  We study the local Szeg\"o-Weinberger profile in a geodesic ball
$B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$.
This profile is obtained by maximizing the first nontrivial Neumann eigenvalue
$\mu_2$ of the Laplace-Beltrami Operator $\Delta_g$ on $\M$ among subdomains of
$B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic bounds of this
profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we
deduce a local comparison principle depending only on the scalar curvature. Our
study is related to previous results on the profile corresponding to the
minimization of the first Dirichlet eigenvalue of $\Delta_g$, but additional
difficulties arise due to the fact that $\mu_2$ is degenerate in the unit ball
in $\R^N$ and geodesic balls do not yield the optimal lower bound in the
asymptotics we obtain.
