  The aim of this study is to understand to what extent a 1-convex domain with
Levi-flat boundary is capable of holomorphic functions with slow growth. This
paper discusses a typical example of such domain, the space of all the geodesic
segments on a hyperbolic compact Riemann surface. Our main finding is an
integral formula that produces holomorphic functions on the domain from
holomorphic differentials on the Riemann surface. This construction can be seen
as a non-trivial example of $L^2$ jet extension of holomorphic functions with
optimal constant. As its corollary, it is shown that the weighted Bergman
spaces of the domain is infinite dimensional for any weight order greater than
$-1$ in spite of the fact that the domain does not admit any non-constant
bounded holomorphic functions.
