arXiv subjects
Publications and source records attributed to Habib Alizadeh.
The diameter of the spectral pseudometric on the universal cover of the Hamiltonian diffeomorphism group of $\mathrm{Gr}(2,p)$ is shown to be finite whenever $p$ is a prime number. On the other hand, it is shown that the diameter is infinite in the case of $\mathrm{Gr}(2k,2n)$ for all natural numbers $k<n$.
The spectral diameter of a symplectic ball is shown to be equal to its capacity; this result upgrades the known bound by a factor of two and yields a simple formula for the spectral diameter of a symplectic ellipsoid. We also study the relationship between the spectral diameter and packings by two balls.
Given a compact Lagrangian $L$ in a semipositive convex-at-infinity symplectic manifold $W$, we establish a cup-length estimate for the action values of $L$ associated to a Hamiltonian isotopy whose spectral norm is smaller than some $\hbar(L)$. When $L$ is rational, this implies a cup-length estimate on the number of intersection points. This Chekanov-type result generalizes a theorem of Kislev and Shelukhin proving non-displaceability in the case when $W$ is closed and monotone. The method of proof is to deform the pair-of-pants product on Hamiltonian Floer cohomology using the Lagrangian $L$.
We prove a new Hamiltonian extension and consequently a fragmentation result in dimension $4$ for the symplectic manifold $\mathbb{D}^{2}\times \mathbb{D}^{2}$. Polterovich and Shelukhin have recently constructed a family of functionals on the space of time dependent Hamiltonian functions on $S^{2}(1) \times S^{2}(a)$ for certain rational $0 < a < 1$, called Lagrangian spectral estimators. Using our fragmentation result we prove that the restriction of their functionals to the subdomain $\mathbb{D}^{2}(c) \times \mathbb{D}^{2}(a)$ is a uniformly $C^{0}$-continuous functional where $0 < c < 1$. As an application of our results, we show that the complement of a Hofer ball in the group of compactly supported Hamiltonian diffeomorphisms of $\mathbb{D}^{2}(c)\times \mathbb{D}^{2}(a)$ contains a $C^{0}$-open subset. Finally, we show that the aforementioned group equipped with the Hofer distance admits an isometric embedding of an infinite dimensional flat space for suitable values of parameters $c$ and $a$.
We show that if a diffeomorphism of a symplectic manifold $(M^{2n},ω)$ preserves the form $ω^{k}$ for $0 < k < n$ and is connected to identity through such diffeomorphisms then it is indeed a symplectomorphism.