Weber-Wolf Surface of genus 3 with 5 ends

The Hoffman-Meeks conjecture is possibly the most important open problem in the theory of minimal surfaces. It states that an embedded minimal surface of finite total curvature of genus g can have at most g+2 ends. This conjecture is open even for g=1. In the borderline case we have the catenoid for g=0, the Costa surface for g=1, and Wohlgemuth’s surface for g=2. The Weber-Wolf surface establishes the borderline case for g=3. They have two catenoidal and three planar ends. At the level of each planar end, the connections to the ends below and above are realized by “Costa-saddles”.
Similar surfaces also exist for all odd genera, but pictures are hard to make. The existence of borderline surfaces for even genus g≥4 has never been established, not even numerically.