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 C k 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.