In this paper, we give a rigidity theorem for holomorphic disks associated to holomorphic families of Riemann surfaces, aiming for studying the geometry (the asymptotic boundary behaviors) of such holomorphic disks. Indeed, our rigidity theorem is described with distributions of associated holomorphic disks and orbits of monodromies at infinity.
"A criterion for holomorphic families of Riemann surfaces to be virtually isomorphic." Proc. Japan Acad. Ser. A Math. Sci. 91 (10) 151 - 154, December 2015. https://doi.org/10.3792/pjaa.91.151