  Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian
manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for
$\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove
that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and
Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected
space form with constant sectional curvature $\kappa$ enclosing the same volume
as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where
$\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov
eigenvalues of $\Omega$ and $\Omega^*$ respectively, and $C=C(n,\kappa, K, d)$
is an explicit constant. When $\kappa=K$, 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.
