Paper · Nov 10, 2017

Uniformly compressing mean curvature flow

Wenhui Shi, Dmitry Vorotnikov

math.APWider flowsarXiv:1711.03864v2
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Abstract

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto's Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are however restricted to the 1D case of evolution of loops.

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Submitted
Nov 10, 2017
Last revised
Sep 17, 2018 (v2)
Subjects
math.AP
Mean curvature flowStability & uniqueness

Its neighbourhood

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)3Baspinar 2016 Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)4Baspinar 2016 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)5Kopfer 2016 Is Mean Curvature Flow a Gradient Flow? (Zhonggan Huang, 2022)1Huang 2022 The Wasserstein distance for Ricci shrinkers (Franciele Conrado and Detang Zhou, 2023)2Conrado 2023 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)7Li 2025 Local well-posedness of the skew mean curvature flow for large data (Jiaxi Huang and Daniel Tataru, 2025)8Huang 2025 Well-posedness of mean curvature flow (Yongheng Han, 2026)6Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Is Mean Curvature Flow a Gradient Flow?

    Zhonggan HuangWider flows

    Topics Mean curvature flow · Shared terms “gradient flow”, “Riemannian structure”, “Michor”

  2. The Wasserstein distance for Ricci shrinkers

    Franciele Conrado and Detang Zhou

    Topics Stability & uniqueness · Shared terms “probability measures”, “Wasserstein”

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

    Emre Baspinar and Giovanna CittiWider flows

    Topics Stability & uniqueness · Shared terms “Riemannian structure”

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

    Emre Baspinar and Giovanna CittiWider flows

    Topics Stability & uniqueness · Shared terms “Riemannian structure”

  5. Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

    Eva Kopfer and Karl-Theodor Sturm

    Topics Stability & uniqueness · Shared terms “Wasserstein space”, “probability measures”

  6. Well-posedness of mean curvature flow

    Yongheng HanWider flows

    Topics Stability & uniqueness · Shared terms “posedness”

  7. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows

    Topics Mean curvature flow · Shared terms “gradient flow”

  8. Local well-posedness of the skew mean curvature flow for large data

    Jiaxi Huang and Daniel TataruWider flows

    Topics Stability & uniqueness · Shared terms “posedness”

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) 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)

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.