The barpartial-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in mathbb{C}P^n with n ge 3

In this paper we study the Cauchy–Riemann equation in complex projective spaces. Specifically, we use the modified weight function method to study the $barpartial$ -Neumann problem on pseudoconvex domains in these spaces. The solutions are used to study function theory on pseudoconvex domains via the $barpartial$ -Cauchy problem. We apply our results to prove nonexistence of Lipschitz Levi-flat hypersurfaces in complex projective spaces of dimension at least three, which removes the smoothness requirement used in an earlier paper of Siu.