We investigate surfaces of constant mean curvature with a nontrivial fundamental group in a Weierstraß type representation. As an application we construct a family of complete CMC immersions which are conformally cylinders and have umbilics.
"Construction of non-simply connected CMC surfaces via dressing." J. Math. Soc. Japan 55 (2) 335 - 364, April, 2003. https://doi.org/10.2969/jmsj/1191419120