Paper · Jul 26, 2021

Graphical translators for anisotropic and crystalline mean curvature flow

Annalisa Cesaroni, Heiko Kroener, Matteo Novaga

math.APWider flowsarXiv:2107.12082
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Abstract

In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features.

Comments: 15 pages

Mean curvature flowAnisotropic & nonlocal flowsSelf-shrinkers & translatorsStability & uniqueness

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 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)1Cesaroni 2022 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)3Ma 2022 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)5Gößwein 2023 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)7Chambolle 2024 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)8Bulanyi 2025 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)4Cagnetti 2025 Dynamical stability of various convex graphical translators (Junyoung Park, 2026)2Park 2026 A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group (A. Cesaroni, M. Novaga and P. Pozzi, 2026)6Cesaroni 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga

  2. Dynamical stability of various convex graphical translators

    Junyoung ParkWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”

  3. 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, Stability & uniqueness · Shared terms “graphical translators”

  4. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline mean”

  5. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same author Matteo Novaga

  6. A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group

    A. Cesaroni, M. Novaga and P. PozziWider flows PDF

    Topics Mean curvature flow · Same authors Annalisa Cesaroni, Matteo Novaga

  7. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline”, “anisotropy”

  8. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows PDF

    Topics Anisotropic & nonlocal flows

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.