Propagation of regularity for Monge-Ampère exhaustions and Kobayashi metrics

Abstract

We prove that if a smoothly bounded strongly pseudoconvex domain $$D \subset {\mathbb {C}}^n$$ , $$n \ge 2$$ , admits at least one Monge-Ampere exhaustion smooth up to the boundary (i.e., a plurisubharmonic exhaustion $$\tau : {\overline{D}} \rightarrow [0,1]$$ , which is $${\mathscr {C}}^\infty $$ at all points except possibly at the unique minimum point x and with $$u {:}{=} \log \tau $$ satisfying the homogeneous complex Monge-Ampere equation), then there exists a bounded open neighborhood $${\mathscr {U}}\subset D$$ of the minimum point x, such that for each $$y \in {\mathscr {U}}$$ there exists a Monge-Ampere exhaustion with minimum at y. This yields that for each such domain D, the restriction to the subdomain $${\mathscr {U}}\subset D$$ of the Kobayashi pseudo-metric $$\kappa _D$$ is a smooth Finsler metric for $${\mathscr {U}}$$ and each pluricomplex Green function of D with pole at a point $$y \in {\mathscr {U}}$$ is of class $${\mathscr {C}}^\infty $$ . 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-Ampere exhaustions of regularity $$\mathscr {C}^{k}$$ 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.