Riemannian Penrose inequality in all dimensions

Abstract: We prove the Riemannian Penrose inequality in arbitrary dimension for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary, where the boundary is allowed to have a singular set of Hausdorff dimension at most (n-8). Moreover, the equality holds exactly when the manifold is isometric to the Riemannian Schwarzschild exteriors. Our proof extends Bray's conformal-flow method to higher dimensions, where the outer-minimizing enclosures along the flow may be singular.

• The paper extends the Riemannian Penrose inequality to arbitrary dimensions, proving rigidity by showing equality only for the Schwarzschild exterior.
• It employs a conformal flow method and mass-capacity inequalities to manage singular, almost-minimizing boundaries and ensure ADM mass monotonicity.
• The results affirm key aspects of cosmic censorship in higher dimensions and provide new tools for geometric analysis in mathematical relativity.

Riemannian Penrose Inequality in Arbitrary Dimensions

Introduction and Context

The paper "Riemannian Penrose inequality in all dimensions" (2605.00680) establishes the general validity of the Riemannian Penrose inequality for smooth, complete, asymptotically flat manifolds of dimension $n \geq 3$ with nonnegative scalar curvature, compact outer-minimizing minimal boundary, and boundary singular sets of Hausdorff dimension at most $n-8$. The Penrose inequality relates the ADM mass $m_{\mathrm{ADM}}(M, g)$ to the area of the outer-minimizing minimal boundary $\Sigma$ via

$$m_{\mathrm{ADM}}(M,g) \geq \frac{1}{2} \left( \frac{\mathcal{H}_g^{n-1}(\Sigma)}{n\omega_n} \right)^{\frac{n-2}{n-1}}$$

with equality if and only if $(M, g)$ is a Riemannian Schwarzschild exterior. The result extends previous proofs (Bray, Huisken-Ilmanen, Bray-Lee) beyond $n \leq 7$, overcoming singularities in higher-dimensional outer-minimizing enclosures and leveraging developments in regularity and positive mass theorem proofs.

Conformal Flow Methodology and Generalization

The proof adapts Bray's conformal flow method by constructing a one-parameter family of metrics $g_t$ and hypersurfaces $\Sigma_t$, where $\Sigma_t$ is always the outermost $n-8$0-minimizing enclosure of the initial boundary $n-8$1. The regularity theory ensures $n-8$2 admits a decomposition $n-8$3, with the singular part $n-8$4 of Hausdorff dimension at most $n-8$5 and $n-8$6 a smooth minimal hypersurface.

The method requires:

• Existence and control of the flow in all dimensions, including treating singular boundaries via almost-minimizing properties and Almgren-Bombieri-Tamanini regularity results.
• Use of a weak inverse mean curvature flow where necessary to deal with non-smooth enclosures.
• Extension and regularity of harmonic functions defining the conformal flow, with boundary Hölder estimates to guarantee compactness.

Mass Monotonicity and Mass-Capacity Inequality

The paper proves that the ADM mass is monotone non-increasing along the flow, using a mass-capacity inequality holding for singular boundaries. The following holds for almost-minimizing boundaries (with regular part smooth and minimal and singular set $n-8$7):

$n-8$8

where $n-8$9 is the $m_{\mathrm{ADM}}(M, g)$0-capacity of $m_{\mathrm{ADM}}(M, g)$1. The monotonicity follows from:

• Harmonic conformal perturbations to produce a strictly separated boundary whose regular part is mean-convex.
• Mean-convex smoothing (Gromov's construction) and reduction to smooth mean-convex boundary cases.
• Application of the positive mass theorem for smooth corners and doubling/trick arguments.

The rigidity in the mass-capacity inequality is established: equality implies $m_{\mathrm{ADM}}(M, g)$2 is isometric to the Riemannian Schwarzschild exterior.

Flow Convergence and Rigidity

The normalized conformal flow converges, under a harmonically flat asymptotic end, to the Schwarzschild metric:

$m_{\mathrm{ADM}}(M, g)$3

with horizon area matching the initial minimal boundary area. Smooth convergence of hypersurfaces to the round sphere is obtained via area bounds, regularity, and harmonic capacity estimates. Numerical results are explicit: horizon area $m_{\mathrm{ADM}}(M, g)$4 and ADM mass $m_{\mathrm{ADM}}(M, g)$5 are related as

$m_{\mathrm{ADM}}(M, g)$6

with rigorous control on convergence and rigidity.

Implications and Future Directions

The established validity of the Riemannian Penrose inequality in arbitrary dimensions has deep implications:

• Completes a central geometric conjecture relating gravitation and geometry, underpinning the cosmic censorship hypothesis.
• Extends the positive mass theorem framework to include all dimensions and singular minimal boundaries, paving the way for further applications in geometric analysis and mathematical relativity.
• Provides technical tools (regularity for almost-minimizing boundaries, mass-capacity inequalities with singularities) applicable to minimal surface theory, geometric measure theory, and higher-dimensional general relativity.
• Future developments may include potential generalizations to non-asymptotically flat or time-dependent settings, sharper rigidity classification, and interaction with quantum corrections.

Conclusion

The Riemannian Penrose inequality is proven for all dimensions $m_{\mathrm{ADM}}(M, g)$7 under minimal regularity conditions on the boundary, leveraging modern regularity and positive mass theorem advances. The essential conformal flow methodology, mass monotonicity, and regularity theory are extended to handle singularities, culminating in a sharp, rigid inequality with equality solely for Schwarzschild exteriors. This resolves a longstanding open question in geometric analysis and mathematical relativity, widening the foundational landscape for further explorations of mass, area, and curvature in high-dimensional spaces.