Threshold-Sharp Conformal Scalar Stability on Carter Slabs and Black Hole Exteriors

TL;DR

This paper establishes a sharp stability theory for the conformal scalar-curvature sector on Carter backgrounds, including black hole exteriors, identifying conditions for stability and obstructions.

Contribution

It provides a fully closed bounded-slab theorem with positive conserved energy and identifies the affine threshold obstruction, extending the threshold philosophy to black-hole exteriors.

Findings

Constructed reflecting evolution with positive conserved energy.

Identified the complete affine threshold obstruction.

Proved stability for the conformal scalar-curvature sector only.

Abstract

We prove a threshold-sharp stability theory for the conformal scalar-curvature sector on zero-curvature Carter backgrounds. The main result is a fully closed bounded-slab theorem: the reflecting evolution is constructed, the conserved energy is proved positive, the complete affine threshold obstruction is identified, and all remaining finite-energy dynamics are shown to be uniformly stable with no unstable modes. This is the sharp statement for compact reflecting slabs, where genuine time decay is false in general. We then extend the same threshold philosophy to black-hole exteriors, separating the intrinsic conformal mechanism from the exterior scalar-wave inputs needed for red-shift, local energy, limiting absorption, and zero-frequency control. The framework gives main applications to Kerr, Reissner-Nordstr\"om, slowly rotating weakly charged Kerr-Newman wall exteriors, and extremal horizon-charge obstructions. Our precise result is that it proves stability only for the conformal scalar-curvature sector, not tensorial or nonlinear gravitational stability, and it distinguishes boundedness, qualitative local decay, polynomial decay, and extremal Aretakis-type obstruction without conflating them.