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 $h=0$, $0 0$ 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.