Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T22:00:57.540Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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, ([email protected])
Paolo Piccione
Affiliation:
Departamento de Mátematica, Universidade de São Paulo, Rua do Matão 1010, São PauloSP 05508-090, Brazil, ([email protected]) ([email protected])
Gaetano Siciliano
Affiliation:
Departamento de Mátematica, Universidade de São Paulo, Rua do Matão 1010, São PauloSP 05508-090, Brazil, ([email protected]) ([email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alías, L. J. and Piccione, P., On the manifold structure of the set of unparameterized embeddings with low regularity, Bull. Braz. Math. Soc. 42(2) (2011), 171–183.CrossRefGoogle Scholar
2.Bettiol, R. G., Generic properties of semi-Riemannian geodesic flows, MSc Dissertation, University of Sao Paulo, Brazil (2010).Google Scholar
3.Biliotti, L., Javaloyes, M. A. and Piccione, P., On the semi-Riemannian bumpy metric theorem, J. Lond. Math. Soc. (2) 84(1) (2011), 1–18.CrossRefGoogle Scholar
4.Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, Volume 46 (Academic, 1972).Google Scholar
5.Cervera, V., Mascaró, F. and Michor, P., The action of the diffeomorphism group on the space of immersions, Diff. Geom. Applic. 1(4) (1991), 391–401.CrossRefGoogle Scholar
6.Dancer, E. N., An implicit function theorem with symmetries and its application to nonlinear eigenvalue equations, Bull. Austral. Math. Soc. 21(1) (1980), 81–91.Google Scholar
7.Dancer, E. N., The G-invariant implicit function theorem in infinite dimension, Proc. R. Soc. Edinb. A 92(1-2) (1982), 13–30.CrossRefGoogle Scholar
8.Dancer, E. N., The G-invariant implicit function theorem in infinite dimension, II, Proc. R. Soc. Edinb. A 102(3-4) (1986), 211–220.Google Scholar
9.Daniel, B. and Mira, P., Existence and uniqueness of constant mean curvature spheres in Sols, J. Reine Angew. Math. 2013(685) (2013), 1–32.Google Scholar
10.Eells, J. and Lemaire, L., Two reports on harmonic maps (World Scientific, 1995).CrossRefGoogle Scholar
11.Kapouleas, N., Constant mean curvature surfaces in Euclidean three-space, Bull. Am. Math. Soc. 17(2) (1987), 318–320.Google Scholar
12.Kapouleas, N., Complete constant mean curvature surfaces in Euclidean three-space, Annals Math. (2) 131 (2) (1990), 239–330.Google Scholar
13.Koiso, M. and Palmer, B., Geometry and stability of surfaces with constant anisotropic mean curvature, Indiana Univ. Math. J. 54(6) (2005), 1817–1852.CrossRefGoogle Scholar
14.Mazzeo, R. and Pacard, F., Constant mean curvature surfaces with Delaunay ends, Commun. Analysis Geom. 9(1) (2001), 169–237.Google Scholar
15.Mazzeo, R., Pacard, F. and Pollack, D., Connected sums of constant mean curvature surfaces in Euclidean 3-space, J. Reine Angew. Math. 536 (2001), 115–165.Google Scholar
16.Palais, R. S., Foundations of global nonlinear analysis (W. A. Benjamin, 1968).Google Scholar
17.Pérez, J. and Ros, A., The space of properly embedded minimal surfaces with finite total curvature, Indiana Univ. Math. J. 45(1) (1996), 177–204.Google Scholar
18.Piccione, P. and Tausk, D. V., On the Banach differential structure for sets of maps on non-compact domains, Nonlin. Analysis TMA 46(2) (2001), 245–265.CrossRefGoogle Scholar
19.White, B., The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987), 567–602.CrossRefGoogle Scholar
20.White, B., The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161–200.CrossRefGoogle Scholar