Paper · Feb 4, 2022

Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

Giuseppe Gentile, Boris Vertman

math.DGWider flowsarXiv:2202.02424
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As on arXiv

Abstract

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.

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Submitted
Feb 4, 2022
Subjects
math.DG, math.AP

Comments: 51 pages

Mean curvature flowSingularitiesBoundaries & non-compactRotational symmetry & warped productsPhysics

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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 Parallel spinor flows on three-dimensional Cauchy hypersurfaces (Ángel Murcia and C. S. Shahbazi, 2021)6Murcia 2021 New Examples of Translating Solitons in Generalised Robertson-Walker Geometries (Diego Artacho, Marie-Amélie Lawn and Miguel Ortega, 2022)5Artacho 2022 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)4Bahuaud 2023 Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds (Luen-Fai Tam, 2024)1Tam 2024 A CMC existence result for expanding cosmological spacetimes (Gregory J. Galloway and Eric Ling, 2024)3Galloway 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)2Fujihara 2024 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)7Gomes 2026 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II (Gang Li, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

    Luen-Fai TamWider flows

    Topics Physics, Boundaries & non-compact · Shared terms “prescribed mean”, “compact spaces”

  2. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Rotational symmetry & warped products, Singularities · Shared terms “infinite time”

  3. A CMC existence result for expanding cosmological spacetimes

    Gregory J. Galloway and Eric LingWider flows

    Topics Physics · Shared terms “Cauchy”, “globally hyperbolic”

  4. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al.

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Shared terms “bounded geometry”

  5. New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

    Diego Artacho, Marie-Amélie Lawn and Miguel OrtegaWider flows

    Topics Physics · Shared terms “Robertson Walker”, “Lorentzian”

  6. Parallel spinor flows on three-dimensional Cauchy hypersurfaces

    Ángel Murcia and C. S. Shahbazi

    Topics Physics · Shared terms “Cauchy hypersurface”, “hyperbolic Lorentzian”, “globally hyperbolic”

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

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

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

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

    Gang Li

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

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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