Invertibility along paths of analytic projections

Determine whether invertibility of the analytic projections associated with conformal weldings can be controlled along paths of weldings through the corresponding paths of analytic projections in the BMO topology.

Background

The operator characterization of chord-arc curves suggests a possible approach to the connectedness problem for the chord-arc locus in BMO-Teichmüller space. The proposed approach is to represent paths of weldings by paths of analytic projections and establish stability or control of invertibility along those paths. The paper presents this as a question for further investigation rather than proving such a pathwise result.

References

Namely, one may study paths of weldings through the corresponding paths of analytic projections and ask whether invertibility can be controlled along such paths in the BMO topology.

— Conformal welding of chord-arc curves  (2608.24745 - Tailiang, 25 Aug 2026) in Section 1, Introduction and main results