Minimal Gaussian Curvature Surface

2021/01/17 by Gilat, Tom
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.06673

Abstract

This paper deals with finding surfaces in ℝ³ 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.

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