An end-to-end construction of doubly periodic minimal surfaces

TL;DR

This paper constructs a new family of embedded, doubly periodic minimal surfaces with higher genus using Traizet's regeneration method, expanding the known examples beyond previous limitations.

Contribution

It provides an explicit construction of doubly periodic minimal surfaces of arbitrary genus, demonstrating their convergence to multiple Scherk surfaces and filling a gap in the classification.

Findings

Constructed infinite family of minimal surfaces with genus 2n-1

Proved smooth convergence to multiple Scherk surfaces

Extended known classes of doubly periodic minimal surfaces

Abstract

Using Traizet's regeneration method, we prove that for each positive integer n there is a family of embedded, doubly periodic minimal surfaces with parallel ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal group of translations. The genus 2n-1 family converges smoothly to 2n copies of Scherk's doubly periodic minimal surface. The only previously known doubly periodic minimal surfaces with parallel ends and genus greater than 3 limit in a foliation of Euclidean space by parallel planes.