# Spectrum of the Laplacian on the quotient of $3$-sphere

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 $$m=2,3,\cdots$$.

• Have you tried asking Mark Rubin? – 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 $$\Delta h=\lambda h$$.