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 and , we have the Weierstrass representation and . The López-Ros parameter 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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