Torus With One Catenoid and One Enneper End

In the same way one can grow a catenoid end out of an Enneper surface, one can do this to the Chen-Gackstatter surface. When keeping the reflectional symmetries, this leads to a 1-parameter family of surfaces that limits on one side at the Enneper surface and on the other side at the Chen-Gackstatter surface. They have the largest possible total curvature for 2-ended minimal tori. There are also very likely less symmetric examples.

The Weierstraß data can be given by

$G(z) = \rho (z-1)\frac{\sqrt{z-a}}{\sqrt z \sqrt{z-b}}\qquad\text{and}\qquad dh = \frac{z-1}{z}\, dz$

and one has to solve a 1-dimensional period problem involving elliptic integrals. The solutions are, as one would expect, not very explicit, but there is a curiosity: a=-1 and b=-1/2 provides an exact solution. I have only verified this numerically, and I don’t know whether that particular surface (below) has any special features.

angel-2