Ribaucour transformations for constant mean curvature and linear Weingarten surfaces

Abstract

We provide a method to obtain linear Weingarten surfaces from a given such surface, by imposing a one parameter algebraic condition on a Ribaucour transformation. Our main result extends classical results for surfaces of constant Gaussian or mean curvature. By applying the theory to the cylinder, we obtain a two-parameter family of complete linear Weingarten surfaces (hyperbolic, elliptic and tubular), asymptotically close to the cylinder, which have constant mean curvature when one of the parameters vanishes. The family contains n-bubble Weingarten surfaces which are 1-periodic, have genus zero and two ends of geometric index m, where n/m is an irreducible rational number. Their total curvature vanishes, while the total absolute curvature is 8πn. We also apply the method to obtain families of complete constant mean curvature surfaces, associated to the Delaunay surfaces, which are 1-periodic for special values of the parameter.Introduction.

Linear Weingarten Hypersurfaces in Riemannian Space Forms

“…As a natural generalization of hypersurface with constant scalar curvature or with constant mean curvature, linear Weingarten hypersurface has been studied in many places ( [5,8,9,11,13,15,16]). Recall that a hypersurface in a Riemannian space form is said to be linear Weingarten if its normalized scalar curvature R and mean curvature H satisfy R = aH + b for some constants a, b ∈ R. In [13], Li, Suh and Wei proved the first rigidity result for linear Weingarten hypersurface in S n+1 (1) under the assumption that the hypersurface is compact.…”

On Ribaucour transformations and applications to linear Weingarten surfaces

“…However, they were applied for the first time to obtain minimal surfaces in (Corro et al 2000). More recently, in (Corro et al 2001), the method was extended to linear Weingarten surfaces, providing a unified version of the classical results.…”

“…The reader is referred to (Corro et al 2000(Corro et al , 2001) for proofs and details in the case of the minimal surfaces, linear Weingarten and cmc surfaces. In what follows, we first describe the families of minimal surfaces associated to Enneper surface and to the catenoid.…”

“…For special values of c one gets periodic surfaces in one variable otherwise the surfaces are not periodic in any variable. Details and proofs for the results on linear Weingarten surfaces and cmc-surfaces can be found in (Corro et al 2001). …”

“…In this paper, we will restrict ourselves mainly to the results obtained in (Corro et al 2000(Corro et al , 2001). We will show how Ribaucour some basic geometric aspects which motivated the new definition.…”