Subflexible symplectic manifolds

We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.Comment: Various expository and structural changes. Filled technical gaps pointed out by an anonymous refere

Integrating Lie algebroids via stacks

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into a Weinstein groupoid.Comment: a proof improve

The Weinstein Conjecture for Planar Contact Structures in Dimension Three

In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.Comment: 19 pages, corrected some typo

Extremal values of the (fractional) Weinstein functional on the hyperbolic space

Abstract We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space ℍ n {\mathbb{H}^{n}} . We are primarily interested in the existence of Weinstein functional maximizers or, in other words, existence of extremal functions for the best constant of the Gagliardo–Nirenberg inequality. The main result is that the supremum of the Weinstein functional on ℍ n {\mathbb{H}^{n}} is the same as that on ℝ n {\mathbb{R}^{n}} and the related fact that the said supremum is not attained on ℍ n {\mathbb{H}^{n}} , when functions are chosen from the Sobolev space H 1 ⁢ ( ℍ n ) {H^{1}(\mathbb{H}^{n})} . This proves a conjecture made in [8] (see also [3]). We also prove an analogous version of the conjecture for the Weinstein functional defined with the fractional Laplacian.</jats:p

Flexible Weinstein manifolds

This survey on flexible Weinstein manifolds is, essentially, an extract from our recent joint book.Comment: 41 pages, 8 figure