Costa's Surface
Costa's Surface is a minimal surface, given by a Weierstrass representation, genus 1, 3 ends, embedded.
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aperiodic complete-finite-total-curvature def-weierstrass embedded implemented minimal tradition-classical
About
For over two centuries the only known embedded minimal surfaces of finite topology were the plane, the catenoid and the helicoid, and it was widely assumed there were no others. Costa's surface, found in 1982, was the counterexample: it has genus one and three ends, and it does not cross itself. The picture came before the proof -- it was Hoffman and Meeks who established that it really is embedded, using computer images to see what to prove.
Properties
- Family
- minimal
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- not periodic
- Genus
- 1
- Ends
- 3
- Embedding
- embedded
- Orientable
- yes
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
No (g, dh) pair is stored: the shipped immersion is the Gray/Nylander closed form in Weierstrass elliptic functions (zeta, wp) on the square torus (math_art/minsurf/parametric.py, _costa_xyz) -- outside the elementary expression language. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- C. J. Costa, 'Example of a complete minimal immersion in R^3 of genus one and three embedded ends', Bol. Soc. Bras. Mat. 15 (1984).
- D. Hoffman and W. H. Meeks III, 'A complete embedded minimal surface in R^3 with genus one and three ends', J. Diff. Geom. 21 (1985) -- the proof that Costa's example is embedded.
- C. J. Costa, 'Classification of complete minimal surfaces in R^3 with total curvature 12pi', Invent. Math. 105 (1991) 273-303 -- independent confirmation of the recorded -12pi.