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start [2017/07/19 16:08]
niederkruger 		start [2017/08/29 17:14] (Version actuelle)
alvaro [Symplectic foliations]
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 ===== Foliations in dimension three ===== ===== Foliations in dimension three =====
  
 +A codimension one foliation //F// of a 3-manifold //M// is said to be **taut** if for every leaf λ of //F// there is a circle transverse to //F// which intersects λ. 
  
-The dichotomy between taut/non-taut foliations in dimension 3 gives a very rich theory:+This definition, which is of topological nature, can be alternatively characterised as follows:
  
-Recall that a codimension one foliation //F// of a 3-manifold //M// is **taut** if for every leaf λ of //F// there is a circle transverse to //F// which intersects λ. +**Theorem (Rummler, Sullivan).** For a smooth transversely orientable codimension one foliation //F// in a closed, orientable, smooth manifold //M//, the following conditions are equivalent:
  
-There is the following characterisation of taut foliations.+  - //F// is taut, 
 +  - there is a flow transverse to //F// which preserves some volume form on //M//, 
 +  - there is a closed 2-form on M that is an area form over each leaf, 
 +  - there is a Riemannian metric on //M// for which the leaves of //F// are least area surfaces.
  
-**Theorem (Rummler, Sullivan).** The following conditions are equivalent for transversely orientable codimension one foliations (//M//, //F//) of closed, orientable, smooth manifolds //M//:+Taut foliations in dimension 3 present a rich geometrical and topological behaviour. The following theorem follows from the theory of minimal surfaces:
  
-  - //F// is taut; +**Theorem (Novikov, Rosenberg).** Let //F// be a taut foliation on a closed 3-manifold //M//. The following statements hold:
-  - there is a flow transverse to //F// which preserves some volume form on //M//; +
-  - there is a Riemannian metric on //M// for which the leaves of //F// are least area surfaces.+
  
 +  - any loop transverse to //F// represents a non trivial class in homotopy,
 +  - any leaf of //F// pi_1 injects into the ambient space,
 +  - //M// is irreducible.
  
-====== Symplectic foliations ======+Gabai was able to relate the construction of taut foliations to decompositions of the ambient manifold into sutured pieces and to the construction of essential laminations. In particular:
  
 +**Theorem (Gabai).** Let //M// be a closed irreducible 3-manifold with non zero second real cohomology. Then //M// admits a taut foliation.
  
-One question of this workshop will be to deal with the question if this dichotomy can be extended to higher dimensions by considering symplectic foliations.  We say that a symplectic foliation is [[strong symplectic foliation|strong]] if the leafwise symplectic form arises from a global closed 2-form; otherwise we say that the symplectic foliations is weak. Certain symplectic techniques only extend (naively) to the first setting: Donaldson techniques and cohomological energy estimates for pseudoholomorphic curves do require closeness.+The existence of taut foliations in rational homology spheres is a topic of active research. An outstanding conjecture in 3-dimensional topology states that a rational homology sphere admits a taut foliation if and only if it is not an L-space and if and only if its fundamental group is left orderable. 
 + 
 +Taut foliations also relate to contact structures. The Eliashberg-Thurston confoliation construction shows that any taut foliation (or more generally, a foliation without a Reeb component) can be approximated by a tight (and in particular, fillable) contact structure. 
 + 
 +This rich theory contrasts with the following facts: 
 + 
 +**Theorem (Thurston).** Let //M// be a closed manifold with zero euler class. Then //M// admits a foliation. 
 + 
 +**Theorem (Meigniez).** Let //M// be a closed manifold with zero euler class of dimension at least 4. Then //M// admits a foliation by dense leaves (which is, in particular, taut). 
 + 
 + 
 +====== Symplectic foliations ====== 
 + 
 +The purpose of the workshop is to explore whether one may be able to define interesting classes of foliations in higher dimensions by taking a symplectic viewpoint. We say that a foliation is [[symplectic |symplectic]] if it admits a leafwise symplectic form. If this form arises from a global closed 2-form we say that the foliation is  [[strong symplectic |strong]] or 2-calibrated. Certain symplectic techniques only extend (naively) to the strong setting: Donaldson techniques and cohomological energy estimates for pseudoholomorphic curves do require closeness.
  
 The following is a tentative list of potentially interesting topics for the workshop: The following is a tentative list of potentially interesting topics for the workshop:
Ligne 26: Ligne 45:
   * [[Obstructions to existence of symplectic foliations|Obstructions to existence]]. In the opposite direction, there are no known obstructions to the existence of strong symplectic foliations (apart from the obvious formal ones). Does a Novikov type theorem hold in this setting? What are some reasonable hypothesis for such a statement?   * [[Obstructions to existence of symplectic foliations|Obstructions to existence]]. In the opposite direction, there are no known obstructions to the existence of strong symplectic foliations (apart from the obvious formal ones). Does a Novikov type theorem hold in this setting? What are some reasonable hypothesis for such a statement?
  
-  * The [[Confoliations|confoliation programme]]. The formal data underlying a symplectic foliation and a contact structure is the same. Is it possible to reproduce Eliashberg and Thurston’s confoliation result in higher dimensions? Is it perhaps simpler to carry it out if one assumes strongness?+  * [[The confoliation programme|The confoliation programme]]. The formal data underlying a symplectic foliation and a contact structure is the same. Is it possible to reproduce Eliashberg and Thurston’s confoliation result in higher dimensions? Is it perhaps simpler to carry it out if one assumes strongness?
  
   * [[Pseudoholomorphic curve theory]]. In the strong case, under reasonable assumptions, moduli spaces of pseudoholomorphic curves should be compact manifolds endowed with singular foliations where the leaves correspond to the leafwise moduli spaces. What structure results for strong symplectic foliations can be obtained this way? What about classification results for fillings of contact foliations?   * [[Pseudoholomorphic curve theory]]. In the strong case, under reasonable assumptions, moduli spaces of pseudoholomorphic curves should be compact manifolds endowed with singular foliations where the leaves correspond to the leafwise moduli spaces. What structure results for strong symplectic foliations can be obtained this way? What about classification results for fillings of contact foliations?
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   *    Transverse Hamiltonian dynamics. Strong symplectic foliations are not stable Hamiltonian usually, but there is a well defined notion of Reeb vector field. What can be said about its dynamics?   *    Transverse Hamiltonian dynamics. Strong symplectic foliations are not stable Hamiltonian usually, but there is a well defined notion of Reeb vector field. What can be said about its dynamics?
 +
 +
 +====== Research ideas/questions =====
 +
 +The following is a list of the particular topics and questions brought up by the participants.
 +
 +  *  [[Differential operators and foliations]]
 +
 +  *  [[Transverse submanifolds and foliations]]
 +
 +  *  [[Existence of symplectic foliations]]
 +
 +  *  [[The confoliation programme]]
 +
 +  *  [[Deformations of symplectic foliations]]
 +
 +  *  [[Foliated symplectic topology]]
 +
  
start.1500473315.txt.gz · Dernière modification: 2017/07/19 16:08 de niederkruger

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