Paper · Jul 18, 2022

The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds

Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri

math.DGWider flowsarXiv:2207.08607
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Abstract

We study the asymptotic behaviour of the -capacitary potential and of the weak Inverse Mean Curvature Flow of a bounded set along the ends of an Asymptotically Conical Riemannian manifolds with asymptotically nonnegative Ricci curvature.

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compact

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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 2021, 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 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)2Benatti 2022 Isoperimetry and the properness of weak inverse mean curvature flow (Kai Xu, 2023)7Xu 2023 Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature (Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al., 2024)5Antonelli 2024 Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials (Esther Cabezas-Rivas, Salvador Moll and Marcos Solera, 2024)3Cabezas-Riva 2024 Inverse mean curvature flow with outer obstacle (Kai Xu, 2024)6Xu 2024 Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas (Luca Benatti, Alessandra Pluda and Marco Pozzetta, 2024)1Benatti 2024 Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow (Mattia Fogagnolo, Giorgio Gatti and Alessandra Pluda, 2026)4Fogagnolo 2026 The σ-inverse mean curvature flow and the generalized Penrose conjecture (Conghan Dong, 2026)8Dong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

    Luca Benatti, Alessandra Pluda and Marco PozzettaWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Luca Benatti · Shared terms “capacitary potentials”, “nonnegative Ricci”, “weak inverse”

  2. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same authors Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri · Shared terms “weak inverse”

  3. Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of -capacitary potentials

    Esther Cabezas-Rivas, Salvador Moll and Marcos SoleraWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “capacitary potentials”

  4. Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Mattia Fogagnolo · Shared terms “Riemannian manifolds”

  5. Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

    Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al.Wider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Mattia Fogagnolo · Shared terms “weak inverse”

  6. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Shared terms “weak inverse”

  7. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “weak inverse”

  8. The -inverse mean curvature flow and the generalized Penrose conjecture

    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows

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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. 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.