Paper · Nov 4, 2024

Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters

Shouvik Datta Choudhury

math.DGarXiv:2412.05281
View PDF View HTML arXiv page

As on arXiv

Abstract

This paper explores the evolution and monotonicity of geometric constants within the framework of extended Ricci flows, incorporating variable coupling parameters. Building on Hamiltons foundational Ricci flow and subsequent extensions by List (2008), we introduce modifications to the extended Ricci flow by varying parameters that affect the interaction between the metric and scalar fields. Specifically, we modify the coefficients in the evolution equations governing geometric constants, thereby introducing new degrees of freedom in the analysis. The primary contributions include deriving evolution formulas for the modified geometric constant lambda under the extended and normalized extended Ricci flows, and proving conditions under which monotonicity is maintained.

View PDF View HTML arXiv page
Submitted
Nov 4, 2024
Subjects
math.DG
Ricci flow

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 2021, 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 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)7Streets 2022 Mean curvature flow in an extended Ricci flow background (José N. V. Gomes and Matheus Hudson, 2023)2Gomes 2023 Monotonicity of the Cheeger constant under Ricci flow on spheres (Hollis Williams, 2024)6Williams 2024 Evolution of Functionals Under Extended Ricci Flow (Shouvik Datta Choudhury, 2024)1Choudhury 2024 A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background (José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto, 2024)3Gomes 2024 Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts (Mauricio Angel, 2025)5Angel 2025 Functional Renormalization Group as a Ricci Flow: An 𝓕-Entropy Perspective on Information Metric Dynamics (Jin Mo Bok and Ki-Seok Kim, 2026)4Bok 2026 Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow (Tianyin Ren and Chong Song, 2026)8Ren 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of Functionals Under Extended Ricci Flow

    Shouvik Datta Choudhury

    Topics Ricci flow · Same author Shouvik Datta Choudhury · Shared terms “extended Ricci”, “scalar field”, “deriving”

  2. Mean curvature flow in an extended Ricci flow background

    José N. V. Gomes and Matheus Hudson

    Topics Ricci flow · Shared terms “extended Ricci”, “evolution equation”, “modified”

  3. A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background

    José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto

    Topics Ricci flow · Shared terms “extended Ricci”

  4. Functional Renormalization Group as a Ricci Flow: An -Entropy Perspective on Information Metric Dynamics

    Jin Mo Bok and Ki-Seok Kim

    Topics Ricci flow · Shared terms “coupling”, “modified”

  5. Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts

    Mauricio Angel

    Topics Ricci flow · Shared terms “contributions include”

  6. Monotonicity of the Cheeger constant under Ricci flow on spheres

    Hollis Williams

    Topics Ricci flow

  7. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets

    Topics Ricci flow · Shared terms “coupling”

  8. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song

    Topics Ricci flow · Shared terms “coupling”

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.