Paper · Jul 9, 2019

Convex ancient solutions to mean curvature flow

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

math.DGWider flowsarXiv:1907.03932
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Abstract

X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In this expository article, we present Wang's structure theory and some of its consequences. We shall simplify some of Wang's analysis by making use of the monotonicity formula and the differential Harnack inequality, and obtain an important additional structure result by exploiting the latter. We conclude by showing that various rigidity results for convex ancient solutions and convex translators follow quite directly from the structure theory, including the new result of Corollary 8.3}. We recently provided a complete classification of convex ancient solutions to curve shortening flow by exploiting similar arguments.

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Submitted
Jul 9, 2019
Subjects
math.DG, math.AP

Comments: 24 pages

Mean curvature flowCurve shorteningAncient solutionsSelf-shrinkers & translatorsHarnack inequalitiesLimit spaces & RCDStability & uniqueness

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 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 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)6Bourni 2017 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)1Bourni 2019 Differential Harnack inequalities via Concavity of the arrival time (Theodora Bourni and Mat Langford, 2019)4Bourni 2019 The atomic structure of ancient grain boundaries (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2020)3Bourni 2020 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)2Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)7Bourni 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)5Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)8Bourni 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex ancient”, “Wang”

  2. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows

    Topics Curve shortening, Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  3. The atomic structure of ancient grain boundaries

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex ancient”

  4. Differential Harnack inequalities via Concavity of the arrival time

    Theodora Bourni and Mat LangfordWider flows

    Topics Harnack inequalities · Same authors Theodora Bourni, Mat Langford · Shared terms “differential Harnack”, “directly”

  5. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same author Theodora Bourni · Shared terms “convex ancient”

  6. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex ancient”

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

    Theodora Bourni and Mat LangfordWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  8. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows

    Topics Curve shortening, Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 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.

  3. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.