Paper · Jan 10, 2019

Mean Curvature Flow with Boundary

Brian White

math.DGWider flowsarXiv:1901.03008v4
View PDF View HTML arXiv page

As on arXiv

Abstract

We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.

View PDF View HTML arXiv page
Submitted
Jan 10, 2019
Last revised
Aug 17, 2021 (v4)
Subjects
math.DG
Journal
Ars Inven. Anal.(2021), Paper No. 4, 43 pp

Comments: 43 pages

Mean curvature flow

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 A local regularity theorem for mean curvature flow with triple edges (Felix Schulze and Brian White, 2016)7Schulze 2016 Boundary regularity for mean curvature flows of higher codimension (Qi Ding, J. Jost and Y. L. Xin, 2017)3Ding 2017 Scherk-like Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)8Hoffman 2019 Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries (Brian White, 2021)5White 2021 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)1White 2022 Existence of BV flow via elliptic regularization (Kiichi Tashiro, 2023)2Tashiro 2023 The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow (Brian White, 2024)4White 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)6Tashiro 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows

    Topics Mean curvature flow · Same author Brian White · Shared terms “boundary moving”, “mild hypotheses”

  2. Existence of BV flow via elliptic regularization

    Kiichi TashiroWider flows

    Topics Mean curvature flow · Shared terms “elliptic regularization”

  3. Boundary regularity for mean curvature flows of higher codimension

    Qi Ding, J. Jost and Y. L. XinWider flows

    Topics Mean curvature flow · Shared terms “boundary regularity”

  4. The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow

    Brian WhiteWider flows

    Topics Mean curvature flow · Same author Brian White

  5. Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

    Brian White

    Topics Mean curvature flow · Same author Brian White

  6. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows

    Topics Mean curvature flow · Shared terms “elliptic regularization”

  7. A local regularity theorem for mean curvature flow with triple edges

    Felix Schulze and Brian WhiteWider flows

    Topics Mean curvature flow · Same author Brian White

  8. Scherk-like Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Mean curvature flow · Same author Brian White

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)

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.