We classify all complete embedded minimal surfaces in ℝ3; with three ends of genus g and at least 2g+2 symmetries. The surfaces in this class are the Costa-Hoffman-Meeks surfaces that have 4g+4 symmetries in the case of a flat middle end. The proof consists of using the symmetry assumptions to deduce the possible Weierstrass data and then studying the period problems in all cases. To handle the 1-dimensional period problems, we develop a new general method to prove convexity results for period quotients. The 2-dimensional period problems are reduced to the 1-dimensional case by an extremal length argument.
"On properly embedded minimal surfaces with three ends." Duke Math. J. 107 (3) 533 - 559, 15 April 2001. https://doi.org/10.1215/S0012-7094-01-10735-7