I am trying to solve a physics problem in the context of the AdS/CFT correspondence, which is somewhat related to the question in spectral geometry: "Can one hear the shape of a drum?". The question I am trying to understand is: if the geometry is fixed at the boundary and the behavior of an eigenfunction of the Laplace operator at the boundary is also fixed, what can we tell about the interior geometry? I am trying to understand this in the following particular case.
Let us consider a 2D Riemannian manifold which is asymptotically the Poincaré disk. The situation I have in mind is one where, writing the metric in conformal coordinates \begin{equation} ds^2=e^{2\omega(z)}\left(dz^2+dt^2 \right)\,, \end{equation} we have \begin{equation} \lim_{z \to 0} e^{2\omega(z)} \sim \frac{1}{z^2} \end{equation} Notice moreover that I am assuming the geometry to have a symmetry, namely $\frac{\partial}{\partial t}$ is a Killing vector. Let us now consider an eigenfunction $f(z,t)$ of the Laplace operator in the manifold, namely \begin{equation} \Delta_g f=\lambda f \end{equation} with $\lambda>0$, whose asymptotic behavior can be seen to be given by \begin{equation} \lim_{z \to \infty} f(z,t) \sim z^{\Delta_{-}}A(t)+z^{\Delta_{+}}B(t)\,, \end{equation} with \begin{equation} \Delta_{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+4\lambda} \right)\,. \end{equation} This is fixed by the fact that the geometry is asymptotic to the Poincaré disk. Assume further that $A(t)$ and $B(t)$ are given and that $f(z,t)$ is smooth and does not blow up in the interior. In general $A(t)$ and $B(t)$ are independent functions, but imposing regularity of $f(z,t)$ in the interior relates them. So in some way the relationship between $A$ and $B$ gives us some information about the geometry in the interior. My question is if someone knows if this could entirely fix the geometry or not, in particular in the case where there is a Killing vector $\partial_t$. Even if it doesn't fix the geometry, does it impose important constraints on it? I know this is a somehow wide question but I would also be happy with references to important literature. Thanks!