Variational Minimization on String-rearrangement Surfaces, Illustrated by an Analysis of the Bilinear Interpolation

TL;DR

This paper introduces a variational algorithm to minimize the surface area spanned by boundary curves, demonstrating its effectiveness on specific surfaces and suggesting bilinear interpolation is nearly minimal.

Contribution

The authors propose a novel variational method that simplifies the minimization of surface area by polynomial approximation of mean curvature, applied to string-rearrangement surfaces.

Findings

Significant area reduction for a hemiellipsoid surface.

Less than 0.8% area decrease for bilinear interpolation.

Bilinear interpolation may be close to the minimal surface.

Abstract

In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter $t$ multiplying the numerator $H_{0}$ of mean curvature function defined over the surface. We point out that the integral of the square of the mean curvature with respect to the surface parameter becomes a polynomial in this variational parameter. Finding a zero, if there is any, of this polynomial would end up at the same (minimal) surface as obtained by minimizing more complicated area functional itself. We have instead minimized this polynomial. Moreover, our minimization is restricted to a search in the class of all surfaces allowed by our ansatz. All in all, we have not yet obtained the exact minimal but we do reduce the area for the same fixed boundary. This reduction is significant for a surface (hemiellipsoid) for which we know the exact minimal surface. But for the bilinear interpolation spanned by four bounding straight lines, which can model the initial and final configurations of re-arranging strings, the decrease remains less than 0.8 percent of the original area. This may suggest that bilinear interpolation is already a near minimal surface.