Paper · May 14, 2018

On the existence of translating solutions of mean curvature flow in slab regions

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

math.DGWider flowsarXiv:1805.05173v3
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Abstract

We prove, in all dimensions , that there exists a convex translator lying in a slab of width in (and in no smaller slab) if and only if . We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics and reflection symmetry of translators lying in slab regions.

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Submitted
May 14, 2018
Last revised
Jun 12, 2018 (v3)
Subjects
math.DG

Comments: We have made some updates to the introduction. The mathematical content is unchanged from version 2

Mean curvature flowSelf-shrinkers & translators

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 Type-II singularities of two-convex immersed mean curvature flow (Theodora Bourni and Mat Langford, 2016)2Bourni 2016 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)1Bourni 2017 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)6Bourni 2019 Convex ancient solutions to mean curvature flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)4Bourni 2019 The atomic structure of ancient grain boundaries (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2020)3Bourni 2020 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)7Bourni 2021 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)8Hoffman 2023 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)5Bourni 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Mean curvature flow · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “smaller slab”, “asymptotics”

  2. Type-II singularities of two-convex immersed mean curvature flow

    Theodora Bourni and Mat LangfordWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford · Shared terms “translating solutions”, “convex translator”

  3. The atomic structure of ancient grain boundaries

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “slab regions”, “translating solutions”, “reflection”

  4. Convex ancient solutions to mean curvature flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex translator”

  5. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Self-shrinkers & translators · Same author Theodora Bourni · Shared terms “translating solutions”, “appropriate”

  6. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia

  7. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Mean curvature flow · Same authors Theodora Bourni, Mat Langford

  8. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Self-shrinkers & translators · Shared terms “reflection”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.