On the area of minimal surfaces in a slab

TL;DR

This paper proves that among certain minimal surfaces in a slab, a catenoidal waist minimizes area when the surface's intersections with horizontal planes have consistent orientation.

Contribution

It establishes a minimal area comparison between general minimal surfaces and catenoidal waists under specific intersection orientation conditions.

Findings

Catenoidal waists have minimal area among certain minimal surfaces in a slab.

The area inequality holds when the surface's horizontal intersections are consistently oriented.

The result extends understanding of minimal surface configurations in constrained geometries.

Abstract

Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S)>= Area(W), provided the intersections of S with horizontal planes have the same orientation.