The planar middle end of the Costa surface can be deformed into a catenoidal end. This gives a 1-parameter family of embedded minimal tori, as first announced in a 1987 paper by William Meeks and David Hoffman, described in Celso Costa’s Classification paper and proven to exist and be embedded in the the Finite Total Curvature paper by David Hoffman and Hermann Karcher.

With changing parameter, the separation of the two catenoidal necks becomes more pronounced.

Costa has shown that these are the only embedded, 3-ended minimal tori of finite total curvature. It is an open problem whether there are any other embedded minimal tori of finite total curvature (for infinite curvature, there is the genus one helicoid). It is also an open problem to come up with a conceptual proof of Costa’s result.

##### Resources

Mathematica Notebook

PoVRay Sources

D. Hoffman and W.H. Meeks III: Properties of properly embedded minimal surfaces of finite topology, Bull. Amer. Math. Soc. 17 (1987), 296-300.

C. J. Costa: Classification of complete minimal surfaces in **R**³ with total curvature 12π, Inventiones mathematicae 105 (1991), 273–303.

D. Hoffman, H. Karcher: Complete embedded minimal surfaces of finite total curvature, in Geometry V, Encyclopedia Math. Sci. 90 Springer, (1997) 5-93.

##### Related Surfaces

- Costa Surface
- Dihedralizations
- The Genus One Helicoid