Payne-Pólya-Weinberger inequalities on closed Riemannian manifolds
Mehdi Eddaoudi
TL;DR
The paper investigates Payne–Pólya–Weinberger-type inequalities for Laplace–Beltrami eigenvalues on closed Riemannian manifolds by linking spectral gaps to intrinsic geometry. It develops a topological, vector-field-based construction to produce admissible test functions on the sphere and extends this approach to general manifolds via conformal immersions and the $m$-conformal volume, combined with sharp Sobolev embeddings to control Rayleigh quotients. Key results include a sphere-conformal bound $\lambda_{2k+1}-\left(1+\frac{4}{n-2}\right)\lambda_{2k} \le \frac{1}{n-1}\max_M S_g$, and general inequalities of the form $\overline{\lambda}_{2k+1}-\overline{\lambda}_{2k}\left(1+\frac{8 C^{*2}V_c(m,M,[g])^{2/n}}{(n-2) C^2(M,g) w_n^{2/n}}\right)\le 4n V_c(m,M,[g])^{2/n}$, with variants under positive Yamabe constant, Ricci curvature lower bounds, and isoperimetric constants. These results tie eigenvalue gaps to curvature, conformal invariants, and volume-geometry quantities, offering a framework for constructing metrics with controlled spectral gaps. The methods blend Hopf–Poincaré/topological arguments with sharp Sobolev inequalities and conformal geometry concepts, yielding broadly applicable geometric spectral bounds. The work advances understanding of how intrinsic geometry governs universal eigenvalue inequalities beyond Euclidean domains.
Abstract
Payne-Pólya-Weinberger inequalities are known to be exclusive to bounded Euclidean domains with Dirichlet boundary condition. In this paper, we discuss the corresponding inequalities on Riemannian manifolds of dimension $n \geq3$, and we prove explicit bounds in terms of geometric quantities such as scalar curvature, Yamabe constant, isoperimetric constant and conformal volume.