Helicoidal surfaces with constant anisotropic mean curvature

TL;DR

This paper investigates helicoidal surfaces with constant anisotropic mean curvature, extending the twizzler representation to anisotropic cases and providing a method to generate all such surfaces via quadratures.

Contribution

It generalizes the twizzler representation for helicoidal surfaces to anisotropic mean curvature, enabling systematic construction of these surfaces.

Findings

Extended twizzler representation to anisotropic cases.

Derived quadrature formulas for generating helicoidal surfaces.

Identified conditions for axially symmetric Wulff shapes.

Abstract

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature surfaces can be obtained by quadratures.