Lieb–Thirring Inequalities on Manifolds with Constant Negative Curvature

In this short note, we prove Lieb–Thirring inequalities on manifolds with negative constant curvature. The discrete spectrum appears below the continuous spectrum \([(d-1)^2/4, \infty )\) $[{(d-1)}^2/4,\infty )$ , where d is the dimension of the hyperbolic space. As an application, we obtain a Pólya type inequality with not a sharp constant. An example of a 2D domain is given for which numerical calculations suggest that the Pólya inequality holds for it.