On the duality between rotational minimal surfaces and maximal surfaces

TL;DR

This paper explores the duality between rotational minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space, revealing connections with Bonnet surfaces and Goursat transformations.

Contribution

It introduces a detailed analysis of the duality process for rotational minimal and maximal surfaces, highlighting the role of Bonnet surfaces and Goursat transformations.

Findings

Dual surfaces of congruent rotational minimal/maximal surfaces are often congruent.

The duality process involves a one-parameter rotation group.

Bonnet minimal/maximal surfaces and Goursat transformations naturally arise in this context.

Abstract

We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parameter group of rotations, it appears the family of Bonnet minimal (maximal) surfaces and the Goursat transformations.