Embedded, Doubly Periodic Minimal Surfaces

TL;DR

This paper investigates the existence of embedded doubly periodic minimal surfaces with Scherk-type ends in Euclidean 3-space, extending previous results and identifying new cases of existence and non-existence.

Contribution

It extends the known existence results of doubly periodic minimal surfaces with handles, providing new cases where such surfaces exist or do not.

Findings

Extended existence results of doubly periodic minimal surfaces with handles.

Identified new cases where such surfaces do not exist.

Broadened understanding of the topological configurations possible for these surfaces.

Abstract

We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.