Flat fronts in hyperbolic 3-space

Abstract

We investigate flat surfaces in hyperbolic 3-space with admissible singularities, called flat fronts. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded, and give new examples for which equality holds.

Singularities of flat fronts in hyperbolic space

“…In particular, Izumiya and Takeuchi [2003] proved that the set of developable surfaces whose singularities are only cuspidal edges, swallowtails or cuspidal cross caps are open and dense in the set of noncylindrical developable surfaces, where (u, v) → (u, v 2 , v 3 ) represents a cuspidal edge, (u, v) → (3u 4 + u 2 v, 4u 3 + 2uv, v) a swallowtail, and (u, v) → (u, uv 3 , v 2 ) a cuspidal cross cap. Recently, geometric inequalities for complete flat fronts in hyperbolic 3-space and complete maximal surfaces with certain singularities in Minkowski 3-space were found in [Kokubu et al 2004] and [Fujimori et al 2005]. Kitagawa [1988;1995;2000] has made a deep investigation of flat tori in the 3-sphere.…”

Minimal Darboux transformations

“…By Corollary 3.1, resp [13,Cor 3.4], the Gauss maps n andn of a minimal Darboux pair (f,f ) in Euclidean space form a (degenerate) Darboux pair themselves, hence a curved flat in the space of point pairs in the 2-sphere, see [7] or [11, §3.3.2 or §5.5.20]. Consequently, interpreting the interior of the 2-sphere as a Poincaré ball model of hyperbolic space, (n,n) qualifies as the pair of hyperbolic Gauss maps of a parallel family of flat fronts, see [12]. More precisely, linearization of of the Riccati equations Conversely, the hyperbolic Gauss maps of a flat front in hyperbolic space form a curved flat in the 2-sphere, hence determine a minimal Darboux pair in Euclidean space.…”

A connection between flat fronts in hyperbolic space and minimal surfaces in Euclidean space

“…This canonical Riemann surface structure provides a conformal representation for the immersion X that allows one to represent any flat front in H 3 in terms of holomorphic data (see [6], [7] and [11] for the details).…”

“…In fact, adapting Theorem 2.11 in [11] to our model of hyperbolic space, we have the following holomorphic representation: …”

“…They also share in common the fact that there are many interesting global theorems about their geometry and topology, see for instance [7], [11] and [17]. From the point of view of partial differential equations, both classes are intimately related to the Monge-Ampère equation.…”