AI 中文总结
本文解决了三维球面中双参数螺旋极小曲面族$\text{Hel}_c^h$的紧性判定问题，给出紧成员的刻画条件并计算了对应紧浸入曲面的Willmore能量。
AI 中文摘要
近期，I. Castro、I. Castro-Infantes和J. Castro-Infantes在三维球面$\boldsymbol{\text{S}^3}$中引入了一个双参数螺旋极小曲面族，记为$\boldsymbol{\text{Hel}_c^h}$，其中螺距$h\boldsymbol{\text{≥0}}$，参数$c\boldsymbol{\text{∈[0,1/2)}}$。当$(h,c)\boldsymbol{\text{=(0,0)}}$时，曲面$\boldsymbol{\text{Hel}_0^0}$是全测地球面；当$c\boldsymbol{\text{→1/2}^-}$时的极限曲面为Clifford环面。$c\boldsymbol{\text{=0}}$且$h\boldsymbol{\text{>0}}$的子族由Lawson球面螺旋面构成，而$h\boldsymbol{\text{=0}}$且$0\boldsymbol{\text{<}c\boldsymbol{\text{<}1/2}}$的子族由球面悬链面构成，其紧成员为Otsuki环面。Castro等人指出，确定$\boldsymbol{\text{Hel}_c^h}$何时为紧曲面并非易事。本文解决了这一紧性问题，证明该族的紧成员满足如下刻画：$\boldsymbol{\text{Hel}_c^h}$是紧的$\boldsymbol{\text{⇔}}$当$c\boldsymbol{\text{=0}}$时，$h\boldsymbol{\text{∈Q}}$；当$0\boldsymbol{\text{<}c\boldsymbol{\text{<}1/2}}$时，$h\boldsymbol{\text{∈Q}}$且$q(h,c)\boldsymbol{\text{∈Q}}$，其中$q(h,c)$由显式积分给出。对于$0\boldsymbol{\text{<}c\boldsymbol{\text{<}1/2}}$，该参数化诱导的每个紧商曲面都是环面；对于$c\boldsymbol{\text{=0}}$且$h\boldsymbol{\text{=j/ν>0}}$（最简分数形式），当$j$和$\nu$均为奇数时，参数平面被全自同构群商后为环面，否则为Klein瓶。本文还显式计算了对应紧浸入曲面的Willmore能量。在球面悬链面的每个Lawson关联族中，仅有有限个参数值能产生Willmore能量低于任意给定界的紧螺旋曲面。
英文摘要
Recently, I. Castro, I. Castro-Infantes, and J. Castro-Infantes introduced a two-parameter family of helicoidal minimal surfaces in $\mathbb S^3$, denoted by $\operatorname{Hel}_c^h$, with the pitch $h\geq0$ and $c\in[0,1/2)$. At $(h,c)=(0,0)$, the surface $\operatorname{Hel}_0^0$ is the totally geodesic sphere, while the limiting surface as $c\to1/2^-$ is the Clifford torus. The subfamily $c=0$, $h>0$, consists of the Lawson spherical helicoids, whereas the subfamily $h=0$, $0<c<1/2$, consists of the spherical catenoids, whose compact members are the Otsuki tori. Castro et al. remarked that it is not an easy problem to determine when $\operatorname{Hel}_c^h$ is a compact surface. In this paper, we resolve this compactness problem, namely we prove that the compact members of the family are characterized by \[ \operatorname{Hel}_c^h \text{ is compact} \quad\Longleftrightarrow\quad \begin{cases} h\in\mathbb Q, & c=0,\\[1mm] h\in\mathbb Q\ \text{and}\ q(h,c)\in\mathbb Q, & 0<c<1/2, \end{cases} \] where $q(h,c)$ is given by an explicit integral. For $0<c<1/2$, every compact quotient surface induced by the parametrization is a torus. For $c=0$ and $h=j/ν>0$ written in lowest terms, the quotient of the parameter plane by the full automorphism group is a torus when $j$ and $ν$ are both odd and a Klein bottle otherwise. The Willmore energies of the corresponding compact immersed surfaces are computed explicitly. Along each Lawson associated family of a spherical catenoid, only finitely many parameter values yield compact helicoidal surfaces with Willmore energy below any prescribed bound.
