New proofs of the Torelli theorems for Riemann surfaces
  In this paper, by using the Kuranishi coordinates on the Teichm\"uller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces. 
Kefeng Liu, Quanting Zhao and Sheng Rao
math.DG math.AG