Paper · Feb 2, 2016

A Pohozaev Identity on Warped Product Solitons

Timothy Carson

math.DGarXiv:1602.01055v3
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Abstract

Warped product metrics are a class of Riemannian metrics on cross products which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base has two dimensions, the metric must in fact be a cross product. The theorem is a consequence of a Pohozaev-type identity. This closely parallels results in the study of blow-ups of solutions to the reaction- diffusion equation in .

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Submitted
Feb 2, 2016
Last revised
Mar 10, 2016 (v3)
Subjects
math.DG

Comments: This paper has been withdrawn because of an important sign error in the equation for Rc, below equation 19

Ricci solitonsSingularitiesRotational symmetry & warped products

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 2016, 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 Flow recovering from pinched discs (Timothy Carson, 2017)1Carson 2017 On warped product gradient η-Ricci solitons (Adara M. Blaga, 2017)7Blaga 2017 Painlevé analysis of Ricci solitons over warped products (Alejandro Betancourt de la Parra, 2018)8de la Parra 2018 Ricci flow from some spaces with asymptotically cylindrical singularities (Timothy Carson, 2018)2Carson 2018 Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow (Maxwell Stolarski, 2019)3Stolarski 2019 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)6Gomes 2021 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)5Kotschwar 2023 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)4Yadav 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow recovering from pinched discs

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Same author Timothy Carson · Shared terms “product metrics”

  2. Ricci flow from some spaces with asymptotically cylindrical singularities

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Same author Timothy Carson

  3. Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

    Maxwell Stolarski

    Topics Rotational symmetry & warped products, Singularities · Shared terms “product metrics”

  4. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker

    Topics Rotational symmetry & warped products, Ricci solitons

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

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Ricci solitons

  6. A note on gradient Ricci soliton warped metrics

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

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

  7. On warped product gradient -Ricci solitons

    Adara M. Blaga

    Topics Rotational symmetry & warped products, Ricci solitons

  8. Painlevé analysis of Ricci solitons over warped products

    Alejandro Betancourt de la Parra

    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 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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