Stability of the conformal equations in the defocusing far-from-CMC regime

Establish whether the Einstein–scalar field conformal constraint equations (system (CE)) are stable in the defocusing case (i.e., when B_{τ,ψ} < 0) when the mean curvature τ is far from constant, beyond the constant-mean-curvature and near-constant-mean-curvature regimes.

Background

The authors review known stability results for the Einstein–scalar field conformal constraint equations: stability is established in the focusing case (B_{τ,ψ} > 0) under certain dimensional and regularity conditions, and in the defocusing case (B_{τ,ψ} < 0) for CMC and near-CMC data.

They point out that for the defocusing case with far-from-CMC mean curvature, there is no general result, highlighting a gap in the theory that remains to be resolved.

References

However, when $\mathcal{B}_{\tau, \psi} < 0$, that is the defocusing case, the system is known to be stable in the CMC and near-CMC settings thanks to well-known arguments in . However, when $\tau$ is far--from--CMC, the stability question still remains open.

— Spherically symmetric solutions to the Einstein-scalar field conformal constraint equations  (2602.20099 - Castillon et al., 23 Feb 2026) in Introduction, Section 1 (Stability discussion)