Carleson measures on bounded C-convex domains

In this paper, we completely characterize the Carleson and the vanishing Carleson measures on bounded $C$-convex domains in terms of the Carathéodory (or the Kobayashi) geometry and Bergman geometry. As a corollary, we obtain a sufficient and necessary condition for the composition operators on $C$-convex domains to be bounded on associated Bergman spaces.