MathDB

Bartnik’s Mass Minimizer Conjecture

For every admissible Bartnik boundary datum (Σ,γ,H)(\Sigma,\gamma,H), the Bartnik mass mB(Σ,γ,H)=inf⁡{mADM(M,g):(M,g) is an admissible asymptotically flat extension realizing (Σ,γ,H)}m_B(\Sigma,\gamma,H)=\inf\{m_{\mathrm{ADM}}(M,g):(M,g)\text{ is an admissible asymptotically flat extension realizing }(\Sigma,\gamma,H)\} is attained by an asymptotically flat static vacuum extension.

References

Primary source

arXiv

Additional references

general-relativitygeometric-analysisriemannian-geometry
Posted · edited

Progress summary

Refreshed
Claimed progress

A new 2026 paper proves a conditional local version of the conjecture, while the global question remains unresolved.

Bartnik’s conjecture asks whether the infimum defining the mass is attained by a suitable asymptotically flat static vacuum extension. Its unrestricted form is known to fail in some cases, but important restricted regimes remain open.

Known results

  • Anderson (2019) gave counterexamples showing that the minimization infimum need not be attained.
  • Anderson (2023) proved that the static vacuum extension conjecture is false in general and established nonexistence for certain small positive boundary data.
  • Partial results for existing minimizers cover the stationary, non-time-symmetric setting under energy and dimension hypotheses.

September 2026 local advance

Zhongshan An and Lan-Hsuan Huang report strict local minimality under a strict-stability hypothesis and construct static vacuum extensions for nearby boundary data. This is claimed local progress, not a proof of the global conjecture; the result’s verification was not independently established in the scan.

Current status (as of September 2026): The unrestricted conjecture has counterexamples, while a claimed strict-stability local theorem is unverified and the surviving global restricted problem remains open.

Sources

Solutions 0

No solutions have been posted yet.