Paper · Jun 4, 2020

Geometric Flow Equations for Schwarzschild-AdS Space-time and Hawking-Page Phase Transition

Davide De Biasio, Dieter Lust

hep-tharXiv:2006.03076v2
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Abstract

Following the recent observation that the Ricci flow and the infinite distance swampland conjecture are closely related to each other, we will investigate in this paper geometric flow equations for AdS space-time geometries. First, we consider the so called Yamabe and Ricci-Bourguignon flows and we show that these two flows - in contrast to the Ricci flow - lead to infinite distance fixed points for product spaces like , where denotes d-dimensional AdS space and corresponds to a p-dimensional sphere. Second, we consider black hole geometries in AdS space time geometries and their behaviour under the Yamabe and Ricci-Bourguignon flows. Specifically we will examine if and how the AdS black holes will undergo a Hawking-Page phase transition under the Ricci flow, the Yamabe flow and under the general Ricci-Bourguignon flow.

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Submitted
Jun 4, 2020
Last revised
Jun 17, 2020 (v2)
Subjects
hep-th, gr-qc
DOI
10.1002/prop.202000053

Comments: 35 pages, one additional reference in revised version

Ricci flowYamabe flowPhysics

Its neighbourhood

Connected papers

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.

Connected papers This paper in the centre and the 8 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 P-V Criticality of Conformal Gravity Holography in Four Dimensions (Parthapratim Pradhan, 2016)7Pradhan 2016 Swampland, Gradient Flow and Infinite Distance (Alex Kehagias, Dieter Lust and Severin Lüst, 2019)2Kehagias 2019 Cobordism, Singularities and the Ricci Flow Conjecture (David Martín Velázquez, Davide De Biasio and Dieter Lust, 2022)4Velázquez 2022 Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes (Davide De Biasio, Julian Freigang, Dieter Lust et al., 2022)1De Biasio 2022 On-Shell Flow (Davide De Biasio, 2022)3De Biasio 2022 Geometric flows and the Swampland (Davide De Biasio, 2023)6De Biasio 2023 Navigating string theory field space with geometric flows (Saskia Demulder, Dieter Lust and Thomas Raml, 2024)5Demulder 2024 Convergence of the fractional Yamabe flow for arbitrary initial energy (Jingeon An-Lacroix, Hardy Chan and Pak Tung Ho, 2025)8An-Lacroix 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

    Davide De Biasio, Julian Freigang, Dieter Lust et al.

    Topics Physics · Same authors Davide De Biasio, Dieter Lust · Shared terms “black hole”, “Schwarzschild”, “fixed point”

  2. Swampland, Gradient Flow and Infinite Distance

    Alex Kehagias, Dieter Lust and Severin Lüst

    Topics Physics · Same author Dieter Lust · Shared terms “infinite distance”, “swampland”, “AdS”

  3. On-Shell Flow

    Davide De Biasio

    Topics Physics · Same author Davide De Biasio · Shared terms “geometric flows”, “swampland”

  4. Cobordism, Singularities and the Ricci Flow Conjecture

    David Martín Velázquez, Davide De Biasio and Dieter Lust

    Topics Ricci flow · Same authors Davide De Biasio, Dieter Lust · Shared terms “swampland”, “following”

  5. Navigating string theory field space with geometric flows

    Saskia Demulder, Dieter Lust and Thomas Raml

    Topics Physics · Same author Dieter Lust · Shared terms “infinite distance”, “geometric flows”, “Swampland conjectures”

  6. Geometric flows and the Swampland

    Davide De Biasio

    Topics Physics · Same author Davide De Biasio · Shared terms “geometric flows”, “swampland”, “infinite”

  7. Criticality of Conformal Gravity Holography in Four Dimensions

    Parthapratim Pradhan

    Topics Physics · Shared terms “AdS”, “Hawking Page”, “Page phase”

  8. Convergence of the fractional Yamabe flow for arbitrary initial energy

    Jingeon An-Lacroix, Hardy Chan and Pak Tung HoWider flows

    Topics Yamabe flow, Physics

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) Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 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.

  3. 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.