As on arXiv
Abstract
We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section 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 as limit sets of the orbits of the geodesic flow that do not return to . 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.
Comments: 33 pages, 12 figures; version 2: minor modifications. To appear in Ergodic Theory and Dynamical Systems
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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From curve shortening to flat link stability and Birkhoff sections of geodesic flows
Topics Curve shortening · Same author Marco Mazzucchelli · Shared terms “geodesic flow”, “orientable Riemannian”, “Birkhoff sections”
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Immortal Curve Shortening Flow on Analytic Surfaces
Topics Curve shortening · Shared terms “closed orientable”, “Riemannian surface”
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Non-uniqueness of geodesic limits and a question of Grayson and Gage
Topics Curve shortening · Shared terms “simple closed”, “closed surfaces”, “closed geodesics”
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Non-uniqueness of curve shortening flow
Topics Curve shortening
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Monotonicity of the modulus under curve shortening flow
Topics Curve shortening
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The inverse curve shortening flow on the hyperbolic plane
Topics Curve shortening
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Curve shortening flow with an ambient force field
Topics Curve shortening · Shared terms “sufficiently”
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Curve Shortening Flow of Space Curves with Convex Projections
Topics Curve shortening
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.
- 12C.4Loops, Disks and Curve Shrinking
Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.