New characterizations of the helicoid in a cylinder

Abstract

This paper characterizes a compact piece of the helicoid HC in a solid cylinder C ⊂ R from the following two perspectives. First, under reasonable conditions, HC has the smallest area among all immersed surfaces Σ with ∂Σ ⊂ d1 ∪ d2 ∪ S, where d1 and d2 are the diameters of the top and bottom disks of C and S is the side surface of C. Second, other than HC , there exists no minimal surface whose boundary consists of d1, d2, and a pair of rotationally symmetric curves γ1, γ2 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.