Let (M^n,g), n≥ 3, be an asymptotically flat Riemannian manifold of order tau>(n-2)/2 whose scalar curvature is nonnegative and integrable. The ADM mass is defined by m_ADM=frac12(n-1)omega_n-1lim_r→∞int_S_r(g_ij,i-g_ii,j)nu^j,dS_r. Positive mass conjecture. The ADM mass of (M,g) is nonnegative. Moreover, if the mass is zero, then (M,g) is isometric to Euclidean space (R^n,delta). The conjecture is known in dimensions n≤ 7, for conformally flat manifolds, for spin manifolds, and for Euclidean graphs under the conditions described in the source. The general statement is therefore not open in all cases, but the supplied text does not specify a complete resolution status for the unrestricted formulation.

The positive mass conjecture

Let $(M^n,g)$ , $n\geq 3$ , be an asymptotically flat Riemannian manifold of order $\tau>(n-2)/2$ whose scalar curvature is nonnegative and integrable. The ADM mass is defined by

\mathrm{m}_{\mathrm{ADM}}=\frac{1}{2(n-1)\omega_{n-1}}\lim_{r\to\infty}\int_{S_r}(g_{ij,i}-g_{ii,j})\nu^j\,dS_r.

Positive mass conjecture. The ADM mass of $(M,g)$ is nonnegative. Moreover, if the mass is zero, then $(M,g)$ is isometric to Euclidean space $(\mathbb{R}^n,\delta)$ .

The conjecture is known in dimensions $n\leq 7$ , for conformally flat manifolds, for spin manifolds, and for Euclidean graphs under the conditions described in the source. The general statement is therefore not open in all cases, but the supplied text does not specify a complete resolution status for the unrestricted formulation.

The conjecture asserts that every complete asymptotically flat manifold with nonnegative scalar curvature has nonnegative ADM mass, with equality only for Euclidean space. The supplied sources do not identify a proposer or original date.

Known results

Schoen and Yau proved the theorem for dimensions $n \le 7$ .
Witten proved it in arbitrary dimensions for spin manifolds.
Lam proved it for asymptotically flat graphs in $\mathbb{R}^{n+1}$ .
Completeness is essential: without it, both positivity and the stated rigidity can fail.

2006 all-dimensional proof claim

A 2006 preprint by Lohkamp claims the exact unrestricted Riemannian theorem in all dimensions and without topological assumptions. Later literature describes this as an announced approach or result, while the supplied sources provide no independent verification, referee assessment, or resolution of the claim.

Current status (as of September 2026): The theorem is settled for $n \le 7$ , spin manifolds, and several special geometric classes; Lohkamp’s all-dimensional proof claim matches the unrestricted statement but remains unverified in the supplied evidence.

The positive mass conjecture

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Progress summary

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