Propagation of regularity for Monge-Amp\`ere exhaustions and Kobayashi metrics

We prove that if a smoothly bounded strongly pseudoconvex domain $D \subset \mathbb C^n$,$n \geq 2$, admits at least one Monge-Amp`ere exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $\tau: \overline D \to [0,1]$,whichis$\mathcal C^\infty$ at all points except possibly at the unique minimum point $x$ and with $u:=\log \tau$ satisfying the homogeneous complex Monge-Amp`ere equation), then there exists a bounded open neighborhood $\mathcal{U}\subset D$ of the minimum point $x$, such that for each $y \in \mathcal U$thereexistsaMonge-Amp\grave{e}reexhaustionwithminimumat$y$. This yields that for each such domain $D$, the restriction to the subdomain $\mathcal{U}\subset D$ of the Kobayashi pseudo-metric ${\kappa }_D$ is a smooth Finsler metric for $\mathcal{U}$ and each pluricomplex Green function of $D$ with pole at a point $y\in \mathcal{U}$ is of class ${\mathcal{C}}^{\mathrm{\infty }}$. 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-Amp`ere exhaustions of regularity ${\mathcal{C}}^k$ for some $k\ge 5$. In fact, analogues of the above properties hold for each bounded strongly pseudoconvex complete circular domain with boundary of such weaker regularity.