2026/07/30 by Jianquan Ge, Shilin Li
Mathematics · #math.DG
arxiv created 2026/07/30 · arxiv updated 2026/07/31
Recently, I. Castro, I. Castro-Infantes, and J. Castro-Infantes introduced a two-parameter family of helicoidal minimal surfaces in \mathbb S3, denoted by Helch, with the pitch h≥0 and c∈[0,1/2). At (h,c)=(0,0), the surface Hel00 is the totally geodesic sphere, while the limiting surface as c→1/2- 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 Helch 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 Helch is compact \Longleftrightarrow \begincases h∈\mathbb Q, c=0,
h∈\mathbb Q and q(h,c)∈\mathbb Q, 0<c<1/2, \endcases 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.