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In the previous paper, the authors constructed a complete holomorphicimmersion of the unit disk D into C^2 whose image is bounded. In this paper, weshall prove existence of complete holomorphic null immersions of Riemannsurfaces with arbitrary genus and finite topology, whose image is bounded inC^2. To construct such immersions, we apply the method used by F. J. Lopez toperturb the genus zero example changing its genus.  As an analogue the above construction, we also give a new method to constructcomplete bounded minimal immersions (resp. weakly complete maximal surface)with arbitrary genus and finite topology in Euclidean 3-space (resp.Lorentz-Minkowski 3-spacetime).

Complete bounded holomorphic curves immersed in C^2 with arbitrary genus