arXiv subjects

Lorenzo Rigolli

Publications and source records attributed to Lorenzo Rigolli.

4 recordsLinked to original sources

arXiv2019

An alternative proof of infinite dimensional Gromov's non-squeezing for compact perturbations of linear maps

This paper deals with the problem of generalising Gromov's non squeezing theorem to an infinite dimensional Hilbert phase space setting. By following the lines of the proof by Hofer and Zehnder of finite dimensional non-squeezing, we recover an infinite dimensional non-squeezing result by Kuksin for symplectic diffeomorphisms which are non-linear compact perturbations of a symplectic linear map. We also show that the infinite dimensional non-squeezing problem, in full generality, can be reformulated as the problem of finding a suitable Palais-Smale sequence for a distinguished Hamiltonian action functional.

math.SG↗

arXiv2016

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Given a symplectic surface $(Σ, ω)$ of genus $g \ge 4$, we show that the free group with two generators embeds into every asymptotic cone of $(\mathrm{Ham}(Σ, ω), d_\mathrm{H})$, where $d_\mathrm{H}$ is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.

math.SG↗

arXiv2015

Local middle dimensional symplectic non-squeezing in the analytic setting

We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in $\mathbb{R}^{2n}$. Let $φ: D \hookrightarrow \mathbb{R}^{2n}$ be an analytic symplectic embedding of a domain $D \subset \mathbb{R}^{2n}$ and $P$ be a symplectic projector onto a linear $2k$-dimensional symplectic subspace $V\subset \mathbb{R}^{2n}$. Then there exists a positive function $r_0:D\rightarrow (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $Vol_{2k}(Pφ(B_r(x)),ω^k _{0|V})\geq π^{k} r^{2k}$ holds for every $x \in D$ and for every $r < r_0(x)$. This claim will be deduced from an analytic middle-dimensional non-squeezing result (stated by considering paths of symplectic embeddings) whose proof will be carried on by taking advantage of a work by Álvarez Paiva and Balacheff.

math.SG↗

arXiv2013

Equivalence of two diagram representations of links in lens spaces and essential invariants

In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understand if these invariants are able to distinguish links in $L(p,q)$ covered by the same link in $\mathbf{S}^3$. In order to do so, we generalize the combinatorial definition of Knot Floer Homology in lens spaces to the case of links and we analyze how both the invariants change when we switch the orientation of the link.

math.GT↗