Maximal surfaces in anti-de Sitter 3-manifolds with particles

Minimal diffeomorphism between hyperbolic surfaces with cone singularities

“…If moreover for all i ∈ {1, ..., n}, α i < α ′ i then Ψ is unique. The proof of this result is totally different from the proof in [Tou13]. Here, we study the energy functional over T (Σ p ), the Teichmüller space of Σ p .…”

“…For a marked surface Σ p := Σ \ p where p = (p 1 , ..., p n ) ⊂ Σ and for α := (α 1 , ..., α n ) ∈ 0, 1 2 n such that χ(Σ) + n i=1 (α i − 1) < 0, one can construct the Fricke space of Σ p with cone singularities of angle α, denoted by F α (Σ p ), as the moduli space of marked (where the marking fix each p i ∈ p) hyperbolic metrics on Σ p with cone singularities of angle 2πα i at the p i (see Section 2 for the construction). In a previous paper [Tou13], we proved the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity for each pair of points g 0 , g 1 ∈ F α (Σ p ) (that is when the cone angles of g 1 and g 2 are equal). The proof of this result used the deep relations between three dimensional AdS geometry and hyperbolic surfaces: we showed the existence of a unique surface maximizing the area in some AdS singular spacetimes, and proved that it implies the existence of a unique minimal Lagrangian map (see [Tou13] for more details).…”

“…In a previous paper [Tou13], we proved the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity for each pair of points g 0 , g 1 ∈ F α (Σ p ) (that is when the cone angles of g 1 and g 2 are equal). The proof of this result used the deep relations between three dimensional AdS geometry and hyperbolic surfaces: we showed the existence of a unique surface maximizing the area in some AdS singular spacetimes, and proved that it implies the existence of a unique minimal Lagrangian map (see [Tou13] for more details). In this paper, we address the question of the existence and uniqueness of minimal diffeomorphism between hyperbolic cone surfaces with different cone angles.…”

Constant Gauss curvature foliations of AdS spacetimes with particles

“…Indeed, it is obtained from the complete hyperbolic surface with a cone singularity of angle θ 0 by taking a warped product with R (see e.g. [7,12,22]).…”

“…Moreover, GH Σ,θ can also be parameterized by the cotangent bundle T * T Σ,θ of T Σ,θ , since T * T Σ,θ is homeomorphic to the quotient of the space H Σ,θ of maximal surfaces in germs of AdS manifolds with particles by diffeomorphisms isotopic to the identity fixing each marked point of Σ (see [12,Theorem 5.11]) and there is a bijection between this quotient space and GH Σ,θ (see [22,Theorem 1.4]).…”