ESTIMATES FOR THE ¯∂-NEUMANN PROBLEM AND NONEXISTENCE OF LEVI-FLAT HYPERSURFACES IN ℂP^{n}

JIANGUO CAO MEI-CHI SHAW LIHE WANG

Abstract. Let Ω be a pseudoconvex domain with C^{2}-smooth boundary in ℂP^{n}. We prove that the ¯∂ − NeumannoperatorNexistsfor(p,q)−formsonΩ. Furthermore, there exists a t_{0} > 0 such that the operators N, ¯∂^{∗}N, ¯∂N and the Bergman projection are regular in the Sobolev space W^{t}(Ω̄) for t < t_{0}.

The boundary estimates above have applications in complex geometry. We use the estimates to prove the nonexistence of C^{2,α} real Levi-flat hypersurfaces in ℂP^{n}. We also show that there exist no non-zero L^{2}-holomorphic (p, 0)-forms on any pseudoconcave domain in ℂP^{n} with p > 0.