[Paper Review] Flexible Weinstein manifolds

Kai Cieliebak, Yakov Eliashberg|arXiv (Cornell University)|May 7, 2013

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes that flexible Weinstein manifolds are uniquely determined up to homotopy within their formal class, leveraging Murphy’s h-principle for loose Legendrian knots. The key result is that any two flexible Weinstein structures on the same manifold, formally homotopic, are homotopic through flexible structures, enabling powerful classification and embedding results in symplectic topology beyond dimension 4.

ABSTRACT

This survey on flexible Weinstein manifolds is, essentially, an extract from our recent joint book.

Motivation & Objective

• To establish the uniqueness of flexible Weinstein structures up to homotopy within their formal class.
• To extend the h-principle for loose Legendrian knots to the setting of flexible Weinstein manifolds.
• To provide a framework for classifying symplectic structures on open manifolds via flexible handlebody decompositions.
• To derive applications in symplectic embeddings and contact isotopy problems in high dimensions.
• To clarify the relationship between formal homotopy classes and actual homotopy classes of Weinstein structures.

Proposed method

• Use of the h-principle for loose Legendrian knots to construct flexible Weinstein structures.
• Application of Moser’s stability theorem to deform Liouville forms while preserving symplectic structure.
• Construction of a Weinstein homotopy via a Morse homotopy of the exhausting function, fixed near boundaries.
• Employment of a diffeotopy $ h_t $ to normalize the Liouville form and achieve desired symplectic isotopy.
• Utilization of the completion procedure to relate Weinstein domains to finite-type Weinstein manifolds.
• Application of the relative h-principle to extend homotopies over cylindrical ends and maintain flexibility.

Experimental results

Research questions

• RQ1 Can flexible Weinstein structures on a manifold be uniquely classified up to homotopy within their formal class?
• RQ2 What are the implications of the h-principle for loose Legendrian knots in the classification of Weinstein structures?
• RQ3 Under what conditions can a smooth embedding of a flexible Weinstein domain be isotoped to a symplectic embedding?
• RQ4 To what extent do symplectic embeddings of flexible Weinstein manifolds depend on the choice of Liouville form?
• RQ5 Can every pseudo-isotopy of a contact manifold be realized as a contact isotopy, given flexibility conditions?

Key findings

• Flexible Weinstein structures on a manifold are unique up to homotopy within their formal class, as established by the h-principle.
• Any two flexible Weinstein structures that are formally homotopic are homotopic through flexible structures, implying classification up to homotopy.
• Every diffeomorphism of a closed contact manifold of dimension $ \geq 5 $ that is pseudo-isotopic to the identity is smoothly isotopic to a contactomorphism.
• For any flexible Weinstein domain $ (W,\omega,X,\phi) $, any Liouville form $ \Lambda $ with $ d\Lambda $ homotopic to $ \omega $ is isotopic to a multiple of the original Liouville form $ \lambda $, up to an exact form.
• Any smooth embedding $ f_0: W \hookrightarrow X $ with $ f_0^*\Omega $ exact and $ df_0 $ homotopic to a symplectic map is isotopic to a symplectic embedding $ f_1: (W,\varepsilon\omega) \hookrightarrow (X,\Omega) $ for small $ \varepsilon > 0 $.
• If the Liouville vector field dual to $ \Lambda $ is complete, then the symplectic embedding exists for arbitrarily large $ \varepsilon $, extending the range of applicability.