I'm interested in Laplace Beltrami operators $$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $g$ on $M$.
For any other smooth metric $\widetilde g$ on $M$, we can identify the square integrable functions with respect to its associated volume form with our original $L^2$ above, via the unitary map
$U:f\longmapsto \sqrt{|g|/|\widetilde g|}f.$
Under this unitary identification the Laplace-Beltrami operators corresponding to the various smooth metrics on $M$ yield a family of operators on a common domain of definition.
It has been proved that the eigenvalues of this family depend continuously on the metrics.
Is this true for the eigenfunctions (or eigenprojections) as well?
I know that if I consider an analytic one parameter family of metrics the result holds essentially by Kato's perturbation theory. But I am ultimately interested in proving that a certain composition of maps from functions on the manifold to $\mathbb R$ is robust to arbitrary, small (in the $C^\infty$-topology) changes in the metric and one of the maps in the composition is an operator of the form $f(-\Delta_g)$.
This is my first question here, so if I'm transgressing against any etiquette rules, please do let me know. :)