Minimal surfaces in hyperbolic space and maximal surfaces in Anti-de Sitter space

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichm\"uller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichm\"uller distance between the ends of a quasi-Fuchsian manifold $M$ is at most C, then $M$ is almost-Fuchsian. We also prove an estimate maximal surfaces with bounded second fundamental form in Anti-de Sitter space, when the boundary at infinity is the graph of a quasisymmetric homeomorphism $\phi$ of the circle. The supremum of the principal curvatures of the maximal surface is estimated again in a sublinear way, in terms of the cross-ratio norm of $\phi$, if the latter is sufficiently small. This provides an estimate on the maximal distortion of the quasiconformal minimal Lagrangian extension to the disc of a given quasisymmetric homeomorphism. The main ingredients of the proofs are estimates on the convex hull of a minimal/maximal surface and Schauder-type estimates to control principal curvatures.