Handle Addition for doubly-periodic Scherk Surfaces

TL;DR

This paper proves the existence of a new family of embedded doubly periodic minimal surfaces with orthogonal ends, extending classical Scherk surfaces, using induction and the conjugate Plateau method.

Contribution

It introduces a generalized family of embedded doubly periodic minimal surfaces with higher genus and orthogonal ends, expanding the known classes of such surfaces.

Findings

Existence of a new family of minimal surfaces proven.

Embeddedness established via conjugate Plateau method.

Generalization of classical Scherk surfaces to higher genus.

Abstract

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.