Paper · May 21, 2020

Asymptotics for optimal controls for horizontal mean curvature flow

Nicolas Dirr, Federica Dragoni, Raffaele Grande

math.APWider flowsarXiv:2005.10599
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Abstract

The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situation a new type of singularities absent for the Euclidean mean curvature flow occurs, the so-called characteristic points. This paper investigates the asymptotic behaviour of the regularised optimal controls in the vicinity of such characteristic points.

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Submitted
May 21, 2020
Subjects
math.AP, math.OC
Mean curvature flowSingularities

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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 Uniqueness of vanishing viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)7Baspinar 2016 Local non-collapsing of volume for the Lagrangian mean curvature flow (Knut Smoczyk, 2018)5Smoczyk 2018 Precise asymptotics near a generic S¹×ℝ³ singularity of mean curvature flow (Zhou Gang and Shengwen Wang, 2019)6Gang 2019 A stochastic representation for the solution of approximated mean curvature flow (Raffaele Grande, 2021)3Grande 2021 A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow (Antoni Kijowski, Qing Liu, Ye Zhang et al., 2023)8Kijowski 2023 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)1Citti 2024 A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group (A. Cesaroni, M. Novaga and P. Pozzi, 2026)4Cesaroni 2026 Mean-convex horizontal mean curvature flow in the Heisenberg group (Gaia Bombardieri, Mattia Fogagnolo and Valentina Franceschi, 2026)2Bombardieri 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows

    Topics Mean curvature flow · Same authors Nicolas Dirr, Federica Dragoni, Raffaele Grande · Shared terms “horizontal mean”, “Heisenberg group”

  2. Mean-convex horizontal mean curvature flow in the Heisenberg group

    Gaia Bombardieri, Mattia Fogagnolo and Valentina FranceschiWider flows

    Topics Mean curvature flow · Shared terms “horizontal mean”, “characteristic points”, “Heisenberg group”

  3. A stochastic representation for the solution of approximated mean curvature flow

    Raffaele GrandeWider flows

    Topics Mean curvature flow · Same author Raffaele Grande · Shared terms “horizontal mean”, “stochastic”

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

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

    Topics Mean curvature flow · Shared terms “horizontal mean”, “Heisenberg group”

  5. Local non-collapsing of volume for the Lagrangian mean curvature flow

    Knut SmoczykWider flows

    Topics Mean curvature flow · Shared terms “optimal control”

  6. Precise asymptotics near a generic singularity of mean curvature flow

    Zhou Gang and Shengwen WangWider flows

    Topics Singularities · Shared terms “asymptotics”

  7. Uniqueness of vanishing viscosity mean curvature flow solution in two sub-Riemannian structures

    Emre Baspinar and Giovanna CittiWider flows

    Topics Mean curvature flow · Shared terms “characteristic points”, “Heisenberg group”

  8. A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

    Antoni Kijowski, Qing Liu, Ye Zhang et al.Wider flows

    Shared terms “horizontal”, “Heisenberg group”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.