I apologise for (possibly) misunderstanding your reply, but in my case I require the Monge-Ampere equation to be elliptic. The concavity of the equation (as a function of Hermitian matrices) is under question.

When n=3 and the manifold is orientable, it always does (orientable three manifolds are parallelisable)

Thanks for the answer. I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. If not, I guess I would have to learn the scheme stuff....

This is true, but, what if I don't demand that my eigenfunctions lie in L^2 (I just want them to be smooth functions satisfying the differential equation). I mean, in the example I gave, if one can prove that every eigenfunction has a limit as $r\rightarrow \infty$, then it extends to a function on the sphere and hence the eigenvalues will be the same.