Weber-Wolf Surface (genus 3, 5 ends)

Weber-Wolf Surface (genus 3, 5 ends) 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

Weber and Wolf's surface with two catenoidal and three planar ends realizing the borderline case g = 3 of the Hoffman-Meeks conjecture (at most g + 2 ends for an embedded finite-total-curvature surface of genus g); the planar levels connect through Costa saddles. The page states similar surfaces exist for all odd genera; a companion page shows a genus-4 version.

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".
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").
H. Karcher, "Construction of minimal surfaces" (1989) for the symmetrization method (double Enneper, k-noid families);
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, 'Weber-Wolf Surface of genus 3 with 5 ends', minimalsurfaces.blog (local mirror: minsurf/book/.../ch415_weber_wolf_surface_of_genus_3_with_5_ends.md).
M. Weber, 'Weber-Wolf Surface of Genus 4 with 5 Ends', minimalsurfaces.blog (local mirror: minsurf/book/.../ch416_weber_wolf_surface_of_genus_4_with_5_ends.md).