Codimension-1 stability of extremal Kerr with horizon hair (Conjecture)
Establish codimension-1 nonlinear stability of extremal Kerr black holes for vacuum characteristic initial data posed on two null cones, by proving that the solution possesses a complete future null infinity and a regular complete event horizon, remains close to extremal Kerr, asymptotically approaches an extremal Kerr metric, and generically exhibits horizon hair (growth of suitable derivative quantities along the event horizon) under these stability-conditioned data.
References
Conjecture For all vacuum characteristic initial data prescribed on cones as in Fig. 5, assumed sufficiently close to extremal Kerr data with mass M_init=a_init and lying on a codimension-1 “submanifold” M_stable of the moduli space M of initial data, the arising vacuum solution (M, g) satisfies the following properties: (i) (M, g) possesses a complete future null infinity I+, and in fact the future boundary of J-(I+) in M is a regular, future affine complete event horizon H+. (ii) The metric g remains close to the extremal Kerr metric with mass M_init in J-(I+). (iii) The metric g asymptotes, inverse polynomially, to an extremal Kerr metric with mass M_final=a_final ≈ M_init as u → ∞ and v → ∞, in particular along I+ and H+, where u and v are suitably normalised null coordinates. (iv) For generic initial data conditioned to lie on M_stable, then suitable quantities associated to derivatives of the metric grow without bound ("horizon hair") along H+ as v→∞.