TY - JOUR

T1 - The Characteristic Gluing Problem for the Einstein Vacuum Equations

T2 - Linear and Nonlinear Analysis

AU - Aretakis, Stefanos

AU - Czimek, Stefan

AU - Rodnianski, Igor

N1 - Publisher Copyright:
© 2023, Springer Nature Switzerland AG.

PY - 2023

Y1 - 2023

N2 - This is the second paper in a series of papers addressing 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 Cm+2 for any specified integer m≥ 0 in the tangential directions and C2 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).

AB - This is the second paper in a series of papers addressing 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 Cm+2 for any specified integer m≥ 0 in the tangential directions and C2 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).

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U2 - 10.1007/s00023-023-01394-y

DO - 10.1007/s00023-023-01394-y

M3 - Article

AN - SCOPUS:85178925004

SN - 1424-0637

JO - Annales Henri Poincare

JF - Annales Henri Poincare

ER -