Paper · Dec 15, 2018

Constructing Lifshitz spaces using the Ricci flow

R. Cartas-Fuentevilla, A. Herrera-Aguilar, J. A. Herrera-Mendoza

hep-tharXiv:1812.06239v2
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As on arXiv

Abstract

In this work we make use of the Ricci flow equations to show that, by starting from a general ansatz for the metric, we can construct two kinds of Lifshitz spaces in which: (a) the critical exponent coincides with the spatial dimension of the spacetime and therefore adopts discrete values, and (b) the critical exponent is continuous and arbitrary. These results show that Lifshitz spaces are exact solutions to the Ricci flow equations. Moreover, we found that the Ricci flow evolves towards a single fixed point for both cases which coincides with the flat spacetime.

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Submitted
Dec 15, 2018
Last revised
Jun 6, 2019 (v2)
Subjects
hep-th, gr-qc
Journal
Annals of Physics 415 (2020), 168093
DOI
10.1016/j.aop.2020.168093

Comments: 8 pages

Ricci flow

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Connected papers

The 7 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.

Connected papers This paper in the centre and the 7 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Evolution and metric signature change of maximally symmetric spaces under the Ricci flow (R. Cartas-Fuentevilla, A. Herrera-Aguilar and J. A. Olvera-Santamaría, 2017)2Cartas-Fuent 2017 Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow (Keisuke Takasao, 2018)6Takasao 2018 Convergence Stability for Ricci Flow (Eric Bahuaud, Christine Guenther and James Isenberg, 2018)7Bahuaud 2018 Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow (Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al., 2022)1Cartas-Fuent 2022 On-Shell Flow (Davide De Biasio, 2022)5De Biasio 2022 A non-homogeneous generalization of Burgers equations (Francesco Maltese, 2024)4Maltese 2024 The Lₚ chord Minkowski problem for super-critical exponent (Shibing Chen, Qi-Rui Li and Yuanyuan Li, 2025)3Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al.

    Topics Ricci flow · Same authors R. Cartas-Fuentevilla, A. Herrera-Aguilar, J. A. Herrera-Mendoza · Shared terms “Lifshitz”, “critical exponent”, “fixed point”

  2. Evolution and metric signature change of maximally symmetric spaces under the Ricci flow

    R. Cartas-Fuentevilla, A. Herrera-Aguilar and J. A. Olvera-Santamaría

    Topics Ricci flow · Same authors R. Cartas-Fuentevilla, A. Herrera-Aguilar

  3. The chord Minkowski problem for super-critical exponent

    Shibing Chen, Qi-Rui Li and Yuanyuan LiWider flows

    Shared terms “critical exponent”

  4. A non-homogeneous generalization of Burgers equations

    Francesco Maltese

    Shared terms “spatial dimension”, “exact solutions”

  5. On-Shell Flow

    Davide De Biasio

    Topics Ricci flow · Shared terms “constructing”

  6. Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

    Keisuke TakasaoWider flows

    Shared terms “Lifshitz”

  7. Convergence Stability for Ricci Flow

    Eric Bahuaud, Christine Guenther and James Isenberg

    Topics Ricci flow · Shared terms “fixed point”, “continuous”

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.