圆柱螺旋面的新刻画
New characterizations of the helicoid in a cylinder
摘要：
本文从以下两个方面描述了实心圆柱体$C \subset \mathbb{R}^3$中螺旋面$H_C$的紧凑段。首先，在合理条件下，$H_C$在所有浸没表面$Σ$与$\partial Σ\subset d_1 \cup d_2 \cup S$中具有最小的面积，其中$d_1$和$d_2$是$C$的顶盘和底盘的直径，$S$是$C$的侧表面。第二，除了$H_C$之外，不存在其边界由$d_1$、$d_2$和$S$上的一对\textcolor{black}{rotationally symmetric}曲线$γ_1$、$γ_2$组成的极小曲面，所述极小曲面沿着所述极小曲面正交于$S$。当$S$上的边界曲线是一对具有一定高度的螺旋线时，我们也得到了同样的结论。
英文摘要：
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 $Σ$ with $\partial Σ\subset d_1 \cup d_2 \cup S$, 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 $γ_1$, $γ_2$ 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.
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