Bubble tree convergence for harmonic maps
Consider a sequence of harmonic maps \(f_n:\Sigma\to M\) from a compact Riemann surface \((M,h)\) to a compact Riemannian manifold \((M,g)\) with bounded energy \(E(f_n)= {1\over 2}\int_\Sigma|df_n|^2\leq E_0\). Such a sequence has a well-known ``Sacks-Uhlenbeck'' limit consisting of a harmonic map \(f_0:\Sigma\to M\) and some ``bubbles'' -- harmonic maps \(S^2\to M\) obtained by a renormalization process. The set of all bubble maps froms a ``bubble tree'' introduced in the preceding article of \textit{T. H. Parker} and \textit{J. Wolfson} [J. Geom. Anal. 3, No. 1, 63-98 (1993; Zbl 0759.53023)]. The main result of this paper is a precise Bubble Tree Convergence Theorem for a sequence of harmonic maps \(f_n:\Sigma\to M\) with bounded energy. It says that such a sequence of harmonic maps has a bubble tree limit consisting of a harmonic map \(f_0:\Sigma\to M\) and a tree of bubbles \(f_k: S^2\to M\). The author gives a precise construction of this bubble tree and shows two strong convergence statements. The first is that there is no energy loss in the limit, a fact previously shown by \textit{J. Jost} [`Two-dimensional geometric variational problems', Wiley, New York (1991; Zbl 0729.49001)]. The second is that the image of the limit is connected, that is, in the limit there are no necks. As a consequence, the limit preserves homology and homotopy, and the images \(f_n(\Sigma)\) converge pointwise to the image of the bubble tree map. Furthermore, the author gives explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps \(f_n\) when the conformal structure of \(\Sigma\) varies with \(n\), and (ii) when the conformal structure is fixed and \(\{f_n\}\) is a Palais-Smale sequence for the harmonic map energy. Finally, the author considers sequences of harmonic maps from a Riemann surface \((\Sigma,h_n)\), where the metric \(h_n\) is allowed to vary, and shows that the Bubble Tree Convergence Theorem still applies if the corresponding complex structure \(J_n\) stays in a compact region in the moduli space of \(\Sigma\). The problems raised in this paper have stimulated much recent work, and further progress on the ``neck issues'' has been made by \textit{W. Ding} and \textit{G. Tian} [Commun. Anal. Geom. 3, No. 4, 543-554 (1995; Zbl 0855.58016)] and others.
scientific article; zbMATH DE number 1002400