Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDistinguishing between exotic symplectic structures
86
0
0.0
(
0
)
Authors
Richard M. Harris
Ask ChatGPT about the research
No Arabic abstract
We investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish between two examples based on those found in cite{maydanskiy,maydanskiyseidel}, even though their classical symplectic invariants such as symplectic cohomology vanish. We also exhibit, for any $n$, an exact symplectic manifold with $n$ distinct but exotic symplectic structures, which again cannot be distinguished by symplectic cohomology.
rate research
Read More
This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.
We introduce a procedure for gluing Weinstein domains along Weinstein subdomains. By gluing along flexible subdomains, we show that any finite collection of high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a `maximal Weinstein domain also with the same topology. As an application, we produce exotic cotangent bundles containing many closed regular Lagrangians that are formally Lagrangian isotopic but not Hamiltonian isotopic and also give a new construction of exotic Weinstein structures on Euclidean space. We describe a similar construction in the contact setting which we use to produce `maximal contact structures and extend several existing results in low-dimensional contact geometry to high-dimensions. We prove that all contact manifolds have symplectic caps, introduce a general procedure for producing contact manifolds with many Weinstein fillings, and give a new proof of the existence of codimension two contact embeddings.
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is locally conformally Kaehler, and, in fact, Vaisman. This leads to classification results for normal metric contact pairs. In complex dimension two we obtain a new proof of Belguns classification of Vaisman manifolds under the additional assumption that the Kodaira dimension is non-negative. We also produce many examples of manifolds admitting locally conformally symplectic structures but no locally conformally Kaehler ones.
We formulate and prove the analogue of Mosers stability theorem for locally conformally symplectic structures. As special cases we recover some results previously proved by Banyaga.
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D. This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
