Subharmonic variation of Azukawa pseudometrics for balanced domains

Genki Hosono

doi:10.1142/s0129167x21500154

Abstract

We consider the subharmonicity property of the logarithm of Azukawa pseudometrics of pseudoconvex domains under pseudoconvex variations. We prove that such a property holds for the variation of balanced domains. We also give a non-balanced example. The relation of the volume of Azukawa indicatrix and the estimate in the Ohsawa–Takegoshi [Formula: see text]-extension theorem is also discussed.

Highlights

Let Ω ⊂ Cn be a bounded hyperconvex domain
For a fixed point w ∈ Ω, the pluricomplex Green function gΩ,w on Ω with a pole w is defined as gΩ,w := sup{u ∈ P SH(Ω) : u < 0 and u − log | · −w| is bounded above near w}
Pluricomplex Green functions are pluripotential theoretic generalizations of Green functions, which are the solutions of the Laplace equation

Summary

Introduction

For a fixed point w ∈ Ω, the pluricomplex Green function gΩ,w on Ω with a pole w is defined as gΩ,w := sup{u ∈ P SH(Ω) : u < 0 and u − log | · −w| is bounded above near w}. Pluricomplex Green functions are pluripotential theoretic generalizations of Green functions, which are the solutions of the Laplace equation. Pluricomplex Green functions are the solutions of the complex Monge-Ampere equations with a logarithmic pole. To analyze infinitesimal behavior of pluricomplex Green function near its pole, it is useful to consider (the logarithm of) the Azukawa pseudometric. For a fixed point w ∈ Ω and a vector X ∈ Cn, it is defined as AΩ,w(X) := lim sup(gΩ,w(w + λX) − log |λ|).

HOSONO

Variation of sublevel sets of a log-homogeneous plurisubharmonic function

Balanced domains

An example

Relation to the Ohsawa-Takegoshi L2-extension theorem