Intersection theory of punctured pseudoholomorphic curves

By Richard Siefring. 2009

77 citations recorded by the source corpus.

We study the intersection theory of punctured pseudoholomorphic curves in 4‐ dimensional symplectic cobordisms. Using the asymptotic results of the author [22], we first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings [14]. We then turn our attention to curves in the symplectization R M of a 3‐manifold M admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the 3‐manifold and we present conditions that will guarantee the projection of a curve to the 3‐manifold is an embedding. Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition and an application concerning the existence of generalized pseudoholomorphic curves, as introduced by Hofer [7]. 32Q65; 53D42, 57R58

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Citations in the corpus

Algebraic Torsion in Contact Manifolds (2010, methodology): “ check that there are no asymptotic contributions to (R×γn z )∗ux. This is trivially true unless zis one of the end points of x, so assume the latter. Then the deﬁnition of the intersection number in [Sie11] implies that (R× γn z ) ∗ ux = 0 if and only if the asymptotic end of un x approaching γn z ”
The Restricted Three-Body Problem and Holomorphic Curves (2018, background): “If one ﬁxes the asymptotics Siefring [219] managed to deﬁne an intersection number for ﬁnite energy planes which has all the properties one would expect for an intersection number from the closed case.”
Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “This argument can be extended to the present case using the reﬁned asymptotic analysis of Siefring In fact one can improve this to see [26].”