Compactness and Willmore Energy of Helicoidal Minimal Surfaces in the 3-Sphere

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 $h=j/\nu>0$ written in lowest terms, the quotient of the parameter plane by the full automorphism group is a torus when $j$ and $\nu$ 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.