We present a construction of a complete proper holomorphic embedding from any strictly pseudoconvex domain with smooth boundary in \(\mathbb C^n\) into the unit ball of \(\mathbb C^N\), for \(N\) large enough.

An analytic disc is a holomorphic map from the unit disc in $\mathbb{C}$ into some target space, for example $\mathbb{C}^n$ , which extends continuously up to the boundary. I will solve approximately a nonlinear Riemann-Hilbert boundary value problem and then I will present its applications in constructing analytic discs with certain properties.