Jihun Yum, Characterization of Diederich-Fornaess and Steinness Indices in Complex Manifolds

Let Ω be a relatively compact pseudoconvex domain in a complex manifold X with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by

DF(Ω) := sup_ρ { 0 < η < 1 : -(-ρ)^η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },

S(Ω) := inf_ρ { η > 1 : ρ^η is strictly plurisubharmonic on Ω^c ∩ U for some neighborhood U of ∂Ω },

where ρ is a defining function for Ω.

In the previous talk, we have seen that two indices are completely characterized by D’Angelo 1-form when the ambient space is X = Cⁿ. In this talk, we generalize the formulas for a relatively compact pseudoconvex domains in a (general) complex manifold X. Since the formulas do not hold anymore in general, unfortunately, we introduce 4 kinds of each of the Diederich-Fornaess and Steinness indices. Then we give some non-degeneracy conditions for these indices agree. Also, we exam the geometric meaning of the D’Angelo 1-form when the boundary ∂Ω is Levi-flat.