Critical Tori for Mean Curvature Energies in Killing Submersions

TL;DR

This paper investigates mean curvature-based surface energies in Killing submersions, constructing critical tori via symmetry reduction and exploring their relation to curvature-critical curves in 2D forms.

Contribution

It introduces a method to construct vertical tori critical for mean curvature energies in Killing submersions, linking them to curvature-critical curves in Riemannian 2-space forms.

Findings

Constructed vertical tori critical for mean curvature energies.

Established correspondence between tori and critical curves.

Generated rotational tori solutions in Riemannian 3-space forms.

Abstract

We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms. The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.