Sharp Upperbound and a Comparison Theorem for the First Nonzero Steklov Eigenvalue

Let M denote either a noncompact rank-1 symmetric space (M, ds²) such that −4 ≤ K_M ≤ −1 or a complete, simply connected Riemannian manifold (M, g) of dimension n with K_M ≤ k where k = −δ² or 0. Let Ω be a bounded domain in M with smooth boundary ∂Ω = M and ν1(Ω) be the first nonzero Steklov eigenvalue on Ω. In the case M = (M, ds²), we prove

                                  ν1(Ω) ≤ ν1(B(R))