  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.
