Rooftop envelopes and residual plurisubharmonic functions

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 study is motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic.