An upper bound for the first nonzero Steklov eigenvalue

Let ( M n , g ) be a complete simply connected n -dimensional Riemannian manifold with curvature bounds Sect g ≤ κ for κ ≤ 0 and Ric g ≥ ( n − 1) Kg for K ≤ 0. We prove that for any bounded domain Ω ⊂ M n with diameter d and Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ 1 (Ω) ≤ C σ 1 (Ω*), where σ 1 (Ω) and σ 1 (Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, and C = C ( n , κ, K , d ) is an explicit constant. When κ = K , we have C = 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.