For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k \in \mathbb{N}}$ of the Laplace-Beltrami operator $-\Delta_s$ associated to $M_s$ satisfy $$s \mapsto \varphi_k(s)\text{ and }s \mapsto \lambda_k(s) \quad\text{are measurable (or continuous)?}$$
We obviously need some continuity of the $s \mapsto M_s$, but I'm not sure how to phrase it precisely. In my case $M_s$ are Euclidean hypersurfaces but I don't know if this affords much of a simplification to the problem.
There being no explicit representation of the eigenelements means that I am a bit stuck on this problem.
