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On the Equivariant Implicit Function Theorem with Low Regularity and Applications to Geometric Variational Problems

Published online by Cambridge University Press:  18 July 2014

Renato G. Bettiol
Affiliation:
1Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN 16556-1618, USA, (rbettiol@nd.edu)
Paolo Piccione
Affiliation:
Departamento de Mátematica, Universidade de São Paulo, Rua do Matão 1010, São PauloSP 05508-090, Brazil, (piccione@ime.usp.br) (sicilian@ime.usp.br)
Gaetano Siciliano
Affiliation:
Departamento de Mátematica, Universidade de São Paulo, Rua do Matão 1010, São PauloSP 05508-090, Brazil, (piccione@ime.usp.br) (sicilian@ime.usp.br)

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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