Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless symmetric $2$-tensor $h$ (i.e. $\text{div }h=\text{Tr}_gh=0$) on $M$. Here, $$\Delta_L h_{ij}=\Delta h_{ij}+2Rm_{ikjl}h_{kl}-R_{ik}h_{jk}-R_{jk}h_{ik}=\Delta h_{ij}+2Rm_{ikjl}h_{kl}-2(n-1)h_{ij}.$$
Can we find a constant $C=C(n)$ such that $\lambda \ge C?$
This is not a real answer but just a suggestion for where to browse.
What I am aware of is that the sign of the first eigenvalue is related to the (non)existence of variations that preserve the volume but increase scalar curvature, see the recent preprint https://arxiv.org/pdf/2109.09556.pdf by M. Dahl and K. Kroencke. In particular, they cite the following paper that is potentially interesting: H.-D. Cao and C. He,Linear stability of Perelman’sν-entropy on symmetric spacesof compact type, J. Reine Angew. Math.709(2015), 229–246 (available on the arXiv as https://arxiv.org/pdf/1304.2697.pdf ) exhibits examples for which the first eigenvalue of the Lichnerowicz Laplacian is negative. The explicit calculation of the eigenvalue is potentially done in the references of this paper but I am unable to access them.
