Paper · Aug 2, 2016

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

Emre Baspinar, Giovanna Citti

math.APWider flowsarXiv:1608.00761v2
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Abstract

Here we provide uniqueness of vanishing viscosity solutions to sub-Riemannian mean curvature flow problem, which was known only far from characteristic points or under special symmetry condition. We employ vanishing viscosity approach and look for solutions as limit of solutions to approximating flow, which is well defined also at characteristic points, and estimate the rate of convergence of the approximating solutions. The results are provided in the settings of both 3-dimensional rototranslation group and Heisenberg group and they are particularly important due to their relation to surface completion problem of model of the visual cortex.

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Submitted
Aug 2, 2016
Last revised
Jul 31, 2018 (v2)
Subjects
math.AP

Comments: This paper is an older version of the paper arXiv:1610.06031 ("Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures"). Please consider arXiv:1610.06031 as the last version

Mean curvature flow3-manifoldsWeak & singular flowsStability & uniqueness

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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 A diffusion driven curvature flow (Emre Baspinar and Giovanna Citti, 2016)2Baspinar 2016 Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)1Baspinar 2016 Invariant translators of the Heisenberg group (Giuseppe Pipoli, 2018)7Pipoli 2018 Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (Yoshiki Jikumaru and Miyuki Koiso, 2019)8Jikumaru 2019 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)5Deruelle 2022 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)3Citti 2024 Equivalence of weak solution concepts for mean curvature flow (Tim Laux and Anton Ullrich, 2025)6Laux 2025 Mean-convex horizontal mean curvature flow in the Heisenberg group (Gaia Bombardieri, Mattia Fogagnolo and Valentina Franceschi, 2026)4Bombardieri 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures

    Emre Baspinar and Giovanna CittiWider flows

    Topics 3-manifolds, Weak & singular flows, Stability & uniqueness · Same authors Emre Baspinar, Giovanna Citti · Shared terms “viscosity mean”, “Riemannian structure”, “sub Riemannian”

  2. A diffusion driven curvature flow

    Emre Baspinar and Giovanna CittiWider flows

    Topics Weak & singular flows · Same authors Emre Baspinar, Giovanna Citti · Shared terms “visual cortex”

  3. 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 Giovanna Citti · Shared terms “sub Riemannian”, “visual cortex”, “Heisenberg group”

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

    Gaia Bombardieri, Mattia Fogagnolo and Valentina FranceschiWider flows

    Topics Weak & singular flows · Shared terms “characteristic points”, “Heisenberg group”, “Riemannian mean”

  5. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics 3-manifolds, Weak & singular flows, Stability & uniqueness

  6. Equivalence of weak solution concepts for mean curvature flow

    Tim Laux and Anton UllrichWider flows

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “viscosity”

  7. Invariant translators of the Heisenberg group

    Giuseppe PipoliWider flows

    Topics 3-manifolds · Shared terms “sub Riemannian”, “Heisenberg group”

  8. Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

    Yoshiki Jikumaru and Miyuki KoisoWider flows

    Topics 3-manifolds, Weak & singular flows, Stability & uniqueness

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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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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