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 $(d-1{)}^2\mathrm{/}4,\mathrm{\infty })$, 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.