Computing Discrete Minimal Surfaces and Their Conjugates

TLDR

The paper introduces algorithms for computing stable discrete minimal surfaces bounded by fixed or free curves in Euclidean, spherical, and hyperbolic spaces, and for generating conjugate harmonic maps from discrete harmonic maps. The algorithms operate without genus restrictions, handle singular triangulations, and preserve symmetry of boundary curves during conjugation. Applying the method to the identity map produces conjugate minimal surfaces, but often results in unstable solutions to free boundary value problems.

Abstract

Abstract We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations. Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.