Helicoidal surfaces with constant mean curvature

Abstract

We describe the space Σ H of all surfaces in R z that have constant mean curvature HφO and are invariant by helicoidal motions, with a fixed axis, of R Ά . Similar to the case Σ o of minimal surfaces Σ H behaves roughly like a circular cylinder where a certain generator corresponds to the rotation surfaces and each parallel corresponds to a periodic family of isometric helicoidal surfaces.1. Introduction. 1.1. Rotation surfaces in the Euclidean space ϋ! 3 with constant mean curvature have been known for a long time (Delaunay [3]). A natural generalization of rotation surfaces are the helicoidal surfaces that can be defined as follows.Let R 3 have coordinates (x, y, z). Consider the one-parameter subgroup g t :R3 -> R 3 of the group of rigid motions of R 3 given by 9t(%> V, z) = (& cos t + y sin t, -x sin t + y cos t, z + ht) , te(-oo f oo).The motion g t is called a helicoidal motion with axis Oz and pitch h.A helicoidal surface with axix Oz and pitch pitch h is a surface that is invariant by g t , for all t. When h = 0, they reduce to rotation surfaces.The helicoidal minimal surfaces have also been known for quite a long time (see e.g. [6] for details). It is therefore mildly surprising that we do not find in the literature the helicoidal surfaces with constant nonzero mean curvature; in this paper we want to determine explicitly all of them.Our interest in this question comes (aside its naturality) from the fact that there are very few explicit examples of surfaces with nonzero constant mean curvature. To understand certain aspects of such surfaces (behaviour of the Gauss map, stability, etc.) it might prove convenient to have at hand a reasonable supply of explicit examples. It should be mentioned that the techniques used here can also give a complete description of helicoidal surfaces with constant Gaussian curvature (Remark 3.16); this is, however, very simple and probably known.

Classification Results on Surfaces in The Isotropic 3‐Space

“…The rotation surfaces in the Euclidean 3-space (Do Carmo and Dajczer, 1982). These classifications were extended to the ones with prescribed mean and Gaussian curvatures by (Baikoussis and Koufogiorgos, 1998) The helicoidal surfaces also have been studied by many authors as focusing on curvature properties in the Minkowskian 3-space …”

CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP

“…Hence, we obtain Remark 2. Helicoidal surfaces in E 3 with constant Gaussian curvature were obtained in [13].…”

Surfaces in E³ invariant under a one parameter group of isometries of E³

“…(Use Chapter 4, Volume 3 in Spivak 1979 and the first and second fundamental forms of a helicoidal surface in the natural coordinates as described earlier or as may be found in Baikoussis & Koufogiorgos 1997, 1998, Do Carmo & Dajczer 1982. Therefore, as before for ψ, we have that k g , k n , τ g are all non-zero constants (along each individual helix).…”

“…We observe that the constant C is the pitch of the helicoidal motion, h in Baikoussis & Koufogiorgos 1997, Do Carmo & Dajczer 1982, Soyuçok 1995 Changing ψ by a constant, for two constant values ψ 1 and ψ 2 of ψ we consider C 1 and C 2 constants satisfying…”

“…where t is parameter along the orbits of the group of the isometries (straight lines, circles, helices respectively) and s is parameter along the curves perpendicular to the orbits which are geodesics (see Baikoussis & Koufogiorgos 1997, 1998, Do Carmo & Dajczer 1982, Hitt & Roussos 1991, Eisenhart 1909, Soyuçok 1995 (19) gives…”