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In 1982 Chen and Gackstatter [C-G] discovered two minimal surfaces with finite total curvature in R and non-trivial topology. These examples have only one end and the symmetry of the Enneper’s surface. The first one is the only complete minimal torus in R with total curvature −8π (see [B], [L1]). The second Chen-Gackstatter example has genus two, total curvature −12π and only one end. D. Hoffman noted that these surfaces represented examples with the least absolute total curvature 4(k+ 1)π for surfaces with fixed genus k, and conjectured that there should be such examples of every genus. Recently Thayer [T] gave numerical evidence for this conjecture, and Weber and Wolf [WW] and Sato [S] proved this conjecture using very different methods. Thus, the main issue here is to obtain uniqueness results for these kind of surfaces. On the other hand, the classical Chen-Gackstatter examples were generalized in another way, first by Karcher [K], and later by Thayer [T], Sato [S] (see Martin and Rodŕiguez work [M-R] for a new approach including uniqueness results), Lopez [L2] and Weber and Wolf [WW]. These authors have shown immersions with only one end and high topology and symmetry. Except in the case of the Weber and Wolf method, which seems to be very powerful, the other constructions have the same spirit of the Hoffman and Meeks [H-M] generalization of the Costa surface [C]. Though this last method is classical and explicit, it has the advantage of providing some natural uniqueness results (see, for instance, [H-M]). In this paper we use these ideas to show a new family of complete minimal surfaces Mk of genus k, k ≥ 2, k even, that generalizes the Chen-Gackstatter genus two example. Except in the case of k = 2 which corresponds to the

Complete minimal surfaces derived from Chen–Gackstatter genus two example