Estimates for the $\bar\partial$-Neumann problem and nonexistence of Levi-flat hypersurfaces in $CP^n$

Abstract

Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $\Omega$. Furthermore, there exists a $t_0>0$ such that the operators $N$, $\bar\partial^*N$, $\bar\partial N$ and the Bergman projection are regular in the Sobolev space $W^t (\bar{\Omega}) $ for $t<t_0$. The boundary estimates above have applications in complex geometry. We use the estimates to prove the nonexistence of $C^{2, \alpha}$ real Levi-flat hypersurfaces in $\mathbb CP^n$. We also show that there exist no non-zero $L^2$-holomorphic $(p, 0)$-forms on any pseudoconcave domain in $\mathbb CP^n$ with $p > 0$.