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- W2384501249 abstract "In his paper [3], M. Gromov states a compactness property for sets of holomorphic curves, which lies between the convergence of images (as subsets in the range) and the convergence of maps (parameter included): given a sequence /j : (8, Js ) -+ (V, J) of holomorphic maps with bounded areas, there exists a subsequence that converges smoothly away from a finite set of points, at which bubbles develop. This kind of result, which is classical in analytic geometry (E. Bishop's compactness theorem for analytic submanifolds in Kahler manifolds [1]), appeared more recently in analysis on manifolds. In this context, the bubbling off phenomenon was first discovered by J. Sacks and K. Uhlenbeck in their work on harmonic maps of a Riemann surface to a Riemannian manifold [6]. Since then, it has shown up in other variational problems where a noncompact symmetry group arises (see the report by J.P. Bourguignon [2]). In these notes, which follow [3] closely, a proof of Gromov's compactness theorem for closed holomorphic curves is given. Holomorphic curves with boundary are covered only in an easy special case. The first step in the proof is the compactness of cusp-curves, i.e., convergence up to a change of parameter. In the second step, convergence of parametrised curves is obtained as a consequence of the convergence of graphs in 8 x V. There are other approaches to compactness theorems, due to T. Parker and J. Wolfson [5] and Rugang Ye [7J." @default.
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- W2384501249 date "1994-01-01" @default.
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- W2384501249 title "Chapter VIII Compactness" @default.
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