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

Part of: Equations and inequalities involving nonlinear operators Calculus on manifolds; nonlinear operators Spaces and manifolds of mappings Nonlinear functional analysis

Published online by Cambridge University Press:  18 July 2014

Renato G. Bettiol ,
Paolo Piccione  and
Gaetano Siciliano
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)
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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.

Keywords

implicit function theoremgeometric variational problemsCMC hypersurfacesclosed geodesicsharmonic maps

MSC classification

Primary: 46T05: Infinite-dimensional manifolds
Secondary: 47J07: Abstract inverse mapping and implicit function theorems 58C15: Implicit function theorems; global Newton methods 58D19: Group actions and symmetry properties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 53 - 80
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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