Abstract
In this paper, we set up the complete bubble tree theory for approximate harmonic maps from a Riemann surface with tension fields bounded in Zygmund class L ln + L L\ln ^+ L . Some special cases of this theory have previously been used in a number of papers. On the other hand, one can see that this bubble tree theory is not true for the general target manifold if we only assume that the tension fields are bounded in L 1 L^1 uniformly.
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