Related papers: The sharp energy-capacity inequality
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…
We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by Lisi and Rieser recently), study their estimates and…
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 in \cite{Groman2015}. As an application, we…
We introduce the Hofer-Zehnder $G$-semicapacity $c_{HZ}^G(M,\om)$ of a symplectic manifold $(M,\om)$ with respect to a subgroup $G \subset \pi_1(M)$ ($c_{HZ}(M,\om) \leq c^G_{HZ}(M,\om)$) and prove that if $(M,\om)$ is tame and there exists…
Entov and Polterovich considered the concept of heaviness and superheaviness by the Oh-Schwarz spectral invariants. The Oh-Schwarz spectral invariants are defined in terms of the Hamiltonian Floer theory. In this paper, we define heaviness…
We show that, on a closed semipositive symplectic manifold with semisimple quantum homology, any Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the manifold, must…
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical…
Calculating the spectral invariant of Floer homology of the distance function, we can find some kind of superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are…
The Ekeland-Hofer-Zehnder capacity (EHZ capacity) is a fundamental symplectic invariant of convex bodies. We show that computing the EHZ capacity of polytopes is NP-hard. For this we reduce the feedback arc set problem in bipartite…
We introduce a new coisotropic Hofer-Zehnder capacity and use it to prove an energy-capacity inequality for displaceable Lagrangians.
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…
Given a symplectic toric manifold, the moment maps of sub-circle actions can be modified to be admissible functions in the sense of Hofer-Zehnder. By exploiting the relationship between the period of Hamiltonian sub-circle actions of a…
Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1(M; \mathbb{Z})\to\mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a…
Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed…
In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed…
In this paper, we compare the symplectic (co)homology capacity with the spectral capacity in the relative case. This result establishes a chain of inequalities of relative symplectic capacities, which is an analogue of the non-relative…
For a convex domain in the standard Euclidean symplectic space which is invariant under a linear anti-symplectic involution $\tau$ we show that its Ekeland-Hofer-Zehnder capacity is equal to the $\tau$-symmetrical symplectic capacity of it.
Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of…
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…
The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their…