Kapouleas Surface (desingularized catenoids)
Kapouleas Surface (desingularized catenoids) is a minimal surface, given by a Weierstrass representation, immersed.
Open Kapouleas Surface (desingularized catenoids) in the interactive viewer →
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
Kapouleas' finite-total-curvature embedded minimal surfaces with arbitrarily many ends, built by desingularizing the circles of intersection of coaxial catenoids and planes with bent singly periodic Scherk surfaces. All period problems are solved numerically only (no simple existence proof), as the source page states; the k = 2 member is believed to admit no embedded example.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- 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".
- 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").
- 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, 'Kapouleas Surfaces', minimalsurfaces.blog (local mirror: minsurf/book/.../ch387_kapouleas_surfaces.md).
- N. Kapouleas, 'Complete embedded minimal surfaces of finite total curvature', J. Differential Geom. 47 (1997) 95-169.