Paper · May 26, 2026

The -inverse mean curvature flow and the generalized Penrose conjecture

Conghan Dong

math.DGWider flowsarXiv:2605.26504v2
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Abstract

Let be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon satisfies . In this paper, we establish this inequality for each connected component of in the case where is proportional to the metric . Our approach is based on a new geometric evolution, which we call the -inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.

Comments: v2: some typos corrected

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

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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 On the evolution of hypersurfaces along their inverse spacetime mean curvature (Gerhard Huisken and Markus Wolff, 2022)4Huisken 2022 Hawking Mass Monotonicity for Initial Data Sets (Sven Hirsch, 2022)3Hirsch 2022 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)6Benatti 2022 Monotone Quantities for p-Harmonic functions and the Sharp p-Penrose inequality (Liam Mazurowski and Xuan Yao, 2023)5Mazurowski 2023 The Riemannian Penrose Inequality with Matter Density (Hubert Bray and Yiyue Zhang, 2024)7Bray 2024 Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow (Mattia Fogagnolo, Giorgio Gatti and Alessandra Pluda, 2026)8Fogagnolo 2026 The Penrose conjecture for initial data sets satisfying a 2-convexity condition (Conghan Dong, 2026)1Dong 2026 The Penrose inequality with charge for 2-convex initial data sets (Tuan Dolmen, 2026)2Dolmen 2026
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    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Same author Conghan Dong · Shared terms “Penrose conjecture”, “set satisfying”, “data set”

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    Tuan DolmenWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “data set”, “apparent horizon”

  3. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows PDF

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “data set”, “set satisfying”, “dominant energy”

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    Gerhard Huisken and Markus WolffWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “apparent horizon”, “data set”, “asymptotically flat”

  5. Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality

    Liam Mazurowski and Xuan YaoWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “complete asymptotically”, “ADM mass”

  6. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “asymptotically flat”

  7. The Riemannian Penrose Inequality with Matter Density

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    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Penrose”

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

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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.