Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms

Lucas Hsu Rob Kusner John Sullivan

Abstract

We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke’s Surface Evolver [Brakke 1992]. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.