Paper · Feb 8, 2022

Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

Jorge Lira, Fernanda Roing

math.DGWider flowsarXiv:2202.041582022
PDF Abstract page

As on arXiv

Abstract

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.

Comments: 30 pages. This paper sheds light on the relations with the paper arXiv:arch-ive/2202.02424 due to G. Gentile and B. Vertman. They study the prescribed mean curvature flow of non-compact spacelike hypersurfaces in a Generalized Robertson-Walker spacetime (GRW), whereas we study the mean curvature flow of compact hypersurfaces in a GRW with a perpendicular Neumann boundary condition

Mean curvature flowBoundaries & non-compactFree boundary & capillaryLong-time existence & convergence

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 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 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)2Jang 2021 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)7Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)5Gao 2021 Parabolic evolution with boundary to the Bishop family of holomorphic discs (Brendan Guilfoyle and Wilhelm Klingenberg, 2023)4Guilfoyle 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)6Mitake 2023 Characterizations of generalized Robertson-Walker spacetimes concerning gradient solitons (Krishnendu De, Mohammad Nazrul Islam Khan and Uday Chand De, 2024)3De 2024 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)8Eto 2024 Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions (Can Cui and Nung Kwan Yip, 2025)1Cui 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  2. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  3. Characterizations of generalized Robertson-Walker spacetimes concerning gradient solitons

    Krishnendu De, Mohammad Nazrul Islam Khan and Uday Chand De PDF

    Shared terms “Robertson Walker”, “generalized Robertson”, “Walker spacetime”

  4. Parabolic evolution with boundary to the Bishop family of holomorphic discs

    Brendan Guilfoyle and Wilhelm KlingenbergWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  5. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  6. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  7. Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  8. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”

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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.