Disks that are double spiral staircases

TL;DR

This paper explores the possible shapes of embedded minimal disks in three-dimensional space, connecting physical phenomena like soap films and DNA structures to mathematical models of minimal surfaces.

Contribution

It provides a detailed analysis of the shapes of embedded minimal disks in ^3, advancing understanding of their geometric properties and underlying principles.

Findings

Characterization of the possible shapes of embedded minimal disks

Connection between physical phenomena and minimal surface theory

Insights into the geometric constraints of minimal disks

Abstract

What are the possible shapes of various things and why? For instance, when a closed wire or a frame is dipped into a soap solution and is raised up from the solution, the surface spanning the wire is a soap film. What are the possible shapes of soap films and why? Or, for instance, why is DNA like a double spiral staircase? ``What..?'' and ``why..?'' are fundamental questions, and when answered, help us understand the world we live in. Soap films, soap bubles, and surface tension were extensively studied by the Belgian physicist and inventor (the inventor of the stroboscope) Joseph Plateau in the first half of the nineteenth century. At least since his studies, it has been known that the right mathematical model for soap films are minimal surfaces -- the soap film is in a state of minimum energy when it is covering the least possible amount of area. We will discuss here the answer to the question: ``What are the possible shapes of embedded minimal disks in $\RR^3$ and why?''.