The Hoffman-Meeks conjecture predicts in its simplest unsolved case that there is no minimal, embedded torus of finite total curvature with four (or more) ends. But one can try:

This is in fact a finite total curvature surface of genus 1 with four individually embedded catenoidal ends. Additional natural symmetry assumptions have been made to help solving the period problems. However, the surface is not embedded, because the top and bottom catenoidal ends grow faster than the middle ones. One can change the growth rates of these ends but always faces eventually the same problem.

This just shows that these particular examples are not counterexamples to the Hoffman-Meeks conjecture.

These surfaces are described as CCCC-surfaces in Meinhard Wohlgemuth’s 1997 paper, and he cites Pascal Romon’s thesis from 1993 for a proof that they are not embedded.

##### Resources

M. Wohlgemuth: Minimal Surfaces of Higher Genus with Finite Total Curvature, Arch. Rational Mech. Anal. 137 (1997) 1–25

PoVRay Sources