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](https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-35/issue-2/Generic-properties-of-the-eigenvalue-of-the-Laplacian-for-compact/10.2748/tmj/1178229047.full) that the eigenvalues of this family depend continuously on the metrics.


Is this true for the eigenfunctions or eigenprojections as well?


I feel like the answer should be yes, arguing in coordinates that if the coefficients of the operators are close, their resolvents and hence their spectral projections are, but I am not sure how to prove it rigorously.
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 and also a duplicate of [this](https://math.stackexchange.com/questions/4246996/do-laplace-beltrami-eigenfunctions-vary-continuously-with-the-metric) question on stackexchange, so if I'm transgressing against any etiquette rules, please do let me know. :)



Do Laplace-Beltrami eigenfunctions vary continuously with the metric?
EDITED: Added clarification, as pointed out by @TerryTao.


Let $g\_1$, $g\_2$ be Riemannian metrics and $\Delta\_1$, $\Delta\_2$ their respective Laplacians. Let $\lambda\_1$ be an eigenvalue of $\Delta\_1$ and $\lambda\_2$ an eigenvalue of $\Delta\_2$.


I think what can be proved is the following: Given an eigenfunction $u\_2$ of $\Delta\_2$ with eigenvalue $\lambda\_2$, there exists an eigenfunction $u\_1$ of $\Delta\_1$ with eigenvalue $\lambda\_1$ such that
$$ \|u\_2-u\_1\|\_2 \le C(|\lambda\_2-\lambda\_1| + \|g\_2-g\_1\|\_{2,2})\|u\_2\|\_2, $$
where $C$ depends on both $g\_1$ and $\lambda\_1$ (specifically, the spectral gap between $\lambda\_1$ and the other eigenvalues of $\Delta\_1$), $\|\cdot\|\_{2,2}$ is the $W^{2,2}$ Sobolev norm with respect to a background metric $g\_0$, and $\|\cdot\|\_2$ is the $L^2$ norm with respect to $dV\_1$.


Here's a sketch of my proof:


Let $\lambda\_2$ be an eigenvalue of $\Delta\_2$ and $u\_2$ a nontrival eigenfunction. Let $\lambda\_1$ be an eigenvalue of $\Delta\_1$ and $u\_1$ a nontrival eigenfunction.


A straighforward calculation shows that
$$
 (-\Delta\_1+\lambda\_1)(u\_2-u\_1)
 = (\Delta\_2-\Delta\_1)u\_2 - (\lambda\_2-\lambda\_1)u\_2.
$$
Choose $u\_1$ so that, for any eigenfunction $u$ of $\Delta\_1$ with eigenvalue $\lambda\_1$,
$$ \int u(u\_2-u\_1)\,dV\_1 = 0. $$
The claim now follows by elliptic estimates.


