Hofer Zehnder Capacity Research Articles

Hofer–Zehnder capacity of disc tangent bundles of projective spaces

AbstractWe compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces

We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu’s bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian H will give rise to the non-vanishing Gromov–Witten invariant.

Remark on coisotropic Ekeland–Hofer–Zehnder capacity

In this paper, we give a counter-example to show that coisotropic Ekeland–Hofer–Zehnder capacities do not satisfy superadditivity for hyperplane cuts in higher dimension. Next, we show that the coisotropic Ekeland–Hofer–Zehnder capacity relative to Rn,k is not asymptotic equivalent to any normalized symplectic capacity in general case. But we show on the class of centrally symmetric convex domain in R2n, the coisotropic Ekeland–Hofer–Zehnder capacity is asymptotic equivalent to the Ekeland–Hofer–Zehnder capacity. Finally, we give some other results about the coisotropic Ekeland–Hofer–Zehnder capacity.

The symplectic cohomology of magnetic cotangent bundles

We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced by Groman (2023). As an application, we deduce that if N is a closed orientable manifold and is a magnetic form that is not weakly exact, then the \pi_1 1-sensitive Hofer–Zehnder capacity of any compact set in the magnetic cotangent bundle determined by \sigma is finite.

Relative Hofer–Zehnder capacity and positive symplectic homology

We study the relationship between a homological capacity $$c_{\mathrm {SH}^+}(W)$$ for Liouville domains W defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on W: if the positive symplectic homology of W is non-zero, then the capacity yields a finite upper bound to the $$\pi _1$$ -sensitive Hofer–Zehnder capacity of W relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of W has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of W in terms of the homological capacity $$c_{\mathrm {SH}}(W)$$ defined using the full symplectic homology. Applications of these statements to cotangent bundles are discussed and use a result by Abbondandolo and Mazzucchelli in the appendix, where the monotonicity of systoles of convex Riemannian two-spheres in $${\mathbb {R}}^3$$ is proved.

Symplectic homology of fiberwise convex sets and homology of loop spaces

For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove (i) symplectic homology capacity of any convex body is equal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivity property of the Hofer-Zehnder capacity, which is a generalization of a result previously proved by Haim-Kislev.

Hofer–Zehnder capacity and Bruhat graph

We find bounds for the Hofer-Zehnder capacity of coadjoint orbits of compact Lie groups with respect to the Kostant--Kirillov--Souriau symplectic form in terms of the combinatorics of their Bruhat graph. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal to the diameter of a weighted Cayley graph.

Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings

We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer-Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.

On the Symplectic Size of Convex Polytopes

In this paper we introduce a combinatorial formula for the Ekeland–Hofer–Zehnder capacity of a convex polytope in $${\mathbb{R}^{2n}}$$ . One application of this formula is a certain subadditivity property of this capacity.

The sharp energy–capacity inequality on convex symplectic manifolds

In symplectic geometry, symplectic invariants are useful tools in studying symplectic phenomena. One such invariant is the Hofer–Zehnder capacity, which is defined for any subset of a symplectic manifold. On the other hand, we can associate the so-called displacement energy to any subset. Many symplectic geometers tried to relate the Hofer–Zehnder capacity and the displacement energy to study the behaviour of closed orbits of Hamiltonian diffeomorphisms. Usher proved the so-called ( $$\pi _1$$ -sensitive) sharp energy–capacity inequality between the Hofer–Zehnder capacity and the displacement energy for closed symplectic manifolds. In this paper, we consider a certain Floer homology on symplectic manifolds with boundaries (not symplectic homology) and its spectral invariants. Then we extend the $$\pi _1$$ -sensitive sharp energy–capacity inequality to convex symplectic manifolds. As a corollary, we also prove the almost existence theorem of closed characteristics near displaceable hypersurfaces in convex symplectic manifolds. In particular, we prove the existence of closed characteristics on displaceable contact type hypersurfaces in convex symplectic manifolds (the Weinstein conjecture).

Symplectic capacities from Hamiltonian circle actions

Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.

Finiteness of $$\pi _1$$ π 1 -sensitive Hofer–Zehnder capacity and equivariant loop space homology

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer–Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.

Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $\pi_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $\pi_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.

Asymptotic equivalence of symplectic capacities

A long-standing conjecture states that all normalized symplectic capacities coincide on the class of convex subsets of \mathbb R^{2n} . In this note we focus on an asymptotic (in the dimension) version of this conjecture, and show that when restricted to the class of centrally symmetric convex bodies in \mathbb R^{2n} , several symplectic capacities, including the Ekeland–Hofer–Zehnder capacity, the displacement energy capacity, and the cylindrical capacity, are all equivalent up to a universal constant.

Shortest closed billiard orbits on convex tables

Given a planar compact convex billiard table T, we give an algorithm to find the shortest generalised closed billiard orbits on T. (Generalised billiard orbits are usual billiard orbits if T has smooth boundary.) This algorithm is finite if T is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular n-gon R n is 2-bounce for \({n \ge 4}\), with length twice the width of R n . As an application we obtain an algorithm computing the Ekeland–Hofer–Zehnder capacity of the four-dimensional domain \({T \times B^2}\) in the standard symplectic vector space \({{\mathbb{R}}^4}\). Our method is based on the work of Bezdek–Bezdek (Geom. Dedicata 141:197–206, 2009) and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.

Hofer–Zehnder capacity of unit disk cotangent bundles and the loop product

We prove a new finiteness result for the Hofer–Zehnder capacity of certain unit disk cotangent bundles. It is proved by a computation of the pair-of-pants product on Floer homology of cotangent bundles, combined with the theory of spectral invariants. The computation of the pair-of-pants product is reduced to a simple key computation of the Chas–Sullivan loop product.

Symplectic capacities on surfaces

We classify capacities on the class Symo(2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Symo(2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form ωα · d + 1, where α is an ordinal and d ≥ 0 is an integer. It also turns out that if S1 and S2 are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Symo(2) of infinite genus differs from the Hofer–Zehnder capacity by a non-negative finite or infinite constant.

From symplectic measurements to the Mahler conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler’s conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer–Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants

We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in $\mathbb{R}^{2n}$ that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacity-dependent constant. In $\mathbb{R}^2$, this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of $S^2$ and $\mathbb{RP}^2$ that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.

Many closed symplectic manifolds have infinite Hofer–Zehnder capacity

We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T 2 n T^{2n} ( n ≥ 2 n\geq 2 ) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K 3 K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3 3 . The examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the “neck” in the symplectic sum have no closed characteristics. We conjecture that any closed symplectic four-manifold with b + > 1 b^+>1 admits symplectic forms with a similar property.