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By Robert J. Adler

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Then dev(p, ·) parameterizes in arc-length the geodesic ray of H3 with end-points ρ(D(p)) ∈ P and D(p) (see Fig. 4). 5. COMPLEX PROJECTIVE STRUCTURES ON SURFACES 33 ρ(D(p)) D(p) Figure 4. The construction of the H-hull. 3. [41] dev is a C1,1 developing map for a hyperbolic structure on S × (0, +∞). Moreover it extends to a map 3 dev : S˜ × (0, +∞] → H such that dev|S×{+∞} is a developing map for the complex projective structure on ˜ S. We call such a hyperbolic structure the H-hull of S and denote it by H(S).

In fact, fix a point p0 ∈ L and consider a geodesic arc c transverse to the leaf l0 through p0 . There exists a neighbourhood K of p such that if a geodesic li meets K then it cuts c. Orient c arbitrarily and orient any geodesic li cutting c in such a way that respective positive tangent vectors at the intersection point form a positive base. Now for x ∈ L ∩ K define v(x) as the unitary positive tangent vector of the leaf through x at x. The following lemma ensures that v is a 1-Lipschitz vector field on L ∩ K (see [29] for a proof).

Conversely, by a classical result of Choquet-Bruhat and Geroch [27], given a scalar product g and a symmetric bilinear form b on S satisfying the GaussCodazzi equation, there exists a unique (up to isometries) maximal globally hyperbolic Lorentzian structure of constant curvature κ on S × R, such that - S × {0} is a Cauchy surface; - The first and the second fundamental form of S × {0} are respectively g and b. 2. Let us make precise what maximal means in this context. A constant curvature globally hyperbolic spacetime M is said maximal if every isometric embedding of M into a constant curvature spacetime M sending a Cauchy surface of M onto a Cauchy surface of M is an isometry.

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