Pseudoholomorphic curve

In mathematics, specifically in topology and geometry, a pseudoholomorphic curve is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy-Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov-Witten invariants, which play a crucial role in type IIA string theory.

Applications in symplectic topology

Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when the almost complex structure J interacts with a symplectic form omega. We say that J is omega-tame iff

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:omega(v, J v) > 0

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for all nonzero tangent vectors v. Tameness implies that the formula

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:(v, w) = rac{1}{2}left(omega(v, Jw) + omega(w, Jv) ight)

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defines a Riemannian metric on X. Gromov showed that, for a given omega, the space of omega-tame J is nonempty and contractible. He used this theory to prove a nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders.

Related Topics:
Riemannian metric - Contractible

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Gromov showed that certain moduli spaces of pseudoholomorphic curves (satisfying additional specified conditions) are compact. This compactness result, now greatly generalized, makes possible the definition of Gromov-Witten invariants, which count pseudoholomorphic curves in symplectic manifolds.

Related Topics:
Moduli space - Compact

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Pseudoholomorphic curves are also used to construct Floer homology, which Andreas Floer (and later authors, in greater generality) used to prove the famous conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows.

Related Topics:
Floer homology - Andreas Floer - Vladimir Arnol'd - Hamiltonian flow

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