Willmore-like functionals for surfaces in 3-dimensional Thurston geometries

TL;DR

This paper introduces analogues of the Willmore functional tailored for each of the Thurston geometries with 4-dimensional isometry groups, demonstrating that constant mean curvature spheres are critical points of these new functionals.

Contribution

The authors develop new Willmore-like functionals specific to Thurston geometries, linking geometric properties with variational principles.

Findings

CMC-spheres are critical points of the new functionals

Analogues extend Willmore functional to Thurston geometries

Provides a variational characterization of CMC-spheres in these geometries

Abstract

We find analogues of the Willmore functional for each of the Thurston geometries with 4-dimensional isometry group such that the CMC-spheres in these geometries are critical points of these functionals.