Perturbations of the harmonic map equation

Thomas Kappeler

Journées équations aux dérivées partielles (2002)

  • Volume: 130, Issue: 4, page 1-9
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Kappeler, Thomas. "Perturbations of the harmonic map equation." Journées équations aux dérivées partielles 130.4 (2002): 1-9. <http://eudml.org/doc/93436>.

@article{Kappeler2002,
abstract = {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.},
author = {Kappeler, Thomas},
journal = {Journées équations aux dérivées partielles},
keywords = {NLS equation; Zakharov-Shabat operators; action-angle variables; symmetries},
language = {eng},
number = {4},
pages = {1-9},
publisher = {Université de Nantes},
title = {Perturbations of the harmonic map equation},
url = {http://eudml.org/doc/93436},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Kappeler, Thomas
TI - Perturbations of the harmonic map equation
JO - Journées équations aux dérivées partielles
PY - 2002
PB - Université de Nantes
VL - 130
IS - 4
SP - 1
EP - 9
AB - 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.
LA - eng
KW - NLS equation; Zakharov-Shabat operators; action-angle variables; symmetries
UR - http://eudml.org/doc/93436
ER -

References

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  1. [BGS] W. Ballmann, M. Gromov, V. Schroeder: Manifolds of nonpositive curvature. Birkhäuser, Basel - Boston, 1985. Zbl0591.53001MR823981
  2. [BH] M. Bridson, A. Haefliger: Metric spaces of non-positive curvature. Springer, Berlin - New York, 1999. Zbl0988.53001MR1744486
  3. [CMS] K. Cieliebak, I. Mundet i Riera, D. Salamon: Equivariant moduli problems, branched manifolds and the Euler class. ETHZ preprint, 2001. Zbl1026.58010MR1953244
  4. [CS] C. Croke, V. Schroeder: The fundamental group of compact manifolds without conjugate points. Comment. Math. Helv. 61 (1986), 161 - 175. Zbl0608.53038MR847526
  5. [ES] J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), p. 109 - 160. Zbl0122.40102MR164306
  6. [Gr] M. Gromov: Pseudo-holomorphic curves on almost complex manifolds. Invent. Math. 82 (1985), p. 307 - 347. Zbl0592.53025
  7. [Ha] P. Hartman: On homotopic harmonic maps. Can. J. Math. 19 (1967), p. 673 - 687. Zbl0148.42404MR214004
  8. [KKS1] T. Kappeler, S. Kuksin, V. Schroeder: Perturbations of the harmonic map equation. Preprint Series, Insitute of Mathematics, University of Zurich, 2001. MR2003212
  9. [KKS2] T. Kappeler, S. Kuksin, V. Schroeder: Poincaré inequality for maps to closed manifolds of negative sectional curvature. In preparation. 
  10. [KL] T. Kappeler, J. Latschev: Counting solutions of perturbed harmonic map equations. In preparation. 
  11. [Ku] S. Kuksin: On double-periodic solutions of quasilinear Cauchy-Riemann equations. CPAM 49 (1996), p. 639 - 676. Zbl0855.58048MR1387189
  12. [Sm] S. Smale: An infinite dimensional version of Sard's theorem. Amer. J. Math. 87 (1965), p. 861 - 866. IX-8 Zbl0143.35301MR185604
  13. [SY] R. Schoen, S.T. Yau: Compact group actions and the topology of manifolds with non-positive curvature. Topology 18 (1979) Zbl0424.58012MR551017

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