I'm looking for the approach to compare PDE solutions on the Remannian manifolds when those solutions are obtained under two different metrics. To be more specific, suppose we have two Riemannian manifolds $(M,g_{1})$ and $(M,g_{2})$, partial differential operator $D$ is dependent on the metric. Then is there any way (I'm not having in mind any specific way) to compare solutions $u$, o the equation $Du=0$?
I know it is rather vague question. I don't even know in what way those solutions to compare. Any advice, link to some literature would be highly appreciated.
Lets assume you have a diffeomorphism $\phi:M_1 \mapsto M_2 $ then I would map the solution on $M_2$ back to $M_1$ via the inverse mapping $\phi^{-1}$ and compare the two solutions.
I hope I understood your problem correctly ?