New characterizations of the helicoid in a cylinder

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 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.