Paper · Nov 4, 2024

Evolution of Functionals Under Extended Ricci Flow

Shouvik Datta Choudhury

math.DGarXiv:2411.03353
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Abstract

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time derivative of integrals of the form for various powers , we explore the intricate interplay between geometric quantities and scalar functions without making any assumptions about the manifold, the scalar field , or the function .

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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 Ricci curvature integrals, local functionals, and the Ricci flow (Yuanqing Ma and Bing Wang, 2021)5Ma 2021 Mean curvature flow in an extended Ricci flow background (José N. V. Gomes and Matheus Hudson, 2023)2Gomes 2023 Robinson-Trautman solutions with scalar hair and Ricci flow (Masato Nozawa and Takashi Torii, 2023)8Nozawa 2023 Novel Boundary Conditions for the Ricci Flow (Rasmus Jouttijärvi, 2024)4Jouttijärvi 2024 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)6Kokotov 2024 Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters (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 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)7Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters

    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”, “time derivative”, “expression”

  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. Novel Boundary Conditions for the Ricci Flow

    Rasmus Jouttijärvi

    Topics Ricci flow · Shared terms “compact Riemannian”, “Riemannian manifolds”

  5. Ricci curvature integrals, local functionals, and the Ricci flow

    Yuanqing Ma and Bing Wang

    Topics Ricci flow · Shared terms “Riemannian manifolds”

  6. Determinants of pseudo-laplacians and for spinor bundles over Riemann surfaces

    Alexey Kokotov and Dmitrii Korikov

    Topics Ricci flow · Shared terms “explicit expression”

  7. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Ricci flow · Shared terms “Riemannian manifolds”

  8. Robinson-Trautman solutions with scalar hair and Ricci flow

    Masato Nozawa and Takashi Torii

    Topics Ricci flow · Shared terms “scalar field”, “time derivative”

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.