https://neckpinch.com/authors/marco-mazzucchelli/New arXiv papers by Marco Mazzucchelli, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2408.11938arxiv:2408.11938Wed, 21 Aug 2024 00:00:00 GMTMarcelo R. R. Alves, Marco Mazzucchelli — We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under C⁰-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed connected orientable Riemannian surfaces: for surfaces of positive genus, the existence of a contractible simple closed geodesic γ forces the existence of infinitely many closed geodesics intersecting γ in every primitive free homotopy class of loops; for the 2-sphere, the existence of two disjoint simple closed geodesics forces the existence of a third one intersecting both. The final result asserts the existence of Birkhoff sections for the geodesic flow of any closed connected orientable Riemannian surface.Curve shorteningStability & uniqueness https://arxiv.org/abs/2403.19281arxiv:2403.19281Thu, 28 Mar 2024 00:00:00 GMTPhilippe Bolle, Marco Mazzucchelli, Andrea Venturelli — A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow.Other curvature flowsInverse mean curvature flow https://arxiv.org/abs/2204.11977arxiv:2204.11977Mon, 25 Apr 2022 00:00:00 GMTGonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli, Benjamin H. Schulz — We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section Σ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in ∂Σ as limit sets of the orbits of the geodesic flow that do not return to Σ. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.Curve shortening