Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

Abstract

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any flat metric on $S$ with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally hyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is given by $h$. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the hyperbolic surface, from which the flat metric can be easily recovered. We show next that given two such flat metrics on the surface, there exists a unique pair of future- and past-complete flat globally hyperbolic maximal (2+1)-spacetimes with the same holonomy, in which the flat metrics embed respectively as convex polyhedral Cauchy surfaces. The proof follows from convexity properties of the total length of the associated balanced geodesic cellulations over Teichmüller space.

Figures (5)

• Figure 1: The left hand side of the picture is a part of a balanced cellulation, around a face $f$. The green arrows in this picture are unit tangent vectors to the edges of the cellulation. In the picture in the middle, Euclidean strips are glued to every edge appearing in the left-hand side picture, in a direction orthogonal to the direction of the edge, and of width the weight of the edge. By the balance condition, the empty parts between the strips may be filled by convex flat polygons (colored brown). Gluing those flat polygons by identifying two segments that bound the same strip, produces a flat metric on the sphere (picture on the right hand side). These faces are glued around a vertex, and the sum of the interior angles of the faces at this vertex is equal to the sum of the exterior angles of the spherical convex polygon $f$. By the Gauss--Bonnet Formula, this sum plus the area of $f$ is equal to $2\pi$.
• Figure 2: A four-legged picture of the four corners around an edge $e$.
• Figure 3: On the left picture are two corners defining rows in $\mathcal{R}_r$, the doted edge between the two vertices having zero length. On the right is the corresponding corner defining a row in $\bar{\mathcal{R}}_r$. Lemma \ref{['zerocorn']} says that those rows are linearly dependent.
• Figure 4: For the proofs of Lemma \ref{['horiz']} and \ref{['vertic']}, the positions of rays in the intrinsic geometry of $\partial C$, which is isometric to the Euclidean plane. The idea of both proofs is to show that the ray $\chi'_1$ belongs to the correct side with respect to the plane $\Pi_0$, which is spanned by the rays $\chi_1$ and $\chi_2$ in space.
• Figure 5: To Lemma \ref{['lem:continuite Ut']}: vectors $U_{e,v}(0)$ for $t>0$ and $t<0$ are different.

Theorems & Definitions (113)

• Theorem 1.1: Alexandrov Theorem, extrinsic version
• Theorem 1.2: Alexandrov Theorem, intrinsic version
• Theorem I: Hyperbolic Alexandrov Theorem, intrinsic version
• Theorem I': Hyperbolic Alexandrov Theorem, extrinsic version
• Theorem II'
• Theorem II
• Theorem 1.3: Iskhakov, iskhakov
• Theorem 1.4
• Theorem 1.5
• Definition 2.1