Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres

Abstract

Let $\Omega \subset \mathbb{C}$ be a bounded simply connected Lipschitz domain with conformal metric $g=\omega |dz{|}^2$, where $\omega \in C^2(\Omega )\cap L^{\infty }(\Omega )$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\le K$, $K>0$, and $KM<4\pi$, where $M={\int }_{\Omega }\omega dA$, then the Neumann eigenvalues, numbered by $0={\mu }_1(\Omega ;\omega )<{\mu }_2(\Omega ;\omega )\le {\mu }_3(\Omega ;\omega )\le \cdots$, satisfy $\frac{1}{{\mu }_2(\Omega ;\omega )}+\frac{1}{{\mu }_3(\Omega ;\omega )}\ge \frac{2}{{\mu }_2(D_K(M))}$. Here $D_K(M)$ is the geodesic disk of area $M$ in the surface of constant curvature $K$. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford--Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.

arXiv comment38 pages, no figures