Yau's conjecture on smooth Reinhardt domains

For every integer $n\ge 1$ , if a Reinhardt domain $\Omega\subset\mathbb{C}^n$ has smooth boundary and its Bergman metric is complete and Einstein, then $\Omega$ is biholomorphic to the unit ball $\mathbb{B}^n\subset\mathbb{C}^n$ .

References

Primary source

arXiv

Additional references

Yau's conjecture on smooth Reinhardt domains — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed manuscript claims to settle the conjecture by showing that only the ball can support the relevant complete metric.

The conjecture concerns rigidity for smooth Reinhardt domains carrying complete Bergman–Einstein metrics. The latest report claims a complete resolution, but gives no referee verification.

August 2026 claimed resolution

A manuscript reported on August 24, 2026, claims that no unbounded smooth Reinhardt domain admits a complete Bergman–Einstein metric and that the ball is the unique possibility. This would settle the conjecture, but the manuscript is explicitly unrefereed.

Current status (as of August 2026): An unrefereed manuscript claims the conjecture is solved, but the result remains unverified.

Sources

No solutions have been posted yet.