Rooftop envelopes and residual plurisubharmonic functions

Abstract

Given a negative plurisubharmonic function ϕ in a bounded pseudoconvex domain of C n , we introduce and study its residual function g ϕ determined by the asymptotic behavior of ϕ near its singularity points, both inside the domain and on its boundary. For certain choices of ϕ, the function g ϕ 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.

Plurisubharmonic Interpolation and Plurisubharmonic Geodesics

“…More results on boundary behavior for functions from other Cegrell classes and their residual functions can be found in [33] (see also the last section of this paper).…”

“…The following was proved in [33], and it can be checked that the proof of assertions (i) and (ii) with the new definition of small unbounded locus remains unchanged. Theorem 14.…”

“…Rooftop envelopes were explicitly introduced in [22] for quasi-psh functions on compact Kähler manifolds, and then the technique was developed in [12,14,15,21,23,42] and others. In the local context, they were considered first in [28] for functions in the Cegrell class F 1 and then in [33] for arbitrary psh functions, bounded from above.…”

“…At the moment, however, it is known to hold only under some assumptions on u. To present them, we need the following This differs from the definition of small unbounded locus used in [33], where the requirement was pluripolarity of L u , that is, existence of a function V ≡ −∞ which is psh in a neighborhood of D and V(ζ) = −∞ for all ζ ∈ L u . The present definition does not change L u ∩ D, while it allows the boundary part L u ∩ ∂D to be much bigger than pluripolar; such sets are called b-pluripolar [53].…”

“…For the local setting of plurisubharmonic functions on bounded domains, the geodesics were considered in the (unpublished) preprint [27] and, independently, in [28,29], and then in [4,[30][31][32][33]. Comparing with the compact manifold case, two main difficulties arise.…”