Strong ellipticity of mixed-conformal boundary conditions

Establish whether generic mixed-conformal boundary conditions for Euclidean Einstein gravity retain strong ellipticity when full Dirichlet data are imposed on a finite collection of measure-zero joints.

Background

The gravitational path integral is treated semiclassically, so the one-loop determinant requires a strongly elliptic gauge-fixed boundary-value problem. Full conformal boundary conditions are described as elliptic, whereas Dirichlet boundary conditions generally are not.

Mixed-conformal conditions interpolate between these choices by imposing conformal data away from joints and full intrinsic metric data on the joints. The authors expect strong ellipticity to survive this modification, but explicitly state that they have not proved it.

References

We expect, though we do not prove, that generic mixed-conformal boundary conditions retain strong ellipticity, because they differ from the full conformal boundary conditions only on the joints, which are finite in number and measure-zero in volume.

— Quantum State of a Gravitating Spacetime Region  (2609.10684 - Bousso et al., 9 Sep 2026) in Section 2.3, subsection “Mixed-Conformal Boundary Conditions”