In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by their period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that we use are to solve a Jenkins-Serrin problem over a strip domain and to consider the conjugate minimal surface to the graph. To construct the Jenkins Serrin graph, we solve in fact the maximal surface equation and use an other conjugation technic.
"Adding one handle to half-plane layers." J. Differential Geom. 84 (2) 389 - 407, February 2010. https://doi.org/10.4310/jdg/1274707318