Paper · Jun 24, 2020

The Pohozaev-Schoen Identity on Asymptotically Euclidean Manifolds: Conservation Identities and their applications

Allan Freitas, Rodrigo Ávalos

math.DGarXiv:2006.13982
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Abstract

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur-type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.

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Submitted
Jun 24, 2020
Subjects
math.DG

Comments: Comments are welcome

Ricci solitonsStability & uniquenessBoundaries & non-compact

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 A Pohozaev Identity on Warped Product Solitons (Timothy Carson, 2016)4Carson 2016 Ricci flow on asymptotically Euclidean manifolds (Yu Li, 2016)1Li 2016 The spinorial energy for asymptotically Euclidean Ricci flow (Julius Baldauf and Tristan Ozuch, 2022)2Baldauf 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)3Carron 2022 Rigidity results for shrinking and expanding Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)6Leandro 2022 Triviality Results and Conjugate Radius Estimation of Ricci Solitons (Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu, 2023)7Shaikh 2023 On steady and expanding Ricci solitons with asymptotic symmetries (Michael B. Law, 2025)8Law 2025 Hamilton's identity and rigidity of complete gradient solitons (Antonio W. Cunha, Antonio N. Silva and William Wylie, 2025)5Cunha 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 on asymptotically Euclidean manifolds

    Yu Li

    Shared terms “Euclidean manifolds”, “asymptotically Euclidean”

  2. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch

    Topics Boundaries & non-compact · Shared terms “asymptotically Euclidean”, “euclidean Ricci”, “Euclidean manifolds”

  3. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Boundaries & non-compact · Shared terms “Euclidean manifolds”, “asymptotically Euclidean”

  4. A Pohozaev Identity on Warped Product Solitons

    Timothy Carson

    Topics Ricci solitons · Shared terms “Pohozaev”, “identity”

  5. Hamilton's identity and rigidity of complete gradient solitons

    Antonio W. Cunha, Antonio N. Silva and William Wylie

    Topics Stability & uniqueness, Ricci solitons · Shared terms “identity”

  6. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons

  7. Triviality Results and Conjugate Radius Estimation of Ricci Solitons

    Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “restriction”

  8. On steady and expanding Ricci solitons with asymptotic symmetries

    Michael B. Law

    Topics Boundaries & non-compact, Stability & uniqueness, 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.