Complete bounded holomorphic curves immersed in $\mathbb{C}^2$ with arbitrary genus

Recently, a complete holomorphic immersion of the unit disk $\mathbb{D}$ into $\mathbb{C}^2$ whose image is bounded was constructed by the authors. In this paper, we shall prove the existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology whose image is bounded in $\mathbb{C}^2$.

As an analogue to the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surfaces) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime).