For an open Riemann surface R of finite genus, we shall consider conformal embeddings of R into closed ones of the same genus. The closing induces the Riemann's period matrix T. Masakazu Shiba and Hiroshi Yamaguchi recently showed that each diagonal element of T describes a close disk M in the upper half plane.

 In this talk, we shall consider the deforming open Riemann surface R(t) of finite genus with complex parameter t, and show the close relation between pseudoconvexity and the closed disk M(t) for R(t).

 This talk is based on a joint work with H.Yamaguchi.