Upper bounds for Steklov eigenvalues on surfaces
  We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper. 
Alexandre Girouard and Iosif Polterovich
math.SP math.DG