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Compact constant mean curvature surfaces with low genus. (English) Zbl 0898.53009

The authors describe numerical experiments and show a lot of graphical images that suggest the existence of certain new compact surfaces of constant mean curvature (CMC). They come in three dihedrally symmetric families, with genus ranging from 3 to 5, 7 to 10, and 3 to 9, respectively; they are further surfaces with the symmetry of Platonic polyhedra and genera 6, 12, and 30. The authors use the algorithm of B. Oberknapp and K. Polthier [in: Visualization and mathematics, Proc. Int. Workshop Berlin-Dahlem 1995 (Springer, Berlin), 141-161 (1997; Zbl 0898.53001)], which generalizes an algorithm for discrete harmonic maps and minimal surfaces by U. Pinkall and K. Polthier [Exp. Math. 2, 15-36 (1993; Zbl 0799.53008)], which defines a discrete version of Lawson’s conjugate surface method. There are two steps: minimizing area (in fact, discrete Dirichlet energy) in the 3-sphere, and conjugating the discrete surface to a CMC surface in Euclidean 3-space.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-04 Software, source code, etc. for problems pertaining to differential geometry
39A12 Discrete version of topics in analysis
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