Abstract
We discuss Morse inequalities for homotopic critical maps of the energy functional with a potential term. For a generic potential this gives a lower bound on the number of homotopic critical maps in terms of the Betti numbers of the moduli space of harmonic maps. Other applications include sharp existence results for maps with prescribed tension field and pseudo-harmonic maps. Our hypotheses are that the domain and target manifolds are closed and the latter has non-positive sectional curvature.
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References
Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, 319, xxii+643 pp. Springer-Verlag, Berlin (1999)
Bialy M. and Polterovich L. (1995). Hopf-type rigidity for Newton equations. Math. Res. Lett. 2: 695–700
Bialy M. and MacKay R.S. (2000). Variational properties of a nonlinear elliptic equation and rigidity. Duke Math. J. 102: 391–401
Chen Q. (1999). Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations. Calc. Var. Partial Differential Equations 8: 91–107
Chen W. and Jost J. (2004). Maps with prescribed tension fields. Comm. Anal. Geom. 12: 93–109
Eells J. and Sampson J.H. (1964). Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86: 109–160
Fardoun A., Ratto A. and Regbaoui R. (2000). On the heat flow for harmonic maps with potential. Ann. Global Anal. Geom. 18: 555–567
Gottlieb D.H. (1969). Covering transformations and universal fibrations. Illinois J. Math. 13: 432–437
Hartman P. (1967). On homotopic harmonic maps. Canad. J. Math. 19: 673–687
Jost J. and Yau S.-T. (1993). A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170: 221–254
Kappeler T., Kuksin S. and Schroeder V. (2003). Perturbations of the harmonic map equation. Commun. Contemp. Math. 5: 629–669
Kappeler T. and Latschev J. (2005). Counting solutions of perturbed harmonic map equations. Enseign. Math. 51(2): 47–85
Kokarev, G., Kuksin, S.: Quasilinear elliptic differential equations for mappings between manifolds, I. (Russian) Algebra i Analiz, 15, 1–60 (2003) (Translation in St. Petersburg Math. Journal 15, 469–505 (2004))
Kokarev G. (2003). On the compactness property of the quasilinearly perturbed harmonic map equation. Sbornik: Mathematics 194: 1055–1068
Kokarev, G.: Elements of Qualitative Theory of Quasilinear Elliptic Partial Differential Equations for Mappings Valued in Manifolds. PhD Thesis, Heriot-Watt University (2003)
Kokarev G. and Kuksin S. (2007). Quasi-linear elliptic differential equations for mappings of manifolds, II. Ann. Global Anal. Geom. 31: 59–113
Kokarev, G.: On pseudo-harmonic maps in conformal geometry. (Preprint) (2007)
Palais R.S. and Smale S. (1964). A generalized Morse theory. Bull. Amer. Math. Soc. 70: 165–172
Peng X. and Wang G. (1998). Harmonic maps with a prescribed potential. C. R. Acad. Sci. Paris Sr. I Math. 327: 271–276
Schoen, R., Yau, S.-T.: Compact group actions and the topology of manifolds with non-positive sectional curvature. Topology 18, 361–380 (1979) (Errata in 21, 483 (1982))
Sunada T. (1979). Rigidity of certain harmonic mappings. Invent. Math. 51: 297–307
Uhlenbeck K. (1981). Morse theory by perturbation methods with applications to harmonic maps. Trans. Amer. Math. Soc. 267: 569–583
Weber J. (2002). Perturbed closed geodesics are periodic orbits: index and transversality. Math. Z. 241: 45–82
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Kokarev, G. A note on Morse inequalities for harmonic maps with potential and their applications. Ann Glob Anal Geom 33, 101–113 (2008). https://doi.org/10.1007/s10455-007-9073-9
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DOI: https://doi.org/10.1007/s10455-007-9073-9