The Diederich-Fornaess index and the dbar-Neumann problem

A domain $\Omega$ in $\mathbb C^n$ is said to be pseudoconvex if $-\log(-\delta(z))$ is plurisubharmonic in $\Omega$, where $\delta$ is a signed distance function of $\Omega$. The study of global regularity of the dbar-Neumann problem on bounded pseudoconvex domains is dated back to the 1960s. However, a complete understanding of the regularity is still absent. On the other hand, the Diederich-Fornaess index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the dbar-Neumann problem and the Diederich-Fornaess index. Indeed, it has been a long-lasting open question whether the trivial Diederich-Fornaess index implies global regularity. In this talk, we will introduce the backgrounds and motivations. We will also answer this open question by a recent result of Straube and me.