Paper · Sep 16, 2016

Stability of sin-cones and cosh-cylinders

Klaus Kroencke

math.DGarXiv:1609.05074v2
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As on arXiv

Abstract

This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-cones, the stability properties of such warped products are now completely determined by spectral properties of the Laplacian and the Einstein operator of the fiber. For cosh-cylinders, we are furthermore able to prove a convergence result for the Ricci flow starting in a small neighbourhood. As an interesting class of examples, we determine the stability properties of sin-cones over symmetric spaces of compact type.

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Submitted
Sep 16, 2016
Last revised
May 29, 2018 (v2)
Subjects
math.DG, math.AP
Journal
Ann. Sc. Norm. Super. Pisa, Cl. Sci. 18 (2018) no. 3, 1155-1187
DOI
10.2422/2036-2145.201610_012

Comments: 26 pages, published version

Ricci flowScalar curvatureStability & uniquenessRotational symmetry & warped productsEigenvalues & spectrum

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 On warped product gradient η-Ricci solitons (Adara M. Blaga, 2017)2Blaga 2017 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)8Colombo 2018 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)5Kroencke 2020 On noncompact warped product Ricci solitons (Valter Borges, 2022)1Borges 2022 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)6Kroencke 2023 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)3Yadav 2025 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)4Kroencke 2025 Ricci flow for the Bures–Helstrom qubit metric (Andrew Lesniewski, 2026)7Lesniewski 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

    Topics Eigenvalues & spectrum, Rotational symmetry & warped products, Stability & uniqueness · Shared terms “Laplacian”

  2. On warped product gradient -Ricci solitons

    Adara M. Blaga

    Topics Rotational symmetry & warped products, Scalar curvature · Shared terms “completely determined”, “Laplacian”

  3. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker

    Topics Rotational symmetry & warped products, Scalar curvature

  4. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Scalar curvature, Stability & uniqueness · Same author Klaus Kroencke

  5. Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Scalar curvature, Stability & uniqueness · Same author Klaus Kroencke

  6. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz

    Topics Stability & uniqueness · Same author Klaus Kroencke · Shared terms “instability”

  7. Ricci flow for the Bures–Helstrom qubit metric

    Andrew Lesniewski

    Topics Eigenvalues & spectrum, Rotational symmetry & warped products, Scalar curvature · Shared terms “spectral”, “Laplacian”

  8. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  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.