Paper · Mar 19, 2021

Uniformization of compact foliated spaces by surfaces of hyperbolic type

Richard Muñiz, Alberto Verjovsky

math.DGarXiv:2103.10880v3
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Abstract

We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.

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Submitted
Mar 19, 2021
Last revised
Aug 3, 2021 (v3)
Subjects
math.DG

Comments: 15 pages; Section 5 was rewritten to make more comprehensible the proof of Theorem 5.1.;submitted to Proc. Am. Math. Soc

Ricci flowSurfaces

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    Topics Surfaces · Shared terms “uniformization”

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    Topics Surfaces · Shared terms “uniformization”

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    Topics Ricci flow · Shared terms “compact foliated”

  4. A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

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    Topics Surfaces · Shared terms “uniformization”, “Gaussian curvature”

  5. Path Connectivity of Anosov Metrics on Surfaces

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  6. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

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  7. Variations of the harmonic entropy and linear drift on negatively curved surfaces

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    Topics Surfaces

  8. Improved Pseudolocality on Large Hyperbolic Balls

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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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.