A rigidity theorem for holomorphic disks in Teichmüller space

In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains with $C^{2}$-boundaries, but the rigidity property does not hold for product manifolds.