Characteristic gluing to extremal Kerr horizons and the third law of black hole thermodynamics in vacuum

We prove a non-perturbative, non-axisymmetric characteristic gluing theorem for the Einstein vacuum equations: every Boyer-Lindquist sphere in the domain of outer communication of a Kerr black hole, in particular every symmetry sphere of a Schwarzschild exterior, can be glued along a single outgoing null hypersurface to a horizon sphere of an extremal Kerr black hole of sufficiently large mass. The gluing is exact at both ends and matches the full sphere data, including the transversal curvature. This proves a conjecture of Kehle and Unger in the extremal case. We first raise the angular momentum to the extremal value by rotating shear pulses of non-axisymmetric gravitational radiation, and then damp the incoming data exponentially along a long non-expanding neck. At the degenerate target horizon we invert the linearised problem at all angular frequencies, using a factorisation of a fourth-order angular operator into an angular Teukolsky-Starobinsky product. As an application, we construct one-ended asymptotically flat vacuum initial data whose maximal development contains an exactly Schwarzschild apparent horizon and, at a later advanced time, an exactly extremal Kerr event horizon. This is the first rigorous counterexample to the third law of black hole thermodynamics in vacuum, and proves a second conjecture of Kehle and Unger.