Paper · Oct 19, 2016

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

Emre Baspinar, Giovanna Citti

math.APWider flowsarXiv:1610.06031v3
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Abstract

Here we provide a uniqueness result for viscosity solutions to sub-Riemannian mean curvature flow. In this setting the uniqueness cannot be deduced via comparison principle, which is known only for graphs and for radially symmetric surfaces, due to the presence of characteristic points. Here we prove that any viscosity solution is limit of a sequence of solutions of Riemannian flow, and obtain as a consequence uniqueness and comparison principle already known in the approximating riemannian setting. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and Carnot groups of step 2, which are particularly important due to their relation to surface completion problem of model of the visual cortex.

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Submitted
Oct 19, 2016
Last revised
Jun 4, 2018 (v3)
Subjects
math.AP
DOI
10.1137/17M1150797

Comments: arXiv admin note: substantial text overlap with arXiv:1608.00761

Mean curvature flow3-manifoldsWeak & singular flowsStability & uniquenessRotational symmetry & warped products

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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 level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)5Giga 2016 A diffusion driven curvature flow (Emre Baspinar and Giovanna Citti, 2016)2Baspinar 2016 Uniqueness of vanishing viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)1Baspinar 2016 On the rotational symmetry of 3-dimensional κ-solutions (Richard H. Bamler and Bruce Kleiner, 2019)6Bamler 2019 Rotational symmetry of solutions of mean curvature flow coming out of a double cone (Letian Chen, 2021)7Chen 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)8Chen 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)4Laux 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of vanishing 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”, “Riemannian mean”

  4. Equivalence of weak solution concepts for mean curvature flow

    Tim Laux and Anton UllrichWider flows

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

  5. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

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

  6. On the rotational symmetry of 3-dimensional -solutions

    Richard H. Bamler and Bruce Kleiner

    Topics Rotational symmetry & warped products, 3-manifolds, Weak & singular flows

  7. Rotational symmetry of solutions of mean curvature flow coming out of a double cone

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows

  8. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular 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.

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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