Propagation of regularity for Monge-Ampère exhaustions and Kobayashi metrics

We prove that if a smoothly bounded strongly pseudoconvex domain D⊂ Cn, n≥ 2 , admits at least one Monge-Ampère exhaustion smooth up to the boundary (i.e., a plurisubharmonic exhaustion τ: D¯ → [0 , 1] , which is C∞ at all points except possibly at the unique minimum point x and with u: = log τ satisfying the homogeneous complex Monge-Ampère equation), then there exists a bounded open neighborhood U⊂ D of the minimum point x, such that for each y∈ U there exists a Monge-Ampère exhaustion with minimum at y. This yields that for each such domain D, the restriction to the subdomain U⊂ D of the Kobayashi pseudo-metric κD is a smooth Finsler metric for U and each pluricomplex Green function of D with pole at a point y∈ U is of class C∞. The boundary of the maximal open subset having all such properties is also explicitly characterized. The result is a direct consequence of a general theorem on abstract complex manifolds with boundary, with Monge-Ampère exhaustions of regularity Ck for some k≥ 5. In fact, analogues of the above properties hold for each bounded strongly pseudoconvex complete circular domain with boundary of such weaker regularity.