K-noid

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Trinoid Trinoid.png
Trinoid
7-noid 7-noid.png
7-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 . [3] This produces the explicit formula

where is the Gaussian hypergeometric function and denotes the real part of .

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

  1. L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)
  2. N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv: math/0702469 .
  3. Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples" (PDF). Indiana.edu. Archived from the original (PDF) on 2019-07-12. Retrieved 2012-10-05.
  4. 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" (PDF). Math.uni-bonn-de. Retrieved 2012-10-05.
  5. Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. arXiv: 0804.4203 . Bibcode:2008arXiv0804.4203B. doi:10.2140/pjm.1995.171.353. S2CID   11328539.