261 research outputs found
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main
result is Grauert's classification of principal holomorphic fiber bundles over
Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds
and Stein spaces to Oka manifolds. It has emerged as a subfield of complex
geometry in its own right since the appearance of a seminal paper of M. Gromov
in 1989.
In this expository paper we discuss Oka manifolds and Oka maps. We describe
equivalent characterizations of Oka manifolds, the functorial properties of
this class, and geometric sufficient conditions for being Oka, the most
important of which is Gromov's ellipticity. We survey the current status of the
theory in terms of known examples of Oka manifolds, mention open problems and
outline the proofs of the main results.
In the appendix by F. Larusson it is explained how Oka manifolds and Oka
maps, along with Stein manifolds, fit into an abstract homotopy-theoretic
framework.
The article is an expanded version of the lectures given by the author at the
Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive
exposition of Oka theory is available in the monograph F. Forstneric, Stein
Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex
Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56,
Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci.
Toulouse Math. (6), vol. 22, no. 4. This version is identical with the
published tex
Convergence of the solutions of the discounted equation
We consider a continuous coercive Hamiltonian on the cotangent bundle of
the compact connected manifold which is convex in the momentum. If
is the viscosity solution of the discounted equation
where is the critical
value, we prove that converges uniformly, as , to a
specific solution of the critical equation We characterize in terms of Peierls barrier and projected
Mather measures.Comment: 35 page
Runge approximation on convex sets implies the Oka property
We prove that the classical Oka property of a complex manifold Y, concerning
the existence and homotopy classification of holomorphic mappings from Stein
manifolds to Y, is equivalent to a Runge approximation property for holomorphic
maps from compact convex sets in Euclidean spaces to Y.Comment: To appear in the Annals of Mat
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Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry
[no abstract available
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