For every admissible compact initial-data set (Omega,g,k) with boundary Sigma for which the Wang--Yau quasi-local mass M_WY(Sigma) is defined and the relevant energy conditions hold, prove that M_WY(Sigma)≥ 0 using only the geometry and initial data on the compact region Omega, without invoking an asymptotically flat extension or the positive mass theorem.

Schoen’s quasi-local positivity question

Schoen’s question asks for a genuinely quasi-local positivity theorem, without relying on an asymptotically flat extension and the positive mass theorem. Recent work gives results for particular quasi-local masses and data classes, but not the full general statement.

Known results

December 2004: Positivity of quasi-local mass II proves strict positivity for each connected boundary component under a local energy condition, except in the flat rigidity case.
September 2023: A Quasi-Local Mass proves nonnegativity and rigidity for its new gauge-independent mass using only the enclosed compact region.
January 2024: Quasi-local masses in General relativity and their positivity: Spinor approach claims Wang–Yau positivity under additional hypotheses, but the supplied record has no independent verification.

September 2026 partial result

Puskar Mondal and Shing-Tung Yau claim that a compact-fill-in argument establishes strict positivity near strictly convex Euclidean data with transverse-traceless perturbations. This is genuine progress in a nontrivial sector, but it does not prove positivity for general admissible initial data; the claim is unverified.

Current status (as of September 2026): Positivity is established or claimed only for restricted masses, geometries, and perturbative data; Schoen’s general quasi-local positivity question remains open, with the latest partial result unverified.

Schoen’s quasi-local positivity question

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Additional references

Progress summary

Solutions 0