Let Gamma be a closed Jordan curve, let h be its conformal welding, let C_h be the pull-back operator on BMO, and let A_h be the analytic projection of C_h. Determine whether A_h is a bounded isomorphism on BMOA if and only if Gamma is a chord-arc curve.

Semmes’s open problem on conformal welding of chord-arc curves

Let $\Gamma$ be a closed Jordan curve, let $h$ be its conformal welding, let $C_h$ be the pull-back operator on $\mathrm{BMO}$ , and let $A_h$ be the analytic projection of $C_h$ . Determine whether $A_h$ is a bounded isomorphism on $\mathrm{BMOA}$ if and only if $\Gamma$ is a chord-arc curve.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Faber-operator formulation
For a rectifiable Jordan curve $\Gamma$ , determine whether the associated Faber integral operator is a bounded isomorphism on $\mathrm{BMOA}$ if and only if $\Gamma$ satisfies the chord-arc condition.
source: Conformal welding of chord-arc curves

References

Primary source

arXiv

Additional references

Conformal welding of chord-arc curves — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to settle Semmes’s decades-old question about when a curve can arise from conformal welding, but the claim has not been independently verified.

Semmes’s open problem, originating in the 1980s, asks for a precise conformal-welding characterization of chord-arc curves and its relationship with analytic projection and Faber operators.

Known results

A 2024 preprint characterized chord-arc images of BMO embeddings by local absolute continuity and $\log \gamma'\in\mathrm{BMO}^*(\mathbb{R})$ , and related Cauchy projections to BMOA and Faber operators.
A 2015 operator-theoretic characterization connected chord-arc curves with invertibility of $I-\mu S$ , while explicitly leaving a related quasiconformal-map question open.

August 2026 claimed resolution

A preprint listed on August 25, 2026, claims a complete conformal-welding characterization of chord-arc curves and identifies the inverse analytic projection with the Faber integral operator. This would settle Semmes’s problem, but the claim is unrefereed and remains unverified.

Current status (as of August 2026): A preprint claims the problem is solved, but independent verification is absent; the result therefore remains unconfirmed.

Sources

No solutions have been posted yet.

Faber-operator formulation

Primary source

Additional references

Progress summary