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

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MR
DOI: 10.5802/jedp.607
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     author = {Thomas Kappeler},
     title = {Perturbations of the harmonic map equation},
     booktitle = {},
     series = {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 = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.607/}
}
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Thomas Kappeler. Perturbations of the harmonic map equation. Journées équations aux dérivées partielles (2002), article  no. 9, 9 p. doi : 10.5802/jedp.607. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.607/
  • References
  • Cited by

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

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[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

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