On the equivariant implicit function theorem with low regularity and applications to geometric variational problems

TL;DR

This paper establishes an implicit function theorem for Banach manifold functions invariant under Lie group actions with low regularity, enabling new analysis in geometric variational problems.

Contribution

It introduces an equivariant implicit function theorem applicable to non-smooth group actions, extending tools for geometric analysis.

Findings

Applicable to harmonic maps and geodesics

Enables analysis of constant mean curvature hypersurfaces

Handles non-differentiable group actions

Abstract

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-)Riemannian geodesics, and constant mean curvature hypersurfaces.