Minimal Gaussian Curvature Surface

TL;DR

This paper introduces a method to find surfaces in three-dimensional space that closely resemble flat surfaces, span a given contour as a geodesic, and minimize total Gaussian curvature squared, using PDE techniques.

Contribution

It formulates a PDE-based approach to identify minimal Gaussian curvature surfaces with prescribed boundary conditions, reducing the problem to a biharmonic equation.

Findings

Derived a PDE controlling surface curvature via coordinate change

Reduced the problem to solving a biharmonic equation with specific boundary conditions

Provided a system of PDEs characterizing the optimal 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.