2016/06/07 by Steven G. Krantz, Bingyuan Liu, Krantz, Steven G. +3 · 1 citation
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1606.02343
openalex publication_date 2016/06/07 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28
We derive a sufficient condition on a bounded pseudoconvex domain Ω⊂ℂ2 with smooth boundary such that -(-ρ)η is plurisubharmonic on Ω for η>0 arbitrarily close to 1 (the supremum of η is called Diederich-Fornæss index, see Definition (df)). This condition (see Theorem prop) extends a theorem of Fornæss and Herbig in 2007 and only requires restriction on Levi-flat sets of the boundary ∂Ω. Since the condition is on Levi-flat sets, it contains more geometric information. As an application of this new condition, we discuss how the geometry of the Levi-flat sets affects the Diederich-Fornæss index. Among other results, we show that the Diederich-Fornæss index is 1 if only the Levi-flat sets form a real curve transversal to the holomorphic tangent vector fields on ∂Ω (see Theorem [main]). We also give a specific example (see Theorem [example]) on the bounded pseudoconvex domains which verify the application but are neither of finite type nor admit a plurisubharmonic defining function on the boundary.