Paper · Jul 9, 2018

Extension of two-dimensional mean curvature flow with free boundary

Siao-Hao Guo

math.DGWider flowsarXiv:1807.02922
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Abstract

Given a mean curvature flow of compact, embedded surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

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Submitted
Jul 9, 2018
Subjects
math.DG
Mean curvature flow3-manifoldsSurfacesBoundaries & non-compactFree boundary & capillary

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 Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces (Valentina-Mira Wheeler, 2016)8Wheeler 2016 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)1Xu 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)4Wheeler 2016 Singularities of the area preserving curve shortening flow with a Neumann free boundary condition (Elena Mäder-Baumdicker, 2017)3Mäder-Baumdi 2017 Mean convex mean curvature flow with free boundary (Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al., 2019)6Edelen 2019 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)7Chai 2022 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)5Bourni 2022 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)2An-Lacroix 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

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

  2. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows

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

  3. Singularities of the area preserving curve shortening flow with a Neumann free boundary condition

    Elena Mäder-BaumdickerWider flows

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

  4. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary

  5. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows

    Topics Free boundary & capillary

  6. Mean convex mean curvature flow with free boundary

    Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al.Wider flows

    Topics Free boundary & capillary · Shared terms “mean convex”

  7. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary

  8. Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

    Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.