Complex surfaces and interpolation on pseudo-holomorphic cylinders

Details Complex surfaces and interpolation on pseudo-holomorphic cylinders Complex surfaces and interpolation on pseudo-holomorphic cylinders

2010-06-09

arxiv.org/abs/1006.1775v1

Mathematics

Differential Geometry

24 pages

Scientific paper

Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. In the present paper, a similar interpolation theorem for pseudo-holomorphic maps from the cylinder S to an almost-complex manifold (M,J) is proved. Properties of the space of pseudo-holomorphic maps from S to (M,J) are derived, in particular a lower bound for the size of this space (in terms of mean dimension). When M is a moduli space of curves, this gives a construction of non-compact complex surfaces. The methods involve an infinite number of surgeries. On the way, a refinement of the gluing process for two pseudo-holomorphic curves is obtained, establishing the geometric behavior of two glued pseudo-holomorphic curves.

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