February 5, 2007
Joel Fish (NYU): A local Gromov compactness theorem for unparameterized pseudo-holomorphic curves
Abstract:
Since their introduction by Gromov, pseudo-holomorphic curves have been
studied as maps from closed Riemann surfaces into almost complex
manifolds with a taming symplectic form. This parameterized view has
lead to a number of versions of Gromov compactness which are quite
global in nature. For instance, in order to obtain convergence of a
sequence of pseudo-holomorphic curves mapping into a family of
symplectic
manifolds, typically one must first assume the family has uniform
bounds on geometric quantities like curvature, injectivity radius,
energy threshold, etc. This talk will focus on a new approach to
Gromov's compactness theorem, in which the curves are treated as
generalized (unparameterized) surfaces. In particular, we prove a
local compactness theorem which is useful when considering a family of
target manifolds which develop unbounded geometry. This result
recovers for instance compactness in the standard "stretching the
neck" construction. Furthermore we will also provide applications of
the local
result to families of connected sums of contact manifolds in which the
connecting handle degenerates to a point.
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