Paper · Dec 31, 2023

Backward propagation of warped product structures and asymptotically conical shrinkers

Brett Kotschwar

math.DGarXiv:2401.00606
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Abstract

We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.

Comments: 56 pages, no figures

Ricci flowRicci solitonsEinstein metricsBoundaries & non-compactRotational symmetry & warped products

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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)6Gomes 2021 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)4Stolarski 2022 On noncompact warped product Ricci solitons (Valter Borges, 2022)1Borges 2022 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)2Isenberg 2025 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)3Yadav 2025 Geometric properties of second Ricci solitons (Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami, 2025)8Khalili 2025 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)5Gomes 2026 On Ricci solitons whose level hypersurfaces have parallel second fundamental form (Matheus Andrade Ribeiro de Moura Horácio, 2026)7Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On noncompact warped product Ricci solitons

    Valter Borges PDF

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

  2. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Ricci solitons · Shared terms “multiply warped”

  3. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker PDF

    Topics Rotational symmetry & warped products, Ricci solitons

  4. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski PDF

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “conical shrinker”, “asymptotically conical”

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

  6. 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, Ricci solitons

  7. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “multiply warped”, “gradient shrinking”, “Einstein”

  8. Geometric properties of second Ricci solitons

    Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami PDF

    Topics Rotational symmetry & warped products, Ricci solitons

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) 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 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.