Geodesic connectivity and rooftop envelopes in the Cegrell classes

TL;DR

This paper advances the understanding of geodesics and envelopes in the Cegrell classes, proving key uniqueness and comparison principles, and broadening conditions for connectivity and envelope equality in complex analysis.

Contribution

It establishes the most general comparison principle in Cegrell classes, proves uniqueness of solutions with comparable singularities, and weakens conditions for rooftop envelope equality and idempotency.

Findings

Proved solutions with similar singularities are identical.

Established the most general Bedford-Taylor comparison principle.

Showed rooftop equality and idempotency hold under weaker conditions.

Abstract

This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Amp\`ere equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford-Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge-Amp\`ere equation.