Costa-Hoffman-Karcher-Meeks Torus
Costa-Hoffman-Karcher-Meeks Torus is a minimal surface, given by a Weierstrass representation, immersed.
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aperiodic def-weierstrass immersed implemented minimal tradition-classical
Properties
- Family
- minimal
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
The 1-parameter family of embedded minimal tori obtained by deforming the Costa surface's planar middle end into a catenoidal end; by Costa's classification these are the ONLY embedded 3-ended minimal tori of finite total curvature. Distinct from the genus-varying Costa-Hoffman-Meeks record, which fixes the planar middle end and raises the genus.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- H. Karcher, "Construction of minimal surfaces" (1989) for the symmetrization method (double Enneper, k-noid families);
- L. P. Jorge, W. H. Meeks III, Topology 22 (1983) for the k-noids; data after M. Weber, minimalsurfaces.blog ("Symmetrized Double Enneper", "k-Noids with Enneper Ends", "Antiprismatic k-Noids").
- B. Riemann (1867) and F. J. Lopez, A. Ros, J. Differential Geom. 33 (1991) for the (never embedded) finite Riemann family; the symmetrized member follows M. Weber, minimalsurfaces.blog, "Symmetrized Finite Riemann".
- L. Henneberg (1875); R. Kusner, Bull. Amer. Math. Soc. 17 (1987) 291-295; F. J. Lopez, Duke Math. J. 71 (1993) 23-30 -- full citations in the weierstrass symtail engine block.
- M. Weber, 'Costa-Hoffman-Karcher-Meeks Tori', minimalsurfaces.blog (local mirror: minsurf/book/.../ch241_hoffman_karcher_tori.md).
- 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 R3 with total curvature 12pi', Invent. Math. 105 (1991) 273-303.
- D. Hoffman and H. Karcher, 'Complete embedded minimal surfaces of finite total curvature', Geometry V, Encyclopaedia Math. Sci. 90, Springer (1997) 5-93.