A holomorphic fibration is a complex 2-dimensional manifold equipped with a holomorphic projection onto a hyperbolic Riemann surface, where the generic fibre is a closed Riemann surface, though a finite number of fibres may have singularities. The monodromy reflects the topological structure of a fibration.
The analytic structure of a holomorphic fibration can be studied through its classifying map: a holomorphic map from the hyperbolic surface to the moduli space of closed surfaces. The image of this map is called a holomorphic curve in the moduli space.
In this talk, we will explore the shape of a holomorphic curve and demonstrate that, when all peripheral monodromies are of infinite order, the holomorphic curve is quasi-isometrically immersed with respect to the intrinsic Kobayashi distance. Furthermore, by putting an additional limit
on the monodromy, we show that the lift of the holomorphic curve is quasi-isometrically embedded in the Teichmüller space, at least in part.
