arXiv research

Ideal Liouville domains simplify symplectic structures.

The Liouville symplectic form connects various moduli spaces in algebraic geometry.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

Defines symplectic homology for Liouville cobordisms, satisfying Eilenberg-Steenrod axioms.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

Smooth actions of infinite groups linked to homotopy theory.

Study of sectorial decompositions in symmetric products of surfaces for symplectic geometry.

Proves symplectomorphism of noncompact manifolds with embedded rays.

Study relates symplectic homology capacity to periodic orbits in Liouville domains.

The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a c…

Unique symplectic fillings for certain contact manifolds up to diffeomorphism.

On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.

The study finds tight contact structures without fillings in high dimensions.

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-…

Study relates manifold properties to group representations.

Modified geometric quantization simplifies quantum mechanics on manifolds.

The study finds at least two closed orbits for Reeb flows on certain contact manifolds.

For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical map $κ\colon \underrightarrow{H}\underleftarrow{T}(W) \to \underleftarrow{H}\under…

Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension $2n+2$ which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on $\partial W$, which we call \emph{convex open book}, induced b…

On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. Is shown, that Poisson manifolds of n-dimensional multi-symplectic phase space have inducting by (n-1) Hamiltonian k-vector fields, each of which requires of (k)-hamiltonians.

Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent …

Study Markov staircases in symplectic embeddings of rational homology ellipsoids.

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…

Odd-dimensional manifolds have contact maps of non-zero degree.

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation par…

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…

Study singularities of Weinstein skeleta and arboreal singularities.

Extends integrability to cosymplectic manifolds.

Constructs symplectic structures from rational functions on fans.

Defines a new category structure on Weinstein manifolds using h-principle.

We relate Kostant's theorem on the cohomology of a flag manifold $G/B$ with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisso…

Defines Floer homology with DG coefficients for symplectic manifolds.

A correspondence between 1) rank 2 completely integrable systems of Jacobians of algebraic curves and 2) (holomorphically) symplectic surfaces was established in a previous paper by the first author. A more general abelian variety that occurs as a Liouville torus of integrable systems is a prym variety associated to a …

Classifies invariant measures on specific character varieties.

On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $θ$ and the symplectic form $d θ$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and…

Study Liouville currents for Hitchin representations using root flows.

New methods prove non-squeezing in locally conformal symplectic geometry.

In this paper we make the first steps towards developing a theory of intersections of coisotropic submanifolds, similar to that for Lagrangian submanifolds. For coisotropic submanifolds satisfying a certain stability requirement we establish persistence of coisotropic intersections under Hamiltonian diffeomorphisms, ak…

The paper explores the $L^1$-Liouville property on graphs and its connections to stochastic completeness.

The paper extends topological field theory to noncompact surfaces using symmetric powers.

We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the …

Study explores Liouville's theorem and SLP for harmonic functions on cones and surfaces.

Equal diagonal energies proven on Liouville surfaces.

A 'Liouville structure' is a structure isomorphic to a cotangent vector fibration. A Liouville structure is an essential ingredient of every variational formulation of a physical theory. For reasons of interpretation the Liouville structure can not be replaced by the corresponding cotangent fibration. We give a precise…

Article proves Liouville theorem for heat equation in super Ricci flow.

The Liouville theorem is proven for V T-harmonic map heat flow.

Study Liouville action for harmonic maps between Riemann surfaces.