first problem, the K-surface is a graph over a plane. In the second problem, the K-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum allowable Gauss curvature Kmax for these problems. The principal results in this paper are numerical estimates of Kmax for a variety of geometries and boundary data. Using a continuation method, we determine numerically the unique one-parameter family of K-surfaces that exist for K ∈ (0,Kmax). We can compare our numerical estimates for Kmax to the true value when the Ksurface is a subset of a hyperbolic spherical surface of revolution. In this case, we find that our numerical estimates for Kmax are in close agreement with the expected values.