Paper · Apr 22, 2021

Transverse -entropy and transverse Ricci flow for Riemannian foliations

Dexie Lin

math.DGarXiv:2104.10945v2
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Abstract

In this paper, we introduce an entropy functional on Riemannian foliation, inspired by the work of , which is monotonically along the transverse Ricci flow. We relate their gradient flow, via diffeomorphism preserving the foliated structure of the manifold with Riemannian foliation, to the transverse Ricci flow. Moreover, inspired by the work of Fuquan Fang and Yuhao Zhang, we give a necessary condition for codimension 4 Riemannian foliation admitting the transverse Einstein metric.

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Submitted
Apr 22, 2021
Last revised
Sep 19, 2022 (v2)
Subjects
math.DG

Comments: 13 pages

Ricci flowEinstein metrics4-manifolds

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  1. Transverse Ricci solitons on a compact foliated manifold

    Seungsu Hwang, Seoung Dal Jung and Jungwoo Moon

    Topics Ricci flow · Shared terms “transverse Ricci”, “Riemannian foliation”, “foliated”

  2. On a compact Einstein-type manifolds with Riemannian foliations

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    Topics Einstein metrics · Shared terms “Riemannian foliation”, “necessary condition”

  3. Harnack inequalities on totally geodesic foliations with transverse Ricci flow

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    Topics Ricci flow · Shared terms “transverse Ricci”, “Riemannian foliation”

  4. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al.

    Topics Einstein metrics · Shared terms “transverse Ricci”, “foliation”

  5. On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature

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    Topics 4-manifolds, Einstein metrics

  6. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature I

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics 4-manifolds, Einstein metrics

  7. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature II

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics 4-manifolds, Einstein metrics

  8. Ricci solitons with an orthogonally intransitive 2-dimensional Abelian Killing algebra

    Diego Catalano Ferraioli

    Topics 4-manifolds, Einstein metrics

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 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.