Chen-Gackstatter type surfaces in R^4_1: deformation, symmetry, and embeddedness

We find a 2-parameter family of deformations in R^4_1 of the classical Chen-Gackstatter surface explicitly, and show the existence of a larger 4-parameter family of deformations. Each of them still has genus one, a unique end, with total Gaussian curvature $-\int K=8\pi$. On the other hand, a uniqueness theorem is obtained when we assume that the surface has more than 4 symmetries. The problem of embeddedness is also discussed with some partial results.