Investigation and application of the dressing action on surfaces of constant mean curvature

TL;DR

This paper explores the dressing action on constant mean curvature surfaces in Euclidean space, revealing that surfaces with umbilics have trivial isotropy groups, and applies this to surfaces with symmetric metrics.

Contribution

It demonstrates that the isotropy group under dressing is trivial for CMC surfaces with umbilics and applies this to symmetric CMC surfaces.

Findings

Isotropy group is trivial for CMC surfaces with umbilics

Dressing action influences symmetry properties of CMC surfaces

Results aid in understanding symmetry in geometric surface theory

Abstract

We investigate the dressing action on surfaces of constant mean curvature (CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces with umbilics the isotropy group under dressing is always trivial. This result is applied to the investigation of CMC surfaces whose metric is invariant under a group of automorphisms of the parameter domain.