Paper · Feb 25, 2019

Bianchi-convex sets and a new maximum principle for the Ricci flow

Stine Franziska Beitz

math.DGarXiv:1902.09088
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Abstract

We introduce the new notion of Bianchi-convex sets, a generalization of convex sets of algebraic curvature tensors inspired by the second Bianchi identity. It turns out that Hamilton's maximum principle for the Ricci flow can be generalized for Bianchi-convex sets.

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Submitted
Feb 25, 2019
Subjects
math.DG

Comments: 20 pages, 1 figure

Ricci flow

Its neighbourhood

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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 2016, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons

    Stefan Ivanov and Nikola Stanchev

    Shared terms “Bianchi”, “identity”

  2. Some local Maximum principles along Ricci Flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Ricci flow · Shared terms “maximum principle”

  3. Ricci Flow Singularity for Triaxial Bianchi IX Metric

    M. Syafiq Johar

    Topics Ricci flow · Shared terms “Bianchi”

  4. ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

    Brian AllenWider flows

    Shared terms “maximum principle”, “Hamilton”

  5. Maximum Principles and Consequences for -translators in

    José Torres SantaellaWider flows

    Shared terms “maximum principle”

  6. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Shared terms “convex set”

  7. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows

    Shared terms “convex set”

  8. Convex sets evolving by volume preserving fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

    Shared terms “convex set”

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)

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.