The helicoidal surfaces as Bonnet surfaces

Hermite Polynomials And Helicoidal Minimal Surfaces

The main objective of this paper is to construct smooth 1-parameter families of embedded minimal surfaces in euclidean space that are invariant under a screw motion and are asymptotic to the helicoid. Some of these families are significant because they generalize the screw motion invariant helicoid with handles and thus suggest a pathway to the construction of higher genus helicoids. As a bypro...

Linear Weingarten Helicoidal Surfaces in Isotropic Space

Introduced in 1861 [1], a Weingarten surface in the Euclidean three-dimensional space E3 is a surface M, whose mean curvature H and Gaussian curvature K satisfy a non-trivial relation Φ(H, K) = 0. Such a surface was introduced by Weingarten. The class of Weingarten surfaces is remarkably large, and it consists of intriguing surfaces in the Euclidean space: the constant mean curvature surfaces, ...

On helicoidal ends of minimal surfaces

Schlesinger Transformations for Bonnet Surfaces

Bonnet surfaces, i.e. surfaces in Euclidean 3-space, which admits a one-parameter family of isometries preserving the mean curvature function, can be described in terms of solutions of some special Painlev e equations. The goal of this work is to use the well-known Schlesinger transformations for solutions of Painlev e VI equations to create new Bonnet surfaces from a known one.

Constructions of Helicoidal Surfaces in Euclidean Space with Density

Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space R3 with a positive density function eφ, where φ = −x2 − y2, (x, y, z) ∈ R3 and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvat...