Paper · Oct 4, 2022

A maximal element of a moduli space of Riemannian metrics

Yuichiro Taketomi

math.DGarXiv:2210.01483
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Abstract

For a given smooth manifold, we consider the moduli space of Riemannian metrics up to isometry and scaling. One can define a preorder on the moduli space by the size of isometry groups. We call a Riemannian metric that attains a maximal element with respect to the preorder a maximal metric. Maximal metrics give nice examples of self-similar solutions for various metric evolution equations such as the Ricci flow. In this paper, we construct many examples of maximal metrics on Euclidean spaces.

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Submitted
Oct 4, 2022
Subjects
math.DG
Ricci flow

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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 2021, 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. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy

    Shared terms “moduli space”, “isometry group”

  2. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Shared terms “isometry group”, “Riemannian metrics”

  3. On isometry groups of gradient Ricci solitons

    Ha Tuan Dung and Hung Tran

    Shared terms “isometry group”

  4. On the moduli spaces of left invariant metrics on cotangent bundle of Heisenberg group

    Tijana Sukilovic, Srdjan Vukmirovic and Neda Bokan

    Shared terms “moduli space”

  5. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Ricci flow · Shared terms “isometry group”

  6. Stability and moduli space of generalized Ricci solitons

    Kuan-Hui Lee

    Shared terms “moduli space”

  7. Kähler-Ricci solitons with almost maximal symmetry

    Ha Tuan Dung, Catherine Searle and Hung Tran

    Shared terms “isometry group”

  8. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu

    Topics Ricci flow · Shared terms “moduli space”

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.