Perturbations of the harmonic map equation
Journées équations aux dérivées partielles (2002), article no. 9, 9 p.

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

@article{JEDP_2002____A9_0,
     author = {Kappeler, Thomas},
     title = {Perturbations of the harmonic map equation},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {9},
     pages = {1--9},
     publisher = {Universit\'e de Nantes},
     year = {2002},
     doi = {10.5802/jedp.607},
     mrnumber = {1968205},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.607/}
}
TY  - JOUR
AU  - Kappeler, Thomas
TI  - Perturbations of the harmonic map equation
JO  - Journées équations aux dérivées partielles
PY  - 2002
SP  - 1
EP  - 9
PB  - Université de Nantes
UR  - http://www.numdam.org/articles/10.5802/jedp.607/
DO  - 10.5802/jedp.607
LA  - en
ID  - JEDP_2002____A9_0
ER  - 
%0 Journal Article
%A Kappeler, Thomas
%T Perturbations of the harmonic map equation
%J Journées équations aux dérivées partielles
%D 2002
%P 1-9
%I Université de Nantes
%U http://www.numdam.org/articles/10.5802/jedp.607/
%R 10.5802/jedp.607
%G en
%F JEDP_2002____A9_0
Kappeler, Thomas. Perturbations of the harmonic map equation. Journées équations aux dérivées partielles (2002), article  no. 9, 9 p. doi : 10.5802/jedp.607. http://www.numdam.org/articles/10.5802/jedp.607/

[BGS] W. Ballmann, M. Gromov, V. Schroeder: Manifolds of nonpositive curvature. Birkhäuser, Basel - Boston, 1985. | MR | Zbl

[BH] M. Bridson, A. Haefliger: Metric spaces of non-positive curvature. Springer, Berlin - New York, 1999. | MR | Zbl

[CMS] K. Cieliebak, I. Mundet I Riera, D. Salamon: Equivariant moduli problems, branched manifolds and the Euler class. ETHZ preprint, 2001. | MR

[CS] C. Croke, V. Schroeder: The fundamental group of compact manifolds without conjugate points. Comment. Math. Helv. 61 (1986), 161 - 175. | MR | Zbl

[ES] J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), p. 109 - 160. | MR | Zbl

[Gr] M. Gromov: Pseudo-holomorphic curves on almost complex manifolds. Invent. Math. 82 (1985), p. 307 - 347. | Zbl

[Ha] P. Hartman: On homotopic harmonic maps. Can. J. Math. 19 (1967), p. 673 - 687. | MR | Zbl

[KKS1] T. Kappeler, S. Kuksin, V. Schroeder: Perturbations of the harmonic map equation. Preprint Series, Insitute of Mathematics, University of Zurich, 2001. | MR

[KKS2] T. Kappeler, S. Kuksin, V. Schroeder: Poincaré inequality for maps to closed manifolds of negative sectional curvature. In preparation.

[KL] T. Kappeler, J. Latschev: Counting solutions of perturbed harmonic map equations. In preparation.

[Ku] S. Kuksin: On double-periodic solutions of quasilinear Cauchy-Riemann equations. CPAM 49 (1996), p. 639 - 676. | MR | Zbl

[Sm] S. Smale: An infinite dimensional version of Sard's theorem. Amer. J. Math. 87 (1965), p. 861 - 866. IX-8 | MR | Zbl

[SY] R. Schoen, S.T. Yau: Compact group actions and the topology of manifolds with non-positive curvature. Topology 18 (1979) | MR | Zbl

Cité par Sources :