Paper · Oct 21, 2022

Hawking Mass Monotonicity for Initial Data Sets

Sven Hirsch

math.DGWider flowsarXiv:2210.12237v4
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Abstract

We introduce new systems of PDE on initial data sets whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications.

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Submitted
Oct 21, 2022
Last revised
Oct 30, 2025 (v4)
Subjects
math.DG, gr-qc, math.AP

Comments: To appear in Asian J. Math

Mean curvature flowOther curvature flowsInverse mean curvature flowPhysics

Its neighbourhood

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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 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 evolution of hypersurfaces along their inverse spacetime mean curvature (Gerhard Huisken and Markus Wolff, 2022)6Huisken 2022 On the Isoperimetric Riemannian Penrose Inequality (Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri, 2022)8Benatti 2022 Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas (Luca Benatti, Alessandra Pluda and Marco Pozzetta, 2024)5Benatti 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)2Dong 2026 Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets (Jacopo Tenan, 2026)7Tenan 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)3Dolmen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Penrose conjecture for initial data sets satisfying a -convexity condition

    Conghan DongWider flows

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

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

    Conghan DongWider flows

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

  3. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “data set”, “Hawking mass”, “initial data”

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

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows

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

  5. Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

    Luca Benatti, Alessandra Pluda and Marco PozzettaWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Hawking mass”, “monotonicity formula”

  6. On the evolution of hypersurfaces along their inverse spacetime mean curvature

    Gerhard Huisken and Markus WolffWider flows

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

  7. Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets

    Jacopo TenanWider flows

    Topics Physics · Shared terms “data set”, “initial data”, “foliation”

  8. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

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

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. 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.