New characterizations of the helicoid in a cylinder

2021/08/26 by Lee, Eunjoo
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.11705

Abstract

This paper characterizes a compact piece of the helicoid H_C in a solid cylinder C ⊂ ℝ³ from the following two perspectives. First, under reasonable conditions, H_C 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 H_C, there exists no minimal surface whose boundary consists of d₁, d₂, and a pair of \textcolorblackrotationally 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.

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