A New Approach to Rotational Weingarten Surfaces

Abstract

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

Constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$ with small planar boundary

“…Bryant [Br] studied closed SW-surfaces using complex analysis. More recently many new results and examples about Special Weingarten surfaces were produced by different authors [AlEsGa,BuOr,CaCa,CoFeTe,EsMe,FeGaMi,GaMaMi,GaMi1,GaMi2,GaMiTa,KuSt,SaTo1,SaTo2]. We point out that, in most of the cited works, W is expressed in terms of the mean curvature H and the Gaussian curvature K, resulting in the symmetry of W with respect to the principal curvatures.…”

Geometry of Enumerable Class of Surfaces Associated with Mylar Balloons

“…and equatorial radius r. By alternating the plus and minus signs in (10) and (11) the two symmetrically lying parts of S M to the XOY-plane and C M to the OX-axis are obtained.…”

“…Later on, these considerations have been explored in concrete settings [9,10] and further extended in [11] to the class of surfaces obeying to the relation…”