The 1-parameter family of minimal surfaces described here appeared first in the 1995 paper Complete Embedded Minimal Surfaces of Finite Total Curvature by David Hoffman and Hermann Karcher.

 

They have been dubbed Finite Riemann surfaces because they look like a chopped down version of Riemann’s minimal surfaces, with the annuli completed by catenoidal ends.

These surfaces are never embedded. This is an illustration of the López-Ros theorem which asserts that a complete, embedded minimal sphere of finite total curvature must be the plane or catenoid.

These surfaces also support the Hoffman-Meeks conjecture that claims that for a complete, embedded minimal surface of finite total curvature, the number of ends cannot  exceed the genus by more than 2.

With P(z)=z^2+3 and Q(z)=z^2-1, we have the Weierstrass representation G(z)=\rho P(z)/Q(z) and dh = P(z)Q(z)\, dz. The López-Ros parameter \rho determines how much the catenoidal ends tilt.

Resources:

David Hoffman, Hermann Karcher: Complete embedded minimal surfaces of finite total curvature at arXiv, page 73. The arXiv version is lacking the images from the paper.

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