Paper · Jul 8, 2022

Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

Tomoki Fujii

math.DGWider flowsarXiv:2207.03653
PDF Abstract page

As on arXiv

Abstract

In this paper, we consider a translating soliton for the inverse mean curvature flow given as a graph of a function on a domain in a unit sphere whose level sets give isoparametric foliation. First, we show that such function is given as a composition of an isoparametric function on the unit sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons for the inverse mean curvature flow.

Comments: 16 pages. arXiv admin note: substantial text overlap with arXiv:2109.14809

Mean curvature flowOther curvature flowsInverse 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 2021, 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 The streamlines of ∞-harmonic functions obey the inverse mean curvature flow (Roger Moser, 2021)4Moser 2021 Graphical translating solitons for the mean curvature flow and isoparametric functions (Tomoki Fujii, 2021)1Fujii 2021 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)5Ma 2022 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)2Neto 2023 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)3Fujii 2023 On potentials whose level sets are orbits (Philippe Bolle, Marco Mazzucchelli and Andrea Venturelli, 2024)7Bolle 2024 Global weak solutions for the inverse mean curvature flow in the Heisenberg group (Adriano Pisante and Eugenio Vecchi, 2024)8Pisante 2024 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)6Raji 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Graphical translating solitons for the mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows PDF

    Topics Self-shrinkers & translators · Same author Tomoki Fujii · Shared terms “isoparametric function”, “graphical translating”, “translating solitons”

  2. Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Inverse mean curvature flow, Self-shrinkers & translators, Other curvature flows · Shared terms “translating solitons”

  3. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows PDF

    Topics Self-shrinkers & translators · Same author Tomoki Fujii

  4. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “level set”

  5. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

    John Man Shun Ma, Yuan Shyong Ooi and Juncheol PyoWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “graphical translating”

  6. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “differential equation”, “ordinary differential”

  7. On potentials whose level sets are orbits

    Philippe Bolle, Marco Mazzucchelli and Andrea VenturelliWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “level set”

  8. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “level set”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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.