On properly embedded minimal surfaces with three ends

Abstract

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.