The Heights Theorem for infinite Riemann surfaces

Quadratic differentials and foliations on infinite Riemann surfaces

“…there is a subsequence of ϕ n that converges uniformly on compact subsets to an integrable holomorphic quadratic differential on X. Since the heights of the limiting quadratic differential are identical to the heights of ϕ, the Heights Theorem [38] gives that the limit is ϕ. The theorem is proved.…”

“…Since X |ϕ n | ≤ D(F n ) ≤ M , we conclude that a subsequence of ϕ n converges to a holomorphic quadratic differential ψ ∈ A(X). Since the heights of F n are equal to the heights of ϕ n and the heights of F are equal to the heights of ϕ, it follows that ψ = ϕ by the Heights Theorem [38]. Therefore every subsequence of {ϕ n } n has a subsequence converging to the same limit ϕ in the L 1 -norm.…”

“…Let B n be the subleaves of the horizontal foliation F ϕ that connect the two boundary components of C n . Note that by [38,Lemma 6.1] we have…”

“…Let w := u + iv = * ϕ(z)dz be the natural parameter of ϕ. where B n stands for the region in C n which is the union of the leaves of B n . Moreover, in the natural parameter the leaves of B n are horizontal arcs and we obtain (see [38] or [7])…”

“…By the push-forward of the transverse measure |Im( √ ϕ)| on the horizontal trajectories, we obtain a map ϕ → µ ϕ from A(X) to the space of measured lamination M L(X). By the Heights Theorem [38], we have that the straightening map ϕ → µ ϕ is injective. We need to characterize the image of A(X).…”