The characteristic gluing problem for the Einstein vacuum equations. Linear and non-linear analysis

TL;DR

This paper addresses the characteristic gluing problem for Einstein vacuum equations, identifying gauge-dependent and invariant obstructions, and constructing solutions with controlled regularity near Minkowski data.

Contribution

It solves the codimension-10 gluing problem for data close to Minkowski space, identifying key charges and constructing solutions with specified regularity.

Findings

Identified gauge-dependent and invariant charges as obstructions.

Constructed solutions with $C^{m+2}$ regularity in tangential directions.

Demonstrated higher-order gluing along bifurcated hypersurfaces.

Abstract

This is the second paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-$10$ characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a $10$-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is $C^{m+2}$ for any specified integer $m\geq0$ in the tangential directions and $C^2$ in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges).