Paper · Jan 16, 2019

Ancient mean curvature flows and their spacetime tracks

Francesco Chini, Niels Martin Møller

math.DGWider flowsarXiv:1901.05481v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We study properly immersed ancient solutions of the codimension one mean curvature flow in -dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic Omori-Yau maximum principle for ancient mean curvature flows.

View PDF View HTML arXiv page
Submitted
Jan 16, 2019
Last revised
Feb 26, 2019 (v2)
Subjects
math.DG, math.AP

Comments: 14 pages; minor edits, added proof details and references

Mean curvature flowAncient solutions

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 Parabolic Omori-Yau maximum principle for mean curvature flow and some applications (John Man Shun Ma, 2017)5Ma 2017 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)4Bourni 2017 Bi-Halfspace and Convex Hull Theorems for Translating Solitons (Francesco Chini and Niels Martin Møller, 2018)1Chini 2018 Construction of High Codimension Ancient Mean Curvature Flows (Douglas Stryker and Ao Sun, 2019)3Stryker 2019 Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -∞ (Douglas Stryker and Ao Sun, 2019)8Stryker 2019 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)6Bourni 2019 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)2Bourni 2021 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)7Han 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bi-Halfspace and Convex Hull Theorems for Translating Solitons

    Francesco Chini and Niels Martin MøllerWider flows

    Topics Mean curvature flow · Same authors Francesco Chini, Niels Martin Møller · Shared terms “halfspace”, “convex hull”, “properly immersed”

  2. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Ancient solutions · Shared terms “convex ancient”

  3. Construction of High Codimension Ancient Mean Curvature Flows

    Douglas Stryker and Ao SunWider flows

    Topics Ancient solutions · Shared terms “Euclidean space”

  4. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions · Shared terms “convex ancient”

  5. Parabolic Omori-Yau maximum principle for mean curvature flow and some applications

    John Man Shun MaWider flows

    Topics Mean curvature flow · Shared terms “parabolic Omori”, “Omori Yau”, “Yau maximum”

  6. On the construction of closed nonconvex nonsoliton ancient mean curvature flows

    Theodora Bourni, Mathew Langford and Alexander MramorWider flows

    Topics Ancient solutions

  7. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Ancient solutions

  8. Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at

    Douglas Stryker and Ao SunWider flows

    Topics Ancient solutions

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) 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)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.