The Cauchy-Riemann Equations on the Hartogs Triangles

The Hartogs triangle in the complex Euclidean space is an important example in several complex variables. It is a bounded pseudoconvex domain with non-Lipschitz boundary. In this talk, we discuss the extendability of Sobolev spaces on the Hartogs triangle and show that the weak and strong maximal extensions of the Cauchy-Riemann operator agree. These results are related to the Dolbeault cohomology groups with Sobolev coefficients on the complement of the Hartogs triangle (joint work with A. Burchard, J. Flynn and G. Lu).

The Hartogs triangles in the complex projective spaces are examples of non-Lipschitz Levi-flat hypersurfaces. We will discuss some recent progress for the Cauchy-Riemann equations on Hartogs triangles in the complex projective space (joint work with C. Laurent-Thiébaut).