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-Ampère exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $\tau :\overline{D}\to [0,1]$, which is ${\mathcal{C}}^{\infty }$ at all points except possibly at the unique minimum point $x$ and with $u:=\log \tau$ satisfying the homogeneous complex Monge-Ampère equation), then there exists a bounded open neighborhood $\mathcal{U}\subset D$ of the minimum point $x$, such that for each $y\in \mathcal{U}$ there exists a Monge-Ampère exhaustion with minimum at $y$. 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 ${\mathcal{C}}^{\infty }$. 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-Ampère 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.

arXiv comment45 pages - Various misprints, the minimal regularity requirements of Thm.2.8 and a minor point in the proof of Prop. 5.4 have been corrected. The introduction is improved. A subsection on examples is inserted at the end. To appear on Ann. Mat. Pura Appl