Given a finite subgroup $\Gamma$ of $O(4)$ acting freely on $S^3$, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric $2$-tensor on $S^3/\Gamma$ equipped with the standard metric?

More precisely, I am interested in computing explicitly the eigenvalue $\lambda \ge 0$ such that $$ \Delta h +\lambda h=0 $$ where $h$ is a symmetric $2$-tensor on $S^3/\Gamma$ such that $\text{tr}_gh=\text{div}_gh=0$.

Notice that if $\Gamma$ is trivial, it follows from the paper "Symmetric Tensor Eigen Spectrum of the Laplacian on n Spheres" that $\lambda=m^2+2m-2$ for $m=2,3,\cdots$.

  • $\begingroup$ Have you tried asking Mark Rubin? $\endgroup$ – Ryan Budney May 2 '19 at 2:45

Any such eigenvalue must be in $\{m^2+2m-2:m\geq2\}$. Moreover, $\lambda=m^2+2m-2$ for some $m\geq2$ is an eigenvalue if and only if there is a $\Gamma$-invariant symmetric $2$-tensor on $S^3$ such that $\textrm{tr}_gh=\textrm{div}_gh=0$, and $\Delta h=\lambda h$.

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