Math 2.0

Differential geometry Geometric evolution equations Theorem

Riemannian Penrose inequality

Builds on Riemannian manifoldScalar curvatureMinimal surfaceInverse mean curvature flowSchwarzschild metricPositive mass theorem

Statement

Let (M,g)(M, g) be a complete, asymptotically flat, three-dimensional Riemannian manifold with nonnegative scalar curvature. Assume that the boundary ∂M\partial M is an outermost minimal surface, possibly with several components. This means that ∂M\partial M is a closed minimal surface and that no other closed minimal surface in MM encloses it. Let AA be the total area of ∂M\partial M and let mm be the ADM mass of (M,g)(M, g). Then m≥A16π.m \ge \sqrt{\frac{A}{16\pi}}. Equality holds if and only if (M,g)(M, g) is isometric to the spatial Schwarzschild metric of mass mm outside its horizon.

Remarks

Huisken and Ilmanen proved the inequality in 2001 when AA is the area of a largest component of ∂M\partial M, using a weak form of inverse mean curvature flow. Along this flow the Hawking mass of the evolving surface does not decrease, a monotonicity first observed by Geroch. The Hawking mass equals A/16π\sqrt{A/16\pi} on the horizon and tends to at most mm at infinity. Bray proved the version with total area in 2001 by a conformal flow of metrics. For the spatial Schwarzschild metric of mass mm, the horizon is the sphere r=2mr = 2m of area 16πm216\pi m^2, so both sides equal mm. Nonnegative scalar curvature is needed: without it the mass can even be negative.

Going deeper

For a student

The statement is the time-symmetric case of a conjecture of Penrose; the scalar curvature condition encodes nonnegative energy density. It strengthens the positive mass theorem, which gives m≥0m \ge 0 when there is no horizon. The two proofs use different flows: inverse mean curvature flow of surfaces, and a flow of conformal metrics. The general case of the Penrose conjecture, without time symmetry, is still open.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Inverse mean curvature flow, concept, no evidence levelInverse mean curvatur…conceptIsometry of Riemannian manifolds, concept, no evidence levelIsometry of Riemannia…conceptMinimal surface, concept, no evidence levelMinimal surfaceconceptRicci curvature, concept, no evidence levelRicci curvatureconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptScalar curvature, concept, no evidence levelScalar curvatureconceptSchwarzschild metric, example, no evidence levelSchwarzschild metricexamplePositive mass theorem, statement, no evidence levelPositive mass theoremstatementRiemannian Penrose inequality, statement, no evidence levelRiemannian Penrose in…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Inverse mean curvature flow concept
  2. Isometry of Riemannian manifolds concept
  3. Minimal surface concept
  4. Ricci curvature concept
  5. Riemannian manifold concept
  6. Scalar curvature concept
  7. Schwarzschild metric example
  8. Positive mass theorem statement
  9. Riemannian Penrose inequality statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemannian_Penrose_inequality
About this page
TagQ5EWVTA2RW
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Kindstatement, theorem
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MSC53C21, 83C57
StatusThis copy of the site has no build data. Run m2 status Q5EWVTA2RW.
AboutScalar curvature
UsesRiemannian manifold, Scalar curvature, Minimal surface, Inverse mean curvature flow, Schwarzschild metric, Positive mass theorem
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Changed2 November 2026

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