This article uses relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi-Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to super-heavy subsets of Entov-Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behaviour of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.
The paper concerns a $C^0$-rigidity result for the charcteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.
An alternative proof of Eliashberg-Gromov's C^0-rigidity theorem is presented and a new notion of weak Lie brackets for Hamiltonian vector fields is proposed and compared.
This paper deals with the ${\mathcal C}^0$-rigidity of the reduction of coiostropic submanifolds under the action of symplectic homeomorphism. More precisely, we exhibit several situations where a symplectic homeomorphism that takes a coisotropic submanifold to a smooth submanifold (which are then known to be coisotropic by a result of Humilière-Leclercq-Seyfaddini) abides to the non-squeezing property in the reduction.
In this paper we study whether symplectic toric manifolds are symplectically cohomologically rigid. Here we say that symplectic cohomological rigidity holds for some family of symplectic manifolds if the members of that family can be distinguished by their integral cohomology rings and the cohomology classes of their symplectic forms. We show how toric degenerations can be used to produce the symplectomorphisms necessary to answer this question. As a consequence we prove that symplectic cohomological rigidity holds for the family of symplectic Bott manifolds with rational symplectic form whose…
An embedding $φ\colon (M_1, ω_1) \to (M_2, ω_2)$ (of symplectic manifolds of the same dimension) is called $ε$-symplectic if the difference $φ^* ω_2 - ω_1$ is $ε$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $ε$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $ε$ and $E (ε) \to 0$ as $ε\to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman. As in the symplect…
We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a result on the rigidity of the Maslov index for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.
We introduce the notion of a point on a locally closed subset of a symplectic manifold being "locally rigid" with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic submanifolds are distinguished among all smooth submanifolds by the property that all of their points are locally rigid. This yields a simplified proof of the Humilière-Leclercq-Seyfaddini theorem on the $C^0$-rigidity of coisotropic submanifolds. Connections are also made to the "rigid locus" that has previously been used in the study of Chekanov-Hofer pseudometri…
We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by J.-C. Sikorav and Y. Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curv…
We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby Lagrangian Conjecture for $T^* \mathbb{R}^2$. As a corollary, we answer a strong version in dimension $2n=4$ of a question of Eliashberg about linking of Lagrangian disks in $T^* \mathbb{R}^n$, which was previously answered by Ekholm and Smith in dimensions $2n \geq 8$.
We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for…
This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension $4$ symplectic submanifolds ($C^0$-flexibility), while this is impossible for codimension $2$ symplectic submanifolds ($C^0$-rigidity). We also discuss $C^0$-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-…
In this paper we study a generalized symplectic fixed point problem, first considered by J. Moser in \cite{M}, from the point of view of some relatively recently discovered symplectic rigidity phenomena. This problem has interesting applications concerning global perturbations of Hamiltonian systems.
We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend to Ham_c(M\setminus U) from its universal cover. Furthermore, we show these invariants and are continuous with respect to the C^0-topology on Ham_c(M\setminus U). We apply these results to Hofer geometry and establish unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for stably displaceable $U$. We also answer a question of F. Le Roux about $C^0$-continuity properties of the Hofer metric.
Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$, then $\mathbb{D}^n(r)$ is the complex or symplectic $n$-discs of radius $r$. Let $z_j = x_j+iy_j\in\mathbb{C}, j=1,2$ and $\mathbb{D}_{\mathbb{R}}^2=\{(z_1,z_2) : |x_1|^2+|x_2|^2<1,|y_1|^2+|y_2|^2<1\}$ be the real bidisc. In this paper we will prove the following two theorems: 1) If $T\in O(4)$ is an orthogonal transformation on $\mathbb{R}^4$, then $T(\mathbb{D}^2)$ is symplectomorphic to $\mathbb{D}^2$ w.r.t. the standard symplectic form on $\mathbb{R}^4$ if and only if $T$ is unitary or conjugate to unitary. 2) For $r\geq 1$ and $n\geq 2$, …
Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the study of billiards. The main result of the paper is the rigidity of a large class of symplectic embedding problems of lagrangian products in any dimension. This is achieved by showing that the lagrangian products under consideration are symplectomorphic to toric domains, and by using the Gromov width and the cube capacity introduced by Gutt and Hutchings to obtain rigidity.
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the techniques are also used to study the topological rigidity of Hamiltonian loops with compact support on a class of noncompact symplectic manifolds.
In the first part of the article we study Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo's symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the nonlinear string equation and treat it as an infinite-dimensional Hamiltonian system. In this context we are able to apply Kuksin's approximation by finite dimensional Hamiltonian flows and prove …
We prove here a quantitative $h$-principle statement that applies to isotropic embeddings of discs. We then apply it to get $C^0$-flexibility and rigidity results in symplectic geometry. On the flexible side, we prove that a symplectic homeomorphism might take a symplectic disc to a smooth isotropic one. We also get a $C^0$-rigidity result for the action of a symplectic homeomorphism on the reduction of a coisotropic submanifold.
We discuss a link between "hard" symplectic topology and an unsharpness principle for generalized quantum observables (positive operator valued measures). The link is provided by the Berezin-Toeplitz quantization.
We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-…
Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize so…
We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.
Given a symplectic three-fold $(M,ω)$ we show that for a generic almost complex structure $J$ which is compatible with $ω$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class $β$ in $\H_2(M,\Z)$ with $c_1(M).β=0$, provided that the divisibility of $β$ is at most 4 (i.e. if $β=nα$ with $α\in H_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$). Moreover, each such curve is embedded and 4-rigid.
The paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe an hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau-Hadamard-Kolmogorov inequalities on the function-theoretic side.
Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $ϕ_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,ω)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Po…
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundle…
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flow…
We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove rigidity results for the skeleton of the divisor complement.
We provide a $C^0$ counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, we prove a quantitative $h$-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, we give another proof of the Dimitroglou Rizell and Sullivan theorem \cite{RS22} which states that Legendrian knots are preserved by contact homeomorphisms, provided their image is smooth. Moreover, our method gives related exa…
This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of symplectic actions of compact Lie groups on symplectic manifolds.
We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.
This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.
We consider deformations of singular Lagrangian varieties in symplectic spaces. We show the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations. Using this result, we prove that, under some assumptions, a Lagrangian deformation is analytically rigid provided that this is the case infinitesimally. This gives generalisations of recent results by Colin de Verdière concerning Lagrangian curves.
We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367. This is a contact version of the Humilière-Leclercq-Seyfaddini coisotropic rigidity theorem in $C^0$ symplectic geometry, and the proof adapts the author's recent re-proof of that result in arXiv:1912.13043…
For a symplectic manifold $M$ let $\{\cdot,\cdot\}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C^\infty_c(M)$ when $\dim M = 2$ and $p < \infty$, extending previous rigidity results for $p = \infty$ in arbitrary dimension.
For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather …
This article is concerned with the question of whether an energy bound implies a genus bound for pseudo-holomorphic curves in almost complex manifolds. After reviewing what is known in dimensions other than 6, we establish a new result in this direction in dimension 6; in particular, for symplectic Calabi-Yau 6-manifolds. The proof relies on compactness and regularity theorems for J-holomorphic currents.
We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does not embed in a ball of radius strictly smaller than $r_n$. We then use symplectic folding to see that this condition is sharp and to construct some nearly optimal embeddings of ellipsoids and polydiscs into balls and cubes. It is finally shown that any connected symplectic manifold of finite volume may be asymptotically filled with skinny ellipsoids or polydiscs.
In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.
In these notes we discuss Lorentz-Finsler metrics, a notion originated in relativity theory, on certain groups of symplectic and contact transformations. Some basic geometric questions arising in this context concerning distance, geodesics and their conjugate points, and existence of a time function, turn out to be related to a variety of subjects including the contact systolic problem, group quasi-morphisms, the Monge-Ampère equation, and a subtle interplay between symplectic rigidity and flexibility. We discuss these interrelations, providing necessary preliminaries, and formulate a number o…
We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools ar…
This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic hom…
We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as cotangent bundle of closed manifolds...) and we derive some consequences to C^0-symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth Hamiltonians whose flows converge to the identity for the spectral (or Hofer's) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the C^0-rigidity o…
In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity condition, then the manifold $M$ is uniquely determined by fixed point data. In particular we can prove that there is a unique family up to isomorphism of 6-dimensional symplectic manifolds with semi-free $S^1$-action and isolated fixed points.
This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that …
In this paper we consider a geometric variant of Hofer's symplectic energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of $M$ which had "too little" energy, one could embed a large ball into a thin cylinder $M \times B^2$. Thus there is a direct geometric relation betwee…
