Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

Abstract

We classify all rotational surfaces in Euclidean space whose principal curvatures ${\kappa }_1$ and ${\kappa }_2$ satisfy the linear relation ${\kappa }_1=a{\kappa }_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces.

27 pages, 9 figures. This paper is submitted for publication since June 4, 2018