Fourth order approximation of harmonic maps from surfaces



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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© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Mathematisches Institut der Albert-Ludwigs-Universität FreiburgFreiburgGermany

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