Genus three embedded doubly periodic minimal surfaces with parallel ends

TL;DR

This paper constructs a family of genus three embedded doubly periodic minimal surfaces with four parallel ends, providing explicit Weierstrass data and solving the period problem.

Contribution

It introduces a new family of minimal surfaces with explicit data and solves the associated period problem for genus three surfaces.

Findings

Constructed a one-parameter family of genus three minimal surfaces.

Provided explicit Weierstrass data for the surfaces.

Solved the two-dimensional period problem.

Abstract

We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved.