On the Area of Immersed Minimal Annuli in a Slab

TL;DR

This paper investigates the geometric properties and area estimates of immersed minimal annuli in a slab, focusing on their winding numbers and related convexity properties, with implications for understanding minimal surface structures.

Contribution

It introduces a classification of minimal annuli based on winding numbers and derives new area estimates by analyzing length functions and comparing to catenoid covers.

Findings

Convexity of the length function for level curves of minimal annuli

Area bounds for minimal annuli based on winding number

Comparison of annuli areas to catenoid waist areas

Abstract

We organize minimal annuli in a slab based on the winding number of the circles that foliate them and study the area of minimal annuli with given winding number. Specifically, we deduce some results regarding the convexity of the length function of corresponding level curves, and apply them to estimate the area of the annuli by comparing to the area of waist of covers of catenoids.