Almost-fuchsian structures on disk bundles over a surface

Considering an integer \(d>0\), we show the existence of convex-cocompactrepresentations of surface groups into SO(4,1) admitting an embedded minimal map withcurvatures in \((-1,1)\) and whose associated hyperbolic 4-manifolds are disk bundles of degreed over the surface, provided the genus \(g\) of the surface is large enough. We also show that we can realize these representations as complex variation of Hodge structures. This gives examples of quasicircles in \(\mathbb{S}^3\) bounding superminimal disks in \(\mathbb{H}^4\) of arbitrarily small second fundamental form. Those are examples of generalized almost-Fuchsian representations which are not deformations of Fuchsian representations.