Monotone Quantities for p-Harmonic functions and the Sharp p-Penrose inequality

Liam Mazurowski; Xuan Yao

Paper · May 31, 2023

Monotone Quantities for $p$ -Harmonic functions and the Sharp $p$ -Penrose inequality

Liam Mazurowski, Xuan Yao

math.DGWider flowsarXiv:2305.19784v4

View PDF View HTML arXiv page

As on arXiv

Abstract

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $Σ$ . Fix a number $1 < p < 3$ . We derive monotone quantities for $p$ -harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$ -capacity of $Σ$ in $M$ , which was first proved by Xiao using weak inverse mean curvature flow.

View PDF View HTML arXiv page

Submitted: May 31, 2023
Last revised: Jan 19, 2024 (v4)
Subjects: math.DG, math-ph, math.CA

Comments: 22 pages, comments are welcome!

Mean curvature flow Other curvature flows Inverse mean curvature flow 3-manifolds Scalar curvature Boundaries & non-compact Geometric inequalities Physics

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 2021, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

On the Isoperimetric Riemannian Penrose Inequality
Luca Benatti, Mattia Fogagnolo and Lorenzo Mazzieri2022Wider flows

Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “weak inverse”, “asymptotically flat”
The Penrose inequality with charge for 2-convex initial data sets
Tuan Dolmen2026Wider flows

Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “monotonicity formula”, “ADM mass”
The Riemannian Penrose Inequality with Matter Density
Hubert Bray and Yiyue Zhang2024Wider flows

Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”
The $σ$ -inverse mean curvature flow and the generalized Penrose conjecture
Conghan Dong2026Wider flows

Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “complete asymptotically”, “ADM mass”
On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes
Brian Harvie and Ye-Kai Wang2025Wider flows

Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “monotone quantities”, “Penrose inequality”, “Schwarzschild”
Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds
Liam Mazurowski and Jintian Zhu2025Wider flows

Topics Inverse mean curvature flow, Scalar curvature, Boundaries & non-compact · Same author Liam Mazurowski · Shared terms “asymptotically flat”
Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas
Luca Benatti, Alessandra Pluda and Marco Pozzetta2024Wider flows

Topics Inverse mean curvature flow, Physics, Scalar curvature · Shared terms “monotone quantities”, “monotonicity formula”, “weak inverse”
Sharp $p$ -Capacity Estimates via Quermassintegrals in Hyperbolic Space
Xiaoshang Jin, Yao Wan and Jie Xiao2026Wider flows

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

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.
10A.4
Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization
Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

[1] 10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

[2] 10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

[3] 10A.4
Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization
Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

Monotone Quantities for $p$ -Harmonic functions and the Sharp $p$ -Penrose inequality

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters