Paper · Apr 15, 2026

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

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

math.DGarXiv:2604.14366v2
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Abstract

We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we develop a method for constructing explicit warped product solutions to the Ricci-Bourguignon flow and present examples that illustrate the scope and geometric relevance of our results

Comments: 26 pages

Ricci flowBoundaries & non-compactRotational symmetry & warped products

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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 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)1Gomes 2021 On noncompact warped product Ricci solitons (Valter Borges, 2022)4Borges 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)3Kotschwar 2023 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)2Gomes 2024 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)5Yadav 2025 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)8Raji 2025 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II (Gang Li, 2026)6Li 2026 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary (Gang Li, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on gradient Ricci soliton warped metrics

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

    Topics Rotational symmetry & warped products · Same author José N. V. Gomes

  2. 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 PDF

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

  3. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact · Shared terms “sufficient conditions”

  4. On noncompact warped product Ricci solitons

    Valter Borges PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact · Shared terms “warping function”

  5. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker PDF

    Topics Rotational symmetry & warped products

  6. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II

    Gang Li PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact

  7. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary

    Gang Li PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact

  8. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “differential equation”, “partial differential”

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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.