Stationary soap films with vertical potentials

TL;DR

This paper classifies specific cylindrical soap film surfaces with mean curvature related to their distance from a plane, characterizing their generating elastic curves through variational principles and describing particular closed solutions.

Contribution

It introduces a classification of cylindrical surfaces with mean curvature as an nth power of distance, linking them to generalized elastic curves with a variational characterization.

Findings

Complete description of n-elastic curves

Identification of particular closed elastic curves

Connection between surface geometry and variational principles

Abstract

We classify cylindrical surfaces in the Euclidean space whose mean curvature is a $n$th-power of the distance to a reference plane. The generating curves of these surfaces, called $n$-elastic curves, have a variational characterization as critical points of a curvature energy generalizing the classical elastic energy. We give a full description of such curves obtaining, in some particular cases, closed curves including simple ones.