paper

Minimal Surface Linear Combinatoin Theorem

arXiv:math/0610706

Abstract

Given two univalent harmonic mappings $f_{1}$ and $f_{2}$ on $D$ , which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_{3} = (1 - s) f_{1} + s f_{2}$ to lift to a minimal surface for $s \in [0, 1]$ . We then construct such mappings from Enneper's surface to Scherk's singularly periodic surface, Sckerk's doubly periodic surface to the catenoid, and the 4-Enneper surface to the 4-noid.