Equivariant bifurcation in geometric variational problems

TL;DR

This paper extends a key equivariant bifurcation theorem to geometric variational problems, introducing a slice theorem for Lie group actions on Banach manifolds, with applications to constant mean curvature hypersurfaces.

Contribution

It generalizes existing bifurcation results to an abstract geometric setting and develops a slice theorem for affine Lie group actions on Banach manifolds.

Findings

Extended equivariant bifurcation theorem for geometric variational problems

Proved a slice theorem for affine Lie group actions on Banach manifolds

Provided examples and counter-examples for bifurcation of constant mean curvature hypersurfaces

Abstract

We prove an extension of a celebrated equivariant bifurcation result of J. Smoller and A. Wasserman, in an abstract framework for geometric variational problems. With this purpose, we prove a slice theorem for continuous affine actions of a (finite-dimensional) Lie group on Banach manifolds. As an application, we discuss equivariant bifurcation of constant mean curvature hypersurfaces, providing a few concrete examples and counter-examples.