Differential geometry Curvature and topology Theorem

Positive mass theorem

Builds on Riemannian manifold Scalar curvature Isometry of Riemannian manifolds

Statement

Let $(M, g)$ be a connected complete Riemannian manifold of dimension $3$ that is asymptotically flat in the following sense. There is a compact set $C \subset M$ and a diffeomorphism from $M \setminus C$ to $\R^3$ minus a closed ball. In the coordinates $x = (x^1, x^2, x^3)$ it gives, for some $q > 1/2$ , $g_{ij} - \delta_{ij} = O(\abs{x}^{-q}), \quad \partial_k g_{ij} = O(\abs{x}^{-q-1}), \quad \partial_k \partial_l g_{ij} = O(\abs{x}^{-q-2}),$ and the scalar curvature $R$ of $g$ is integrable on $M$ . Let the ADM mass of $(M, g)$ be $m = \frac{1}{16\pi} \lim_{r \to \infty} \int_{\abs{x} = r} \sum_{i,j=1}^{3} \bigl(\partial_i g_{ij} - \partial_j g_{ii}\bigr) \frac{x^j}{r} \, dA.$ If $R \ge 0$ on $M$ , then $m \ge 0$ . Moreover, $m = 0$ if and only if $(M, g)$ is isometric to Euclidean space $\R^3$ .

Remarks

The theorem was proved by Schoen and Yau in 1979 using minimal surfaces, and by Witten in 1981 using spinors and a Weitzenböck formula for the Dirac operator. In Schoen and Yau's argument, if $m < 0$ one builds a complete stable minimal surface that is asymptotically a plane. The stability inequality, together with $R \ge 0$ and the Gauss–Bonnet theorem, gives a contradiction. Both methods extend to higher dimensions under extra hypotheses: dimension at most $7$ for minimal hypersurfaces, and spin manifolds for Witten's method.

Example: the Riemannian Schwarzschild metric $g = \bigl(1 + \frac{m}{2\abs{x}}\bigr)^4 \delta$ on $\R^3 \setminus \set{0}$ , with $m > 0$ , has scalar curvature $0$ . It has two asymptotically flat ends, as $\abs{x} \to \infty$ and as $\abs{x} \to 0$ , and each has ADM mass $m$ . The theorem extends to manifolds with several such ends: each end has nonnegative mass.

Completeness is needed: for $m < 0$ the same formula on $\set{\abs{x} > -m/2}$ is scalar-flat and asymptotically flat with mass $m < 0$ . This metric is not complete, because the conformal factor vanishes at $\abs{x} = -m/2$ , which lies at finite distance.

Going deeper

For a student

The Riemannian version above is the time-symmetric case of the positive energy theorem for initial data sets of the Einstein equations, where the condition $R \ge 0$ is the dominant energy condition. The ADM mass is read off from the decay of the metric at infinity and does not depend on the chosen asymptotic coordinates. The theorem is a basic tool in conformal geometry: Schoen used it to finish the solution of the Yamabe problem. The Riemannian Penrose inequality is a sharper version that bounds the mass below by the area of a horizon.

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Isometry of Riemannian manifolds concept
Riemannian manifold concept
Scalar curvature concept
Positive mass theorem statement

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Riemannian Penrose inequality

Tag	`PYRPXHNQTB`
URI	`m2://PYRPXHNQTB`
Kind	statement, theorem
Writing	W0 stub
MSC	53C21, 83C05, 83C40
Status	This copy of the site has no build data. Run `m2 status PYRPXHNQTB`.
About	Scalar curvature
Uses	Riemannian manifold, Scalar curvature, Isometry of Riemannian manifolds
Changed	2 November 2026

Positive mass theorem

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