Abstract
We give a short introduction to discrete flat fronts in hyperbolic space and prove that any discrete flat front in the mixed area sense admits a Weierstrass representation.
Highlights
Flat fronts in hyperbolic space—that is, intrinsically flat surfaces in hyperbolic space that may have certain types of singularities—yield an interesting class of surfaces/fronts for their relations to various areas in differential geometry and, in particular, for the study of singularities of special surfaces: they admit a Weierstrass type representation [10] that is of a rather geometric nature as it is directly linked to the geometry of a pair of hyperbolic Gauss maps that form a curved flat in the integrable systems sense [4]
In a sphere geometric context, these hyperbolic Gauss maps are related to a pair of isothermic sphere congruences that put flat fronts in the context of Demoulin’s Ω surfaces and, in particular, of linear Weingarten surfaces [5]; on the other hand, flat fronts occur in parallel families as orthogonal surfaces to a cyclic system spanned by the pair of hyperbolic Gauss maps, c.f. [4]
Any discrete flat front (X, N ) : Σ20 → H3 × S2,1 obtained from the Weierstrass representation (3.3) is a discrete linear Weingarten net with constant Gauss curvature K ≡ 1
Summary
Flat fronts in hyperbolic space—that is, intrinsically flat surfaces in hyperbolic space that may have certain types of singularities—yield an interesting class of surfaces/fronts for their relations to various areas in differential geometry and, in particular, for the study of singularities of special surfaces: they admit a Weierstrass type representation [10] that is of a rather geometric nature as it is directly linked to the geometry of a pair of hyperbolic Gauss maps that form a curved flat in the integrable systems sense [4] This observation establishes a relation with Darboux pairs of minimal surfaces in Euclidean space [12], c.f. A relation has been established by showing that the nets obtained by the discrete Weierstrass-type representation satisfy the
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