Local geodesics for plurisubharmonic functions

TL;DR

This paper investigates geodesics for plurisubharmonic functions in the Cegrell class, revealing their role in linearizing energy functionals and establishing uniqueness theorems, with applications to capacities and singularities.

Contribution

It introduces a framework for geodesics in the Cegrell class, demonstrating their properties and implications for Monge-Ampère measures and capacity inequalities.

Findings

Geodesics linearize an energy functional similar to Kähler metrics.

Uniqueness of functions in ${\\mathcal{F}_1}$ based on total Monge-Ampère masses.

Reverse Brunn-Minkowski inequality for capacities of multi-circled sets.

Abstract

We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on K\"{a}hler compact menifolds, they linearize an energy functional. As a consequence, we get a uniqueness theorem for functions from ${\mathcal F}_1$ in terms of total masses of certain mixed Monge-Amp\`ere currents. Geodesics of relative extremal functions are considered and a reverse Brunn-Minkowski inequality is proved for capacities of multiplicative combinations of multi-circled compact sets. We also show that functions with strong singularities generally cannot be connected by (sub)geodesic arks.