MathDB

Extended Future Tube Conjecture

Let Ω⊂Rd+1\Omega\subset\mathbb{R}^{d+1} be the Lorentz future cone, let T=Rd+1+iΩT=\mathbb{R}^{d+1}+i\Omega, and let M≥1M\geq 1. For every connected unipotent subgroup G≤SO0(1,d)G\leq\mathrm{SO}_0(1,d), the domain GC⋅TM⊂C(d+1)×MG^{\mathbb{C}}\cdot T^M\subset\mathbb{C}^{(d+1)\times M} is a Stein manifold.

References

Primary source

arXiv

Additional references

complex-analysislie-groups
Posted · edited

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to settle the conjecture for every connected unipotent subgroup, but independent verification is still absent.

The conjecture concerns the Stein property of GC⋅TMG^{\mathbb C}\cdot T^M for connected unipotent subgroups GG of SO0(1,d)\mathrm{SO}_0(1,d). Earlier work in 2003 also claimed a proof of a broader future-tube statement, but the available record does not establish refereed verification.

Known results

  • A July 15, 2003 preprint claimed that SOC(1,n)⋅TN\mathrm{SO}_{\mathbb C}(1,n)\cdot T^N is a domain of holomorphy for arbitrary NN, using geometric invariant theory; it was unrefereed in the retrieved record.

September 2026 unipotent-subgroup preprint

A September 2026 arXiv preprint states that the Stein property is established for GC⋅TMG^{\mathbb C}\cdot T^M when GG is any connected unipotent subgroup of SO0(1,d)\mathrm{SO}_0(1,d), thereby claiming the full stated conjecture. The result is explicitly unrefereed.

Current status (as of September 2026): The full connected-unipotent-subgroup statement is claimed proved, but the proof remains unrefereed and independently unverified.

Sources

Solutions 0

No solutions have been posted yet.