Rooftop envelopes and residual plurisubharmonic functions

TL;DR

This paper introduces a residual function associated with negative plurisubharmonic functions in pseudoconvex domains, exploring its properties and connections to pluricomplex Green functions, motivated by the study of plurisubharmonic geodesics.

Contribution

It defines and analyzes the residual function $g_$ based on the asymptotic behavior of $$, linking it to pluricomplex Green functions and addressing geodesic connectivity.

Findings

Residual function $g_$ captures singularity behavior.

$g_$ coincides with pluricomplex Green functions for certain $$.

Provides insights into plurisubharmonic geodesics.

Abstract

Given a negative plurisubharmonic function $\phi$ in a bounded pseudoconvex domain of ${\mathbb C}^n$, we introduce and study its residual function $g_\phi$ determined by the asymptotic behavior of $\phi$ near its singularity points, both inside the domain and on its boundary. For certain choices of $\phi$, the function $g_\phi$ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic.