Payne-Pólya-Weinberger inequalities on closed Riemannian manifolds

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 $n\ge 3$, and we prove explicit bounds in terms of geometric quantities such as scalar curvature, Yamabe constant, isoperimetric constant and conformal volume.