Abstract
Let (M, g) and (N, h) be twoconnected Riemannian manifolds without boundary (M compact,N complete). Let G ε C ∞(N): ifu: M → N is a smooth map, we consider the functional E G (u) = (1/2) ∫ M [|du|2− 2G(u)]dV M and we study its associated heat equation. Inthe compact case, we recover a version of the Eells–Sampson theorem,while for noncompact target manifold N, we establishsuitable hypotheses and ensure global existence and convergence atinfinity. In the second part of the paper, we study phenomena of blowingup solutions.
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References
Chen, Q.: Liouville theorem for harmonic maps with potential, Manuscripta Math. 95(1998), 507–515.
Chen, Y. and Ding, W. Y.: Blow-up analysis for heat flows of harmonic maps, Inven. Math. 99 (1990), 567–578.
Chen, Y. and Struwe, M.: Existence and partial regularity for heat flows for harmonic maps, Math. Z. 201(1989), 83–103.
Eells, J. and Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 16(1978), 1–68.
Eells, J. and Lemaire, L.: Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524.
Eells, J. and Sampson, J. H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 20 (1964), 109–160.
Fardoun, A. and Ratto, A.: Harmonic maps with potential, in Calculus of Variations and PDE's, Vol. 5, 1997, pp. 183–197.
Hartman, P.: On homotopic harmonic maps, Canad. J. Math. 19(1967), 673–687.
Hong, M.-C.: The Landau–Lifshitz equation with the external field – A new extension for harmonic maps with values in S 2, Math. Z 220(1995), 171–188.
Hong, M.-C. and Lemaire, L.: Multiple solutions from the static Landau–Lifshitz equation from B 2into S 2, Math. Z 220(1995), 295–306.
Jost, J.: Nonlinear Methods in Riemannian and Kählerian Geometry, Birkhäuser, Basel,1991.
Lemaire, L.: On the existence of harmonic maps, Ph.D. Thesis, University of Warwick, 1977.
Schoen, R.: Analytic Aspects of Harmonic Maps, M.S.R.I. Publ. 2, Berlin Springer-Verlag, Berlin, 1984, pp. 321–358.
Schoen, R. and Uhlenbeck, K.: A regularity theory for harmonic maps, J. Differential Geom. 17(1982), 307–335.
Struwe, M.: On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 485–502.
White, B.: Infima of energy functionals in homotopy classes of mappings, J. Differerential Geom. 23(1986), 127–142.
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Fardoun, A., Ratto, A. & Regbaoui, R. On the Heat Flow for Harmonic Maps with Potential. Annals of Global Analysis and Geometry 18, 555–567 (2000). https://doi.org/10.1023/A:1006649025736
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DOI: https://doi.org/10.1023/A:1006649025736