Diederich-Fornæss index, D'Angelo forms, and global regularity in the $\bar\partial$-Neumann problem

Emil Straube (TAMU, College Station)

Nov 20. 2023, 09:30 — 10:20

Let $\Omega$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. My recent work with Liu on the implication $DF(\Omega)=1 \Rightarrow $ global regularity in the $\overline{\partial}$--Neumann problem prompted me to look again at a sufficient condition for global regularity I published in 2008. The condition involves a suitable family of defining functions $\{\rho_{\varepsilon}=e^{h_{\varepsilon}}\rho\}$, where $\rho$ is a fixed defining function, and requires a uniform bound on the family $\{h_{\varepsilon}\}$. Ideally, this bound should not be needed, and I will show a partial result in this direction. This turns out to be enough to give a somewhat new proof/perspective for my result with Liu mentioned above, namely that if $\Omega$ satisfies the condition that its Levi eigenvalues are comparable (equivalently: $\Omega$ satisfies maximal estimates), then $DF(\Omega)=1\Rightarrow$ global regularity. The bridge from information on the DF--index to global regularity properties is provided by a family of one-forms on the boundary of $\Omega$, usually referred to as D'Angelo forms.


 

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Analysis and Geometry in Several Complex Variables (Workshop)
Organizer(s):
Peter Ebenfelt (UC San Diego)
Purvi Gupta (IIS Bengalore)
Bernhard Lamel (U of Vienna)
Nordine Mir (Texas A&M U at Qatar)