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McCarthy, Alan; Schief, Wolfgang K.

On the combinatorics of Demoulin transforms and (discrete) projective minimal surfaces. (English) Zbl 1361.53013

Discrete Comput. Geom. 57, No. 1, 215-230 (2017).
The authors show that the classical Demoulin transformation for a projective minimal surface generates a \(\mathbb Z^2\) lattice of projective minimal surfaces generically which they call a Demoulin lattice. Discussing geometric properties of Demoulin lattices, they introduce the notion of lattice Lie quadrics, associated discrete envelopes and the definition of discrete PMQ-surfaces which are discretizations of either projective minimal surfaces or so-called Q-surfaces. They also prove that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQ-surface constitutes an envelope of the lattice Lie quadrics associated with the other.
Reviewer: Atsushi Fujioka (Osaka)

Cited in 2 Documents

MSC:

53A20 Projective differential geometry
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

Demoulin transformation; Lie quadric; projective minimal surface; discrete differential geometry
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References:

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