k-noid
In differential geometry, a k-noid is a minimal surface with k catenoid openings. In particular, the 3-noid is often called trinoid. The first k-noid minimal surfaces were described by Jorge and Meeks in 1983.[1]
The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").[2]
k-noids are topologically equivalent to k-punctured spheres (spheres with k points removed). k-noids with symmetric openings can be generated using the Weierstrass–Enneper parameterization [math]\displaystyle{ f(z) = 1/(z^k-1)^2, g(z) = z^{k-1}\,\! }[/math].[3] This produces the explicit formula
- [math]\displaystyle{ \begin{align} X(z) = \frac{1}{2} \Re \Bigg\{ \Big(\frac{-1}{kz(z^k-1)} \Big) \Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k)\\ & {}-(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k) \\ &{}-kz^k +k+z^2-1 \Big] \Bigg\} \end{align} }[/math]
- [math]\displaystyle{ \begin{align} Y(z) = \frac{1}{2} \Re \Bigg\{ \Big(\frac{i}{kz(z^k-1)}\Big) \Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k) \\ &{}+(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k)\\ & {}-kz^k+k-z^2-1 ) \Big] \Bigg\} \end{align} }[/math]
- [math]\displaystyle{ Z(z) =\Re \left \{ \frac{1}{k-kz^k} \right\} }[/math]
where [math]\displaystyle{ _2F_1(a,b;c;z) }[/math] is the Gaussian hypergeometric function and [math]\displaystyle{ \Re \{z\} }[/math] denotes the real part of [math]\displaystyle{ z }[/math].
It is also possible to create k-noids with openings in different directions and sizes,[4] k-noids corresponding to the platonic solids and k-noids with handles.[5]
References
- ↑ L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)
- ↑ N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv:math/0702469.
- ↑ Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples". Indiana.edu. http://www.indiana.edu/~minimal/research/claynotes.pdf. Retrieved 2012-10-05.
- ↑ H. Karcher. "Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96". Math.uni-bonn-de. http://www.math.uni-bonn.de/people/karcher/karcherTokyo.pdf. Retrieved 2012-10-05.
- ↑ Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. doi:10.2140/pjm.1995.171.353. Bibcode: 2008arXiv0804.4203B.
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