Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds

Minimal Surface Entropy and Average Area Ratio

“…The same argument as in Theorem 4.3 of [24] (adapted to the case where the minimal discs are not necessarily preserved by some surface group), shows that we can find a neighborhood U of g 0 and ε small so that for each γ ∈ C ε there is a unique non-degenerate area-minimizing disc Σ g (γ) with respect to the metric g so that ∂ ∞ Σ g (γ) = γ. The discs Σ g (γ) and D(γ) are at a bounded Hausdorff distance from each other (independent of g) and if g → g 0 then Σ g (γ) converges to D(γ) uniformly in C 2,α .…”

“…. + a Nm m = 1, so that: φ i m is equivariant with respect to a representation of a Fuchsian subgroup of PSL(2, R) in π 1 (M ) < PSL(2, C) and the laminar measure (24) δ m :=…”

Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

“…On the other hand, the classical result [21] verifies the existence of area-minimizing surface Σ i ⊂ (M, h) in the homotopy class of S i . And based on Theorem 4.3 of [18], there exist a C 2 -neighborhood U 0 of h 0 and N 0 ∈ N, so that when h ∈ U 0 and i ≥ N 0 , Σ i is the unique minimal surface in (M, h) homotopic to S i . Furthermore, let D i (Ω i ) be the lifts of S i (Σ i , respectively) in B 3 .…”