Sharp local estimates for the Szeg\"o-Weinberger profile in Riemannian manifolds

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.