Paper · Jan 31, 2019

Evolution of relative Yamabe constant under Ricci Flow

Boris Botvinnik, Peng Lu

math.DGarXiv:1901.11169
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Abstract

Let be a manifold with boundary given together with a conformal class which restricts to a conformal class on . Then the relative Yamabe constant is well-defined. We study the short-time behavior of the relative Yamabe constant under the Ricci flow on with boundary conditions that mean curvature and . In particular, we show that if the initial metric is a Yamabe metric, then, under some natural assumptions, and is equal to zero if and only the metric is Einstein.

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Submitted
Jan 31, 2019
Subjects
math.DG

Comments: 9 pages

Ricci flowBoundaries & non-compact

Its neighbourhood

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  1. CR Yamabe constant, CR Yamabe flow and its soliton

    Pak Tung Ho and Kunbo WangWider flows

    Topics Boundaries & non-compact · Shared terms “Yamabe constant”

  2. Conformal Ricci flow on asymptotically hyperbolic manifolds

    Peng Lu, Jie Qing and Yu Zheng

    Topics Boundaries & non-compact · Same author Peng Lu · Shared terms “short time”

  3. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Boundaries & non-compact · Shared terms “Yamabe constant”

  4. Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

    Gilles Carron, Jørgen Olsen Lye and Boris VertmanWider flows

    Shared terms “Yamabe constant”

  5. Monotonicity of functionals along conformal Ricci flow

    Fengjiang Li, Peng Lu, Jianhong Wang et al.

    Topics Ricci flow · Same author Peng Lu

  6. Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant

    Jørgen Olsen Lye and Boris VertmanWider flows

    Shared terms “Yamabe constant”

  7. Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

    Tsz-Kiu Aaron Chow

    Topics Boundaries & non-compact · Shared terms “initial metric”, “boundary condition”, “short time”

  8. Novel Boundary Conditions for the Ricci Flow

    Rasmus Jouttijärvi

    Topics Boundaries & non-compact · Shared terms “boundary condition”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.