September 21, 2022 5 p.m. CEST Zoom online
Bingyuan Liu University of Texas Rio Grande Valley

The Diederich-Fornaess index and the $\overline\partial$-Neumann problem

Abstract 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 $\overline\partial$-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 $\overline\partial$-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. We will also answer this open question by a recent result of Straube and me.

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