Discrete Weierstrass-type representations. (English) Zbl 1526.53015
The authors show that the known discrete Weierstrass-type representations of certain surface classes, such as discrete \(L\)-isothermic surfaces, can be understood as applications of the \(\Omega\)-dual transform to light-like Gauss maps in Laguerre geometry, giving rise to further Weierstrass-type representations. This leads to a proof that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps. They also show that the Calapso transformation descends to the Lawson correspondence and gives rise to a discrete analogue of the duality for constant mean curvature \(1\) surfaces in \({\mathbb H}^3\) developed in [M. Umehara and K. Yamada, Tsukuba J. Math. 21, No. 1, 229–237 (1997; Zbl 1027.53010)].
Reviewer: Victor V. Pambuccian (Glendale)
Keywords:
discrete surfaces; discrete Weierstrass representations; discrete isothermic sphere congruences; Calapso transformation; Lawson correspondenceCitations:
Zbl 1027.53010References:
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