Minimal Surfaces and Quaternions

It is well known that any simply connected minimal surface can be realized as the real (or imaginary) part of the integral of a holomorphic curve satisfying a null condition (see https: //en.wikipedia.org/wiki/Weierstrass-Enneper_parameterization). Building on this classical result, I will present a novel construction showing that any such surface can be obtained by suitably conjugating a prototype – namely, a plane. The construction is inspired by the quaternionic representation of spatial Pythagorean-Hodograph curves. This work is the result of a collaboration with Hans-Peter Schröcker, Zbynek Šír, and Jan Vršek (https://arxiv.org/abs/2504.17377).