Paper · Dec 27, 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, Hikaru Yamamoto

math.DGarXiv:2412.19939v3
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Abstract

We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation builds on previous articles by Huisken and the third author, as we apply their techniques to establish new results in this geometric setting. Moreover, under some natural geometric assumptions, the noncompact case is also solved

Comments: Final version, accepted for publication in Communications in Analysis and Geometry

Ricci flowMean curvature flowRicci solitonsBoundaries & non-compact

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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 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)5Gomes 2021 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)8Takahashi 2021 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)4White 2022 Mean curvature flow in an extended Ricci flow background (José N. V. Gomes and Matheus Hudson, 2023)1Gomes 2023 Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters (Shouvik Datta Choudhury, 2024)6Choudhury 2024 Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow (José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa, 2025)2Gomes 2025 On a special class of gradient Ricci solitons (José Nazareno Vieira Gomes and Marcus Antonio Mendonça Marrocos, 2026)7Gomes 2026 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)3Gomes 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow in an extended Ricci flow background

    José N. V. Gomes and Matheus Hudson PDF

    Topics Boundaries & non-compact, Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “extended Ricci”, “monotonicity type”, “Huisken monotonicity”

  2. Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

    José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa PDF

    Topics Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “monotonicity type”, “Huisken monotonicity”, “type formula”

  3. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto PDF

    Topics Boundaries & non-compact · Same authors José N. V. Gomes, Hikaru Yamamoto

  4. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows PDF

    Topics Boundaries & non-compact · Shared terms “Huisken monotonicity”

  5. A note on gradient Ricci soliton warped metrics

    José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro PDF

    Topics Ricci solitons · Same author José N. V. Gomes

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

    Shouvik Datta Choudhury PDF

    Topics Ricci flow · Shared terms “extended Ricci”

  7. On a special class of gradient Ricci solitons

    José Nazareno Vieira Gomes and Marcus Antonio Mendonça Marrocos PDF

    Topics Ricci solitons · Same author José N. V. Gomes

  8. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows PDF

    Topics Boundaries & non-compact · Same author Hikaru Yamamoto · Shared terms “Riemannian metric”

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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.