Minimal Gaussian Curvature Surface

This paper deals with finding surfaces in $\mathbb{R}^3$ which are as close as possible to being flat and span a given contour such that the contour is a geodesic on the sought surface. We look for a surface which minimizes the total Gaussian curvature squared. We show that by a change of coordinates the curvature of the optimal surface is controlled by a PDE which can be reduced to the biharmonic equation with an easy-to-define Dirichlet boundary condition and Neumann boundary condition zero. We then state a system of PDEs for the function whose graph is the optimal surface.