Pseudo-holomorphic maps and bubble trees
This paper proves a strong convergence theorem for sequences of pseudo- holomorphic maps from a Riemann surface to a symplectic manifold \(N\) with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants). The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain ``bubbles on bubbles''. This is a significant step beyond the standard renormalization procedure of \textit{J. Sacks} and \textit{K. Uhlenbeck} [Ann. Math. 113, 1-24 (1981; Zbl 0462.58014)]. The renormalized maps give rise to a sequence of maps from a ``bubble tree'' -- a map from a wedge \(\Sigma\vee S^ 2\vee S^ 2\vee \cdots\to N\). The main result is that the images of these renormalized maps converge in \(L^{1,2}\cap C^ 0\) to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class. While the main focus is on holomorphic maps, the bubble tree construction applies to other conformally invariant problems, including minimal surfaces and Yang-Mills fields.
scientific article; zbMATH DE number 146374
scientific article