Résume | The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. As an application of estimates on homotopy formulas for the $\overline{\partial }$ operator, we consider two generalizations of the Newlander-Nirenberg theorem for domains with strictly pseudoconvex $C^{2}$ boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain D in $C^{n}$ with $C^{2}$ boundary, we show the existence of global holomorphic coordinate systems defined on the closure of D that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. This extends a result of R. Hamilton for strictly pseudoconvex domains with smooth boundary. Using our result, we then prove the existence of local one-sided holomorphic coordinate systems provided that the boundary is strictly pseudoconvex with respect to the given complex structure.
This is joint work with Chun Gan. |