2017/07/27 by Patrizio, Giorgio, Spiro, Andrea
#32G05 #32Q45 #32U35 #32W20 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.09041
We prove that if a smoothly bounded strongly pseudoconvex domain D ⊂ \mathbb 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 \mathcal 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 \mathcal U⊂ D of the minimum point x, such that for each y ∈ \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⊂ D of the Kobayashi pseudo-metric κD is a smooth Finsler metric for \mathcal U and each pluricomplex Green function of D with pole at a point y ∈ \mathcal U is of class \mathcal 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 \mathcal Ck 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.