Sharp local estimates for the Szegö-Weinberger profile in Riemannian manifolds

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We study the local Szegö-Weinberger profile in a geodesic ball Bg(y₀,r₀) centered at a point y₀ in a Riemannian manifold (,g). This profile is obtained by maximizing the first nontrivial Neumann eigenvalue μ₂ of the Laplace-Beltrami Operator Δg on among subdomains of Bg(y₀,r₀) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of at y₀. 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 Δg, but additional difficulties arise due to the fact that μ₂ is degenerate in the unit ball in ⁿ and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.