Pseudo-holomorphic maps and bubble trees
type: Item
instance of: scholarly article
title: Pseudo-holomorphic maps and bubble trees
zbMATH Open document ID: 0759.53023
DOI: 10.1007/BF02921330
author: Thomas H. Parker
author: Jon G. Wolfson
published in: The Journal of Geometric Analysis
publication date: 1993-04-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 58C10
zbMATH DE Number: 146374
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: removable singularity theorem
zbMATH Keywords: conformally invariant
zbMATH Keywords: minimal surfaces
zbMATH Keywords: Yang-Mills fields
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921330
OpenAlex ID: W2028612687
Wikidata QID: Q125780704
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description: scientific article; zbMATH DE number 146374