Costa Minimal Surface

$\begin{figure}\begin{center}\BoxedEPSF{CostaMinimalSurface.epsf}\end{center}\end{figure}$

A complete embedded Minimal Surface of finite topology. It has no Boundary and does not intersect itself. It can be represented parametrically by

$\begin{eqnarray*} x&=& {\textstyle{1\over 2}}\Re\left\{{-\zeta(u+iv)+\pi u +{\p... ...\pi} \ln\left\vert{\wp(u+iv)-e_1\over \wp(u+iv)+e_1}\right\vert, \end{eqnarray*}$

where $\zeta(z)$ is the Weierstraß Zeta Function, $\wp(g_2,g_3;z)$ is the Weierstraß Elliptic Function,

, and the invariants are given by

and

References

Costa, A. ``Examples of a Complete Minimal Immersion in of Genus One and Three Embedded Ends.'' Bil. Soc. Bras. Mat. 15, 47-54, 1984.

do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, 1993.

Gray, A. Images of the Costa surface. ftp://bianchi.umd.edu/pub/COSTAPS/.

Nordstrand, T. ``Costa-Hoffman-Meeks Minimal Surface.'' http://www.uib.no/people/nfytn/costatxt.htm.

Peterson, I. ``The Song in the Stone: Developing the Art of Telecarving a Minimal Surface.'' Sci. News 149, 110-111, Feb. 17, 1996.