Regarding a uniqueness property of singly-periodic Scherk surfaces

TL;DR

This paper provides a new proof that the only smooth, connected, singly-periodic minimal surfaces in three-dimensional space with area growth comparable to two planes are the singly-periodic Scherk surfaces, using Lopez-Ros deformation.

Contribution

It offers an alternative proof of a known uniqueness property of singly-periodic Scherk surfaces via Lopez-Ros deformation.

Findings

Confirmed the uniqueness of singly-periodic Scherk surfaces under specific area growth conditions.

Utilized Lopez-Ros deformation as a novel approach for the proof.

Reinforced the classification of minimal surfaces with certain periodicity and growth properties.

Abstract

Inspired by an argument of Ros [15] -- we use the L\'{o}pez-Ros deformation to give another proof of the fact -- due to Meeks and Wolf [13] -- that the only smooth, connected, singly-periodic minimal surfaces in $\Real^3$ with the area growth of two planes are the singly-periodic Scherk surfaces.