Stability of the Riemannian positive mass theorem in all dimensions

We prove stability of the Riemannian positive mass theorem in all dimensions, extending the Dong-Song stability theorem. For a sequence of complete asymptotically flat manifolds with nonnegative scalar curvature and ADM masses tending to zero, excising domains whose boundary areas tend to zero yields exterior regions converging to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof constructs global coordinates from minimal graphs and controls their Hessians using scalar solutions of the conformal Laplace equation on the associated graph metrics.

Main result

Positive mass stability

Euclidean convergence after excision

For smooth, connected, complete, oriented manifolds of dimension $n\geq3$, without boundary and with finitely many asymptotically flat ends, assume nonnegative integrable scalar curvature and ADM masses tending to zero along distinguished ends. There are exterior regions containing these ends whose boundary areas tend to zero and which converge to Euclidean space in the pointed measured Gromov–Hausdorff topology, using intrinsic length metrics and arbitrary base points in the exterior regions. No spin assumption is required.

Proof architecture

The argument combines minimal graph coordinates with scalar conformal equations to convert small ADM mass into geometric control of exterior regions.

Step 1 · Coordinates

Construct global minimal graphs

Global minimal graph functions define coordinates asymptotic to the identity at the distinguished end. Conformal changes of the associated graph metrics relate scalar curvature to the Hessians of these functions.

Step 2 · Energy estimates

Control coordinate defects by mass

Scalar solutions of the conformal Laplace equation, together with the qualitative positive mass inequality, give weighted Hessian energy estimates in terms of the original ADM mass.

Step 3 · Excision

Recover Euclidean geometry outside small boundaries

Level selection produces exterior regions with small boundary area and small metric distortion. A Euclidean excision argument then yields pointed measured Gromov–Hausdorff convergence with respect to the intrinsic metrics.