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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.

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The references given in this answer to a related issue provide at least a partial answer to your question.

On a complete Riemannian manifold $(M,g)$, the Laplacian associated with $g$ is essentially self-adjoint on $C_0^\infty(M) \subset L^2(M)$. If $M$ is also compact, then its resolvent is also a compact operator. Therefore, under these two additional assumptions you will satisfy the hypotheses of the main theorem on page 1 in the second article linked there:

  • Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416. (pdf).

Cases (A)-(F) and (N) answer your question at a selection of regularity scales. I hope this helps.

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