Paper · Jun 6, 2025

Variational Nonlinear and Nonlocal Curvature Flows

Daniele De Gennaro

math.APWider flowsarXiv:2506.05951
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Abstract

We prove that the minimizing movements scheme á la Almgren-Taylor-Wang converges towards level-set solutions to a nonlinear version of nonlocal curvature flows with time-depending forcing term, in the rather general framework of variational curvatures introduced in. The nonlinearity involved is assumed to satisfy minimal assumptions, namely continuity, monotonicity, and vanishing at zero. Under additional assumptions only on the curvatures involved, we establish uniqueness for level-set solutions.

Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsStability & 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 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)3Chambolle 2022 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)2Chambolle 2024 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)5Kholmatov 2024 Deforming Locally Convex Curves into Curves of Constant k-order Width (Laiyuan Gao, Horst Martini and Deyan Zhang, 2024)8Gao 2024 A mean curvature flow arising in adversarial training (Leon Bungert, Tim Laux and Kerrek Stinson, 2024)4Bungert 2024 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)7Bulanyi 2025 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)1Cagnetti 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)6Kubin 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  2. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Daniele De Gennaro

  3. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Weak & singular flows · Same author Daniele De Gennaro · Shared terms “minimizing movements”, “movement scheme”, “level set”

  4. A mean curvature flow arising in adversarial training

    Leon Bungert, Tim Laux and Kerrek StinsonWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “movement scheme”, “minimizing movements”

  5. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Other curvature flows · Shared terms “minimizing movements”

  6. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “movement scheme”, “minimizing movements”

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

    Bohdan Bulanyi and Berardo RuffiniWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “forcing term”, “depending”

  8. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature 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.

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.