Paper · Oct 14, 2026

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

A. Cesaroni, M. Novaga, P. Pozzi

math.DGWider flowsarXiv:2610.20054v2
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Abstract

The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group , and prove the existence of a flat flow which is locally -Hölder continuous in time, with respect to the -distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.

Comments: 35 pages

Mean curvature flowWeak & singular flows

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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 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)2Cesaroni 2022 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)7Chambolle 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)5Julin 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)8Gößwein 2023 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)6Bellettini 2024 Global weak solutions for the inverse mean curvature flow in the Heisenberg group (Adriano Pisante and Eugenio Vecchi, 2024)4Pisante 2024 Mean-convex horizontal mean curvature flow in the Heisenberg group (Gaia Bombardieri, Mattia Fogagnolo and Valentina Franceschi, 2026)1Bombardieri 2026
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  1. Mean-convex horizontal mean curvature flow in the Heisenberg group

    Gaia Bombardieri, Mattia Fogagnolo and Valentina FranceschiWider flows PDF

    Topics Weak & singular flows · Shared terms “horizontal mean”, “Heisenberg group”

  2. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

    Topics Mean curvature flow · Same authors A. Cesaroni, M. Novaga

  3. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows PDF

    Topics Mean curvature flow · Same authors A. Cesaroni, M. Novaga

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

    Adriano Pisante and Eugenio VecchiWider flows PDF

    Topics Weak & singular flows · Shared terms “Heisenberg group”, “sub Riemannian”

  5. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Weak & singular flows · Shared terms “discrete scheme”, “flat flow”, “movement scheme”

  6. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “movement scheme”, “geometric evolution”

  7. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

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

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

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Same authors M. Novaga, P. Pozzi

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) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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

  2. 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.