Geometry & Topology Seminar: The Diederich–Fornæss index and the ∂ ̄-Neumann problem

Abstract: In the study of complex geometry, a deep connection exists between the geometry of a domain and the analysis of functions within it. This talk explores domains in C^n that are pseudoconvex—the natural generalization of convex sets from Euclidean space.
The study of global regularity of ∂ ̄-Neumann problem on bounded pseudo- convex domains is dated back to the 1960s. However, a complete understanding of the regularity is still absent. On the other hand, the Diederich–Fornæss index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship among global regularity of the ∂ ̄-Neumann problem, the boundary geometry and the Diederich–Fornæss index. Indeed, it has been a long- lasting open question whether the trivial Diederich–Fornæss index implies global regularity. In this talk, we will introduce the backgrounds and motivations. The main theorem of the talk proved recently by Emil Straube and me answers this open question for (0, n − 1) forms.

Host: Quo-Shin Chi