Generalized Helicoidal Surfaces in Euclidean 5-space

In this paper, we study generalized helicoidal surfaces in Euclidean 5-space. We obtain the necessary and sufficient conditions for generalized helicoidal surfaces in Euclidean 5-space to be minimal, flat or of zero normal curvature tensor, which are ordinary differential equations. We solve those equations and discuss the completeness of the surfaces.

Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space

In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density

In this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.