We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.

F13.54> OE 3 ) ' where e OE 1 = 1 2 (1 \Gamma eg 2 )e!; e OE 2 = i 2 (1 + eg 2 )e!; e OE 3 = e ge!; (0.1) are holomorphic one-forms on f M satisfying 3 X i=1 j e OE j j 2 6= 0: 1 Singly Periodic Minimal Surfaces 139 140 Francisco Martin and Domingo Rodriguez We label the pair (eg; e !) as the Weierstrass representation of f M . We can work with the quotient M = f M=T , and so we have a minimal immersion x

We give a new characterization of the periodic Callahan-Hoffman-Meeks surfaces in terms of their symmetries.