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

TL;DR

This paper classifies all rotational surfaces in Euclidean space where the principal curvatures satisfy a linear relation, providing a variational characterization and identifying surfaces similar to Delaunay surfaces.

Contribution

It offers a complete classification of such surfaces and introduces a variational approach based on the generating curve.

Findings

Identified all rotational surfaces with linear principal curvature relations.

Discovered surfaces with properties similar to Delaunay surfaces.

Provided a variational characterization of these surfaces.

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.