In this paper we study the Cauchy-Riemann equation in complex projective spaces. Specifically, we use the modified weight function method to study thē ∂-Neumann problem on pseudoconvex domains in these spaces. The solutions are used to study function theory on pseudoconvex domains via the∂-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.
IntroductionIn this paper we study the∂-Cauchy problem and the∂-closed extension problem for forms on domains in complex hermitian manifolds. These problems were first studied in the paper by Kohn and Rossi [18] (see also [10]), who proved the holomorphic extension of smooth CR functions and the∂-closed extension of smooth forms from the boundary b of a strongly pseudoconvex domain to the whole domain . The L 2 theory of these problem has been obtained for pseudoconvex domains in C n or, more generally, for domains in complex manifolds with strongly plurisubharmonic weight functions (see Chap. 9 in [5] and the references therein). In this paper we study these problems on pseudoconvex domains in complex hermitian manifolds when such weight functions are not available, for instance, on a pseudoconvex domain in the complex projective space CP n .