30 pages. In the revised version,as an application of our main result, we construct a full packing of each of the Grassmanians Gr^+(2,R^5) and Gr^+(2,R^6) by two equal symplectic balls
We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the moment map preimages of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmanians Gr^+(2,R^5) and Gr^+(2,R^6) by two equal symplectic balls.
The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for torus actions. The main ingredient in the proof is the microlocal analysis of the equivariant Szego kernels.
In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic one, or they are all nonzero and the moment map behaves as in the toric $b$-symplectic case studied in [GMPS].
In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,ω)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,ω)$ satisfies the hard Lefschetz property.
We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on $Ham(T^2)$ and some of them are Calabi.
This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of …
On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimi…
The Chekanov torus was the first known \emph{exotic} torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in $S^2 \times S^2$ and a monotone Lagrangian torus which had been introduced before Chekanov's construction \cite{Chekanov}. We prove that the monotone Lagrangian torus fiber in a certain Gelfand--Zeitlin system is Hamiltonian isotopic to the Chekanov torus in $S^2 \times S^2$.
We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in…
We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact Hamiltonian torus manifolds to define geometric quantization from the view point of index theory. We give two applications. The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for Abelian case. The other is a Danilov-type formula for toric case in the non-compact setting, which shows that this geometric …
Suppose given an holomorphic and Hamiltonian action of a compact torus $T$ on a polarized Hodge manifold $M$. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of $T$ on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight $\boldsymbolν$ the $\boldsymbolν$-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through $\boldsymbolν$, we give a gometric interpretation of the isotypical components associated to the…
Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compact Kähler manifold $M$, with nowhere vanishing moment map. Given an integral coadjoint orbit $\mathcal{O}$ for $G$, under transversality assumptions we shall consider two naturally associated 'conic', reductions. One, which will be denoted $\overline{M}^G_{\mathcal{O}}$, is taken with respect to the action of $G$ and the cone over $\mathcal{O}$; another, which will be denoted $\overline{M}^T_{\boldsymbolν}$, is taken with respect to the action of the standard maximal torus $T\leqslant G$ and the ray $\mathbb{R}_+\,\imath\…
We prove the invariant of the symplectic capacity for the Zakharov system on a torus. If the Zakharov solution map is well-defined, then it can be regarded as a symplectomorphism. Thus, we first show the global well-posedness via the local well-posedness and the conservation law. The invariant of the symplectic capacity can be obtained using an approximation method. Many authors use an approximation method to obtain the nonsqueezing theorem, instead of an invariant of the symplectic capacity. However, the conditions of the Hamiltonian system introduced by Kuksin can be relaxed by a new modifie…
We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T \in {\mathbb R}^n$ is the period of the acting $n$-torus.
We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of …
Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell--Liberman theorem. As an application, this confirms a conjecture raised by Battaglia--Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calc…
We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.
We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based on a neck-stretching argument, Gromov's foliation theorem, and the Cieliebak-Schwingenheuer criterion.
We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.
Let K be a simple and simply connected compact Lie group. We call a (twisted) quasi-Hamiltonian K-manifold M a quasi-Hamiltonian model space if it is multiplicity free and its momentum map is surjective. We explicitly identify the subgroups of the Lie algebra of the maximal torus of K, which, by F. Knop's classification of multiplicity free quasi-Hamiltonian manifolds, are in one-to-one correspondence with the isomorphism classes of quasi-Hamiltonian model K-spaces.
