Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in $\mathbb{R}^{n}$, spheres, and closed orientable surfaces

We establish lower bounds for sums of reciprocals of Laplacian eigenvalues for bounded Euclidean domains, spheres and surfaces. For bounded Euclidean domains, we recover a sharp inequality that extends Bucur and Henrot’s upper bound on the second Neumann eigenvalue. For spheres and surfaces, we obtain new Li- and Yau-type bounds and we discuss them in terms of conformal volume.