Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space

Abstract

We define discrete flat surfaces in hyperbolic 3-space H 3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H 3 , and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H 3 , and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.

Lie geometry of flat fronts in hyperbolic space

“…Note that an imaginary parameter still yields a real geometry, as the semi-Riemannian symmetric space of point pairs in a conformal n-sphere is self dual; a change of sign of the curved flat parameter does not change the geometry of the curved flat, see[5, §5.5.19] 6. This, in fact, shows that flat fronts are Guichard surfaces, which were shown to be Ω-surfaces in[2].…”

Discrete linear Weingarten surfaces with singularities in Riemannian and Lorentzian spaceforms

“…For a K ≡ 1 surface with Weierstrass-type representation in H 3 , it was shown in [19] that, without loss of generality, |κ (m,n),(m,n+1) | > 1 and |κ (m,n),(m+1,n) | < 1 for all m and n, which we note in the following theorem: Theorem 6.1. In the case of a K ≡ 1 surface in H 3 with Weierstrass-type representation so that the horizontal edges have principal curvatures with absolute value greater than 1, the first inequality in Definition 6.1 is equivalent to the definition of singular vertices for discrete flat surfaces in H 3 as given in [19].…”

“…This is precisely what allowed for the description of singularities of discrete flat (i.e. K ≡ 1) surfaces in H 3 as given in [19]. Here we develop that into a definition without reliance on a Weierstrass type representation, extending it to all discrete surfaces in any M 3 with nonzero constant Gaussian curvature.…”

“…First we give Weierstrass-type representations for discrete surfaces using the more symmetric form of the base equation as in §6 of [19].…”

“…We can define singularities on discrete flat (i.e. K ≡ 1) surfaces in H 3 , now without referring to caustics as in [19], by simply using the condition…”