Richard Hind

New obstructions to symplectic embeddings

In this paper we establish new restrictions on the symplectic embeddings of basic shapes in symplectic vector spaces. By refining an embedding technique due to Guth, we also show that they are sharp.

Symplectic geometry and the uniqueness of Grauert tubes

Abstract. ((Without Abstract)).

Symplectic embeddings of ellipsoids in dimension greater than four

We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m‐dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6, if the ratio of the areas of any two axes is sufficiently large then the ellipsoid is flexible in the sense that it fully fills a ball. We also show that the same property holds in all dimensions for sufficiently thin ellipsoids E.1;:::;a/. A consequence of our study is that in arbitrary dimension a ball can be fully filled by any sufficiently large number of identical smaller balls, thus generalizing a result of Biran valid in dimension 4. 53D35, 57R17

Ellipsoid embeddings and symplectic packing stability

We prove packing stability for any closed symplectic manifold with rational cohomology class. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some volume preserving embeddings in dimension 4.

Some optimal embeddings of symplectic ellipsoids

We construct symplectic embeddings of ellipsoids of dimension

STEIN FILLINGS OF LENS SPACES

2n \ge 6

Intrinsic vector-valued symmetric form for simple mechanical control systems in the nonzero velocity setting

into the product of a 4-ball or 4-dimensional cube with Euclidean space. A sequence of these embeddings can be shown to be optimal.

Geometric analysis of a class of constrained mechanical control systems in the nonzero velocity setting

We describe a foliation by finite energy holomorphic curves of some symplectic manifolds which are constructed from Stein manifolds with Lens space boundaries. One application is that all such Stein manifolds bounded by the same contact Lens space are equivalent up to Stein homotopy.

HOLOMORPHIC FILLING OF ℝP3

We obtain an intrinsic vector-valued symmetric bilinear form that can be associate with an underactuated simple mechanical control system. We determine properties of the form which serve as necessary conditions for driving underactuated simple mechanical control systems to rest. We also determine properties of the form that serve as sufficient conditions for driving a simple mechanical systems underactuated by one control to an epsiv-neighborhood of rest from an arbitrary initial configuration and velocity. These conditions are computable and coordinate invariant. We focus on the case where the symmetric form is real-valued and indefinite on the entire configuration manifold. Our technical results give rise to a nonlinear control law that drives these systems to an epsiv-neighborhood of rest given an arbitrary initial configuration, velocity and epsiv > 0.

On non-pure forms on almost complex manifolds

Abstract We obtain an intrinsic vector-valued symmetric bilinear form that can be associated with an underactuated constrained mechanical control system. We determine properties of the form that serve as sufficient conditions for driving a constrained mechanical system underactuated by one control to an ∈-neighborhood of rest from an arbitrary initial configuration and velocity. We also determine properties of the form which serve as necessary conditions. These conditions are computable and coordinate invariant.