New characterizations of the helicoid in a cylinder

TL;DR

This paper provides new characterizations of the helicoid within a cylinder, showing it minimizes area among certain surfaces and is uniquely determined by boundary conditions involving symmetric curves.

Contribution

It introduces novel minimal surface characterizations of the helicoid in a cylinder, emphasizing boundary conditions and minimality properties that are unique to the helicoid.

Findings

Helicoid has minimal area among surfaces with specified boundary conditions.

No other minimal surface with similar boundary conditions exists besides the helicoid.

Unique characterization of the helicoid via boundary curves and orthogonal intersections.

Abstract

This paper characterizes a compact piece of the helicoid $H_C$ in a solid cylinder $C \subset \mathbb{R}^3$ from the following two perspectives. First, under reasonable conditions, $H_C$ has the smallest area among all immersed surfaces $\Sigma$ with $\partial \Sigma \subset d_1 \cup d_2 \cup S$, where $d_1$ and $d_2$ are the diameters of the top and bottom disks of $C$ and $S$ is the side surface of $C$. Second, other than $H_C$, there exists no minimal surface whose boundary consists of $d_1$, $d_2$, and a pair of \textcolor{black}{rotationally symmetric} curves $\gamma_1$, $\gamma_2$ on $S$ along which it meets $S$ orthogonally. We draw the same conclusion when the boundary curves on $S$ are a pair of helices of a certain height.