Noether invariants for constant mean curvature surfaces in 3-dimensional homogeneous spaces

TL;DR

This paper derives explicit formulas for Noether invariants like flux and torque for constant mean curvature surfaces in 3D homogeneous spaces, analyzing their behavior under isometries and providing concrete examples.

Contribution

It introduces explicit formulas for Noether invariants in homogeneous spaces and explores their transformation properties and interpretations.

Findings

Explicit formulas for flux and torque invariants.

Behavior of invariants under isometry group actions.

Examples demonstrating invariant computations and interpretations.

Abstract

We give explicit formul{\ae} for Noether invariants associated to Killing vector fields for the variational problem of minimal and constant mean curvature surfaces in 3-manifolds. In the case of homogeneous spaces, such invariants are the flux (associated to translations) and the torque (associated to rotations). Then we focus on homogeneous spaces with isometry groups of dimensions 3 or 4 and study the behavior of these invariants under the action of isometries. Finally, we give examples of actual computations and of interpretations of these invariants in different situations.