The construction of doubly periodic minimal surfaces via balance equations

TL;DR

This paper constructs new families of embedded, doubly periodic minimal surfaces using balance equations and Traizet's regeneration method, revealing complex structures with large genus and multiple ends.

Contribution

It introduces a novel application of balance equations to generate extensive families of doubly periodic minimal surfaces with arbitrary genus and ends.

Findings

Existence of many new 3D families of minimal surfaces.

Surfaces have a foliation by vertical planes as a limit.

Construction allows for arbitrarily large genus and number of ends.

Abstract

Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann surfaces, whose components are copies of C* with finitely many nodes. We derive the balance equations for the location of the nodes and exhibit solutions that allow for surfaces of arbitrarily large genus and number of ends in the quotient.