Congruence of minimal surfaces

TL;DR

This paper presents a method to determine when two minimal surfaces are congruent, including how to find their correspondence, and applies it to conjugate and polynomial minimal surfaces.

Contribution

It introduces a new method for proving congruence of minimal surfaces and demonstrates its effectiveness on conjugate and polynomial examples.

Findings

Conjugate minimal surfaces coincide and match their associated surfaces.

The method successfully shows congruence among certain degree 6 minimal polynomial surfaces.

The approach provides a systematic way to identify congruence in minimal surface families.

Abstract

An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for minimal surfaces. Here we explain not only how this method works but also how we can find the correspondence between the minimal surfaces, if they are congruent. We show that two families of minimal surfaces which are proved to be conjugate actually coincide and coincide with their associated surfaces. We also consider another family of minimal polynomial surfaces of degree 6 and we apply the method to show that some of them are congruent.