Paper · May 13, 2021

A stochastic representation for the solution of approximated mean curvature flow

Raffaele Grande

math.APWider flowsarXiv:2105.06393
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Abstract

The evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with application in IT and neurogeometry (see Citti-Franceschiello-Sanguinetti-Sarti, 2016). Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we define a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr-Dragoni-von Renesse, 2010.

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Submitted
May 13, 2021
Subjects
math.AP
Mean curvature flow

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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)6Baspinar 2016 Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)5Baspinar 2016 Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow (Yoshikazu Giga and Norbert Požár, 2017)7Giga 2017 Asymptotics for optimal controls for horizontal mean curvature flow (Nicolas Dirr, Federica Dragoni and Raffaele Grande, 2020)2Dirr 2020 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)8Bahadursingh 2024 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)3Bombardieri 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 author Raffaele Grande · Shared terms “horizontal mean”, “Citti”, “Sarti”

  2. Asymptotics for optimal controls for horizontal mean curvature flow

    Nicolas Dirr, Federica Dragoni and Raffaele GrandeWider flows

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

  3. 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”, “Riemannian mean”, “approximation”

  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”, “sub Riemannian”

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

    Emre Baspinar and Giovanna CittiWider flows

    Topics Mean curvature flow · Shared terms “sub Riemannian”, “Riemannian mean”, “Riemannian setting”

  6. 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 “sub Riemannian”, “Riemannian mean”, “viscosity”

  7. Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Mean curvature flow · Shared terms “approximated”, “approximation”, “viscosity”

  8. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows

    Topics Mean curvature flow · Shared terms “stochastic”, “differential equation”

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)

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.