April 25, 2024

The Diederich--Forn\ae ss index and the $\bar{\partial}$-Neumann problem

A domain $\Omega\subset\mathbb{C}^n$ 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 $\bar{\partial}$-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--Forn\ae ss index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the $\bar{\partial}$-Neumann problem and the Diederich--Forn\ae ss index. Indeed, it has been a long-lasting open question whether the trivial Diederich--Forn\ae ss 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.

About the speaker

Dr. Bingyuan Liu is an Assistant Professor from the School of Mathematical and Statistical Sciences at the University of Texas Rio Grande Valley. He is looking for research students for thesis projects and dissertations. Contact him for more info!