Paper · May 1, 2019

On the -flow by -Laplace approximation: new estimates via fake distances under Ricci lower bounds

Luciano Mari, Marco Rigoli, Alberto Giulio Setti

math.DGWider flowsarXiv:1905.00216v6
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Abstract

In this paper we show the existence of weak solutions of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the -Laplace equation, and relies on new gradient and decay estimates for -harmonic capacity potentials, notably for the kernel of . These bounds, stable as , are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of .

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Submitted
May 1, 2019
Last revised
Jun 5, 2023 (v6)
Subjects
math.DG, math.AP
Journal
Amer. J. Math. 144, Number 3 (2022), 779-849. Corrigendum on Amer. J. Math 145, Number 3 (2023), 667-671
DOI
10.1353/ajm.2022.0016

Comments: 62 pages, new version. We correct a mistake in our proof of Lemma 2.17. Although we have to strengthen the assumptions therein and, accordingly, in Theorem 2.22, all of our results on the existence and properties of the IMCF are not affected. Minor changes, with no influence elsewhere in the paper, regard Lemma 3.3, Proposition 4.3 and Lemma 5.3

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsLimit spaces & RCDGeometric inequalities

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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 Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (Fangcheng Guo, Guanghan Li and Chuanxi Wu, 2016)1Guo 2016 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)7Colombo 2018 Recent rigidity results for graphs with prescribed mean curvature (Bruno Bianchini, Giulio Colombo, Marco Magliaro et al., 2020)5Bianchini 2020 The streamlines of ∞-harmonic functions obey the inverse mean curvature flow (Roger Moser, 2021)3Moser 2021 Horizontal inverse mean curvature flow in the Heisenberg group (Jingshi Cui and Peibiao Zhao, 2023)8Cui 2023 Global weak solutions for the inverse mean curvature flow in the Heisenberg group (Adriano Pisante and Eugenio Vecchi, 2024)2Pisante 2024 Sharp p-Capacity Estimates via Quermassintegrals in Hyperbolic Space (Xiaoshang Jin, Yao Wan and Jie Xiao, 2026)4Jin 2026 Local regularity and p-harmonic approximation for weak inverse mean curvature flow (Wei-Chun Chen and Chun-Kai Lien, 2026)6Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

    Fangcheng Guo, Guanghan Li and Chuanxi WuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Laplace”, “isoperimetric”, “lower bound”

  2. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “distance associated”, “Moser”, “level set”

  3. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “level set”, “approximation”

  4. Sharp -Capacity Estimates via Quermassintegrals in Hyperbolic Space

    Xiaoshang Jin, Yao Wan and Jie XiaoWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “capacity”

  5. Recent rigidity results for graphs with prescribed mean curvature

    Bruno Bianchini, Giulio Colombo, Marco Magliaro et al.Wider flows

    Topics Mean curvature flow · Same authors Luciano Mari, Marco Rigoli

  6. Local regularity and p-harmonic approximation for weak inverse mean curvature flow

    Wei-Chun Chen and Chun-Kai LienWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “approximation”, “Moser”

  7. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Mean curvature flow · Same authors Luciano Mari, Marco Rigoli

  8. Horizontal inverse mean curvature flow in the Heisenberg group

    Jingshi Cui and Peibiao ZhaoWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Weak & singular flows · Shared terms “Moser”, “level set”

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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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.