On the self-CPG curves and the Bj\"orling problem

TL;DR

This paper investigates special symmetric curves called self-CPG curves in the context of the Bj"orling problem, exploring their properties and the resulting minimal surfaces, especially those that are self-adjoint.

Contribution

It introduces the concept of self-CPG curves and analyzes their role in generating minimal surfaces with specific symmetry properties.

Findings

Self-CPG curves produce minimal surfaces with symmetrical properties.

The adjoint surface of a self-CPG minimal surface contains another self-CPG curve.

The paper characterizes minimal surfaces generated by self-CPG curves that are self-adjoint.

Abstract

Schwartz's solution to the Bj\"orling problem leads to an equivalence class of spatial strips S(t)=(c(t),n(t)) which produce equivalent minimal surfaces. For the particular case when the generating strip S(t) belongs to some plane E and c(t) is symmetric with respect to some straight line in E, the symmetries of the minimal surface permit us to identify another planar curve ~c(t) that we call the CPG curve to c(t). A simple symmetric argument shows that self-CPG curves produce minimal surfaces whose adjoint surface contains another self-CPG curve. We ask for minimal surfaces generated by self-CPG curves which are self-adjoints.