Paper · Sep 11, 2020

Combinatorial Ricci flow on cusped 3-manifolds

Xu Xu

math.GTarXiv:2009.05477
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Abstract

Combinatorial Ricci flow on a cusped -manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact -manifolds with boundary for finding complete hyperbolic metrics on cusped -manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped -manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow through the potential singularities using Luo-Yang's extension. It is shown that the existence of a complete hyperbolic metric on a cusped -manifold is equivalent to the convergence of the extended combinatorial Ricci flow, which gives a new characterization of existence of a complete hyperbolic metric on a cusped -manifold dual to Casson and Rivin's program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped -manifolds.

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Submitted
Sep 11, 2020
Subjects
math.GT

Comments: 19 pages

Ricci flowSingularities3-manifoldsSurfacesBoundaries & non-compactDiscrete & graph Ricci

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  1. Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Ricci”, “decorated hyperbolic”, “hyperbolic polyhedral”

  2. Combinatorial Ricci flow on compact 3-manifolds with boundary

    Xu Xu

    Topics Discrete & graph Ricci, 3-manifolds, Boundaries & non-compact · Shared terms “combinatorial Ricci”, “complete hyperbolic”, “Luo”

  3. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Ricci”, “hyperbolic metric”, “complete hyperbolic”

  4. Convergence of combinatorial Ricci flows to degenerate circle patterns

    Asuka Takatsu

    Topics Discrete & graph Ricci, Surfaces · Shared terms “combinatorial Ricci”, “Luo”, “Chow”

  5. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al.

    Topics Discrete & graph Ricci, 3-manifolds, Boundaries & non-compact · Shared terms “combinatorial Ricci”, “Rivin”, “characterization”

  6. A new class of discrete conformal structures on surfaces with boundary

    Xu Xu

    Topics Discrete & graph Ricci, Surfaces, 3-manifolds · Shared terms “combinatorial Ricci”, “Luo combinatorial”, “hyperbolic metrics”

  7. Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Surfaces, 3-manifolds · Shared terms “combinatorial”, “polyhedral metrics”, “Luo”

  8. Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci, Surfaces, Singularities · Shared terms “combinatorial”, “polyhedral”, “Luo”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.