Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ is smooth in time, or $k = \omega$, when the one-parameter family $g_t$ is real analytic in time. Now, as the metric varies, the Laplacian associated to the metric varies, and hence its spectrum also varies in time. My question is, if $g_t$ is $C^k$ in time, are the eigenvalues of the Laplacian also $C^k$ in time for $k$ integer, $k = \infty$ or $k = \omega$? In case such results are well-known (which I am assuming they are), what is a good reference to learn about such results? Thanks for any guidance.
Edit after Umberto Lupo's answer: I was somewhat more interested in the compact case, which Umberto's answer and the paper he cites cover very nicely. I would, however, be highly interested in learning about possible results in the non-compact setting as well.