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

TL;DR

The paper demonstrates that regularity properties of Monge-Ampère exhaustions in strongly pseudoconvex domains extend locally around their minima, leading to smooth Kobayashi metrics and Green functions in those regions.

Contribution

It establishes the propagation of regularity for Monge-Ampère exhaustions and related metrics in strongly pseudoconvex domains, with explicit boundary characterization.

Findings

Existence of local neighborhoods where Monge-Ampère exhaustions are smooth at all points.

Smoothness of Kobayashi pseudo-metric and Green functions in these neighborhoods.

Extension of properties to domains with boundary of weaker regularity.

Abstract

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 $\tau: \overline D \to [0,1]$, 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-Amp\`ere equation), then there exists a bounded open neighborhood $\mathcal U\subset D$ of the minimum point $x$, such that for each $y \in \mathcal U$ there exists a Monge-Amp\`ere 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-Amp\`ere exhaustions of regularity $\mathcal C^{k}$ for some $k \geq 5$. In fact, analogues of the above properties hold for each bounded strongly pseudoconvex complete circular domain with boundary of such weaker regularity.