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Marco Mazzucchelli

Publications and source records attributed to Marco Mazzucchelli.

At least 37 records · Page 2Linked to original sources
arXiv2021

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

math.DG↗
arXiv2020

The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.

math.SG↗
arXiv2020

On the spectral characterization of Besse and Zoll Reeb flows

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic Euclidean spaces, we give a sufficient condition for the Besse property via the Ekeland-Hofer capacities.

math.SG↗
arXiv2019

Waist theorems for Tonelli systems in higher dimensions

We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Mañé critical energy level, there is a waist on every energy level just above the Mañé critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Mañé critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.

math.DS↗
arXiv2018

A min-max characterization of Zoll Riemannian metrics

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

math.DG↗
arXiv2018

A characterization of Zoll Riemannian metrics on the 2-sphere

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

math.DG↗
arXiv2018

Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Mañé critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. We show that, under a suitable criticality assumption on a given Randers metric, after rescaling its magnetic part with a sufficiently large multiplicative constant, the new metric admits infinitely many closed geodesics, one of which is a waist. Examples of critical Randers metrics include the celebrated Katok metric.

math.DS↗
arXiv2018

Minimal boundaries in Tonelli Lagrangian systems

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of non-reversible Finsler metrics on the 2-sphere possessing infinitely many closed geodesics.

math.DS↗
arXiv2018

On Tonelli periodic orbits with low energy on surfaces

We prove that, on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian $L$ possesses a periodic orbit that is a local minimizer of the free-period action functional on every energy level belonging to the low range of energies $(e_0(L),c_{\mathrm{u}}(L))$. We also prove that almost every energy level in $(e_0(L),c_{\mathrm{u}}(L))$ possesses infinitely many periodic orbits. These statements extend two results, respectively due to Taimanov and Abbondandolo-Macarini-Mazzucchelli-Paternain, valid for the special case of electromagnetic Lagrangians.

math.DS↗
arXiv2017

Closed geodesics with local homology in maximal degree on non-compact manifolds

We show that, on a complete and possibly non-compact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth under iteration forces the existence of infinitely many closed geodesics. For closed manifolds, this was a theorem due to Hingston.

math.DG↗
arXiv2017

Boundary and lens rigidity for non-convex manifolds

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.

math.DG↗
arXiv2017

Isometry-invariant geodesics and the fundamental group, II

We show that on a closed Riemannian manifold with fundamental group isomorphic to $\mathbb{Z}$, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.

math.DG↗
arXiv2016

On the existence of infinitely many closed geodesics on non-compact manifolds

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.

math.DG↗
arXiv2016

Marked boundary rigidity for surfaces

We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and same marked boundary distance, extending a result of Croke and Otal.

math.DG↗
arXiv2016

The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range $(e_0,e_1)$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, $e_0=0$ is the minimal energy of the system).

math.DS↗
arXiv2016

The Morse index of Chaperon's generating families

This is an expository paper devoted to the Morse index of Chaperon's generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott's iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linearized Hamiltonian flow in the symplectic group. We will conclude the survey with a proof that the classical Morse index from Tonelli Lagrangian dynamics coincides with the Maslov index.

math.SG↗