Sharp upper bound and a comparison theorem for the first nonzero Steklov   eigenvalue
  In this paper we prove that given a volume, among all domains with smooth boundary in rank-1 symmetric spaces of noncompact type, geodesic balls maximizes the first nonzero Steklov eigenvalue. We also prove a comparison result for the first nonzero Steklov eigenvalue for domains in simply connected Riemannian manifolds with certain curvature bounds. 
Binoy, G. Santhanam
math.DG