The treadmillsled of a curve

TL;DR

This paper explores properties of the TreadmillSled operator on curves, particularly its role in characterizing helicoidal surfaces with constant curvature and minimal surfaces, providing new insights into their geometric behavior.

Contribution

The paper offers new properties of the TreadmillSled operator, clarifies its geometric implications, and characterizes minimal helicoidal surfaces using this operator.

Findings

TreadmillSled of a minimal helicoidal surface's profile is a hyperbola or the x-axis.

Helicoidal surfaces with constant Gauss curvature have TreadmillSled profiles on a vertical semi-line.

Answers to previous questions about the TreadmillSled's geometric restrictions are provided.

Abstract

In a previous paper the author introduced the notion of TreadmillSled of a curve, which is an operator that takes regular curves in R^2 to curves in R^2. This operator turned out to be very useful to describe helicoidal surfaces, for example, it provides an interpretation for the profile curve of helicoidal surfaces with constant mean curvature similar to the well known interpretation of the profile curve of Delaunay's surfaces using conics. Recentely, Palmer and Kuhns used the TreadmillSled to classify all helicoidal surfaces with constant anisotropic mean curvature coming from axially symmetric anisotropic energy density. Also the author proved that an helicoidal surface different from a cylinder has constant Gauss curvature if and only if the TreadmillSled of its profile curve lies in a vertical semi line contained in the lower or upper half plane and not contained in the y-axis... Why not the whole vertical line? and why the semi-line cannot be contained in the y-axis? In this paper we provide several properties of the TreadmillSled operator, in particular we will answer the questions in the previous sentence. Finally, we prove that the TreadmillSled of the profile curve of a minimal helicoidal surface is either a hyperbola or a the x-axis. The latter case occurs only when the surface is a helicoid.