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Fourth order approximation of harmonic maps from surfaces

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Abstract

Let \((M^2,g)\)be a compact Riemannian surface and let \((N^n,h)\)be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝl. We consider a sequence \(u_\epsilon \in C^\infty (M,N) (\epsilon \to 0\) of critical points of the functional \(E_\epsilon(u)= \int_M (|Du|^2+\epsilon |\Delta u|^2)\) with uniformly bounded energy. We show that this sequence converges weakly in \(W^{1,2}(M,N)\) and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that \(N=S^{l-1} \) ↪ ℝl.

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Authors and Affiliations

  1. Mathematisches Institut der Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104, Freiburg, Germany

    Tobias Lamm

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  1. Tobias Lamm
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Correspondence to Tobias Lamm.

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Mathematics Subject Classification (2000) 58E20, 35J60, 53C43

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Lamm, T. Fourth order approximation of harmonic maps from surfaces. Calc. Var. 27, 125–157 (2006). https://doi.org/10.1007/s00526-005-0001-1

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  • DOI: https://doi.org/10.1007/s00526-005-0001-1

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