Différences

Ci-dessous, les différences entre deux révisions de la page.

--- pseudoholomorphic_curve_theory [2017/07/05 18:24]
niederkruger
+++ pseudoholomorphic_curve_theory [2017/07/21 14:48] (Version actuelle)
niederkruger
@@ Ligne 1: / Ligne 1: @@
 === Pseudo holomorphic curves ===
-Since Gromov published his '85 paper, pseudo-holomorphic curves have been one of the main tool in symplectic topology to prove "rigidity".
+Ever since Gromov published his '85 paper, pseudo-holomorphic curves have been one of the main tools in symplectic topology to prove "rigidity" results.
+Discussions related to holomorphic curves could be inspired by the following theorem in dimension 3.
+**Theorem (Novikov Reebless).** Let //M// be a 3-manifold and let //F// be a Reebless foliation.
+Then
+  - every leaf λ is π_1-injective; and
+  - every transverse loop γ is essential in π_1.
+Could some similar result be proved with holomorphic curves?  Even if it is not possible to state such a result directly using only topological notions like fundamental groups or homology, maybe one could still define some type of Floer type invariants for symplectic foliations to get a generalization of the Novikov statement.
+Should one consider holomorphic curves in the leaves of the foliation?  Should one do the symplectization of the foliated manifold to study holomorphic curves there like in an SFT-style?

symp-foliations