Minimal Gaussian Curvature Surface

Abstract

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.

This work has been included in and superceded by Smooth Surfaces via Nets of Geodesics at arXiv:2109.01429

Cited by in corpus (1)

• Smooth Surfaces via Nets of Geodesics