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

Abstract

We study the local Szegö–Weinberger profile in a geodesic ball $$B_g(y_0,r_0)$$ centered at a point $$y_0$$ in a Riemannian manifold $$({\mathcal {M}},g)$$ . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $$\mu _2$$ of the Laplace–Beltrami Operator $$\Delta _g$$ on $${\mathcal {M}}$$ 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 $${\mathcal {M}}$$ 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 $$\mathbb {R}^N$$ and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.