Paper · Mar 1, 2022

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Katherine Castro, César Rosales

math.DGWider flowsarXiv:2203.00301
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As on arXiv

Abstract

For a Riemannian manifold , possibly with boundary, we consider the Riemannian product with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with non-empty boundary and contained within certain regions of with suitable weights. Our results include half-space and Bernstein-type theorems in weighted cylinders. We also generalize to this setting some classical properties about the confinement of a compact minimal hypersurface to certain regions of Euclidean space according to the position of its boundary. Finally, we show interesting situations where the statements are applied, some of them in relation to the singularities of the mean curvature flow.

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Submitted
Mar 1, 2022
Subjects
math.DG

Comments: 18 pages, no figures. Final version accepted in J. Math. Anal. Appl

Mean curvature flowSingularitiesStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

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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 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)2Han 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)4Chen 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)5Chu 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)6Mramor 2024 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)3Li 2025 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)1Chen 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)8Bourni 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)7Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  2. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “minimal hypersurface”

  3. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows

    Topics Minimal & CMC surfaces · Shared terms “Bernstein type”, “type theorem”, “half space”

  4. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  5. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Topics Minimal & CMC surfaces

  6. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Minimal & CMC surfaces, Singularities · Shared terms “possibly”

  7. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness

  8. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “minimal hypersurface”

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

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  2. 6C.8
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  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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