Local geodesics for plurisubharmonic functions

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 Kahler compact manifolds, 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–Ampere 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 arcs.