Avoiding integers in the spectrum of the Laplacian of a Riemann surface Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant curvature $-1$. Recall that, from Teichmüller theory, this is a finite-dimensional subspace of the space of all metrics on $\Sigma$, and actually has dimension $6gen(\Sigma)-6$. For a given metric $g$, denote by $Spec(\Delta_g)=\{0=\lambda_0<\lambda_1\leq\lambda_2\leq\dots\}$ the spectrum of the Hodge-Laplacian of $g$, acting on $C^\infty(\Sigma)$.

Question 1: Given a natural number $n\in\mathbb N$, does there exist a hyperbolic metric $g\in\mathcal M(\Sigma)$ such that $n\notin Spec(\Delta_g)$?

I have attempted various ``deformation'' arguments to try to prove that if some $g\in\mathcal M(\Sigma)$ has $n$ as an eigenvalue, then a small perturbation $g'\in\mathcal M(\Sigma)$ would no longer have that eigenvalue, but without success... Instead of avoiding any given natural number, one can try the (seemingly) easier task:

Question 2: Does there exist a hyperbolic metric $g\in\mathcal M(\Sigma)$ such that $n\in Spec(\Delta_g)$ only for finitely many numbers $n\in\mathbb N$?

Or even more,

Question 3: Does there exist a hyperbolic metric $g\in\mathcal M(\Sigma)$ such that $\mathbb N\not\subset Spec(\Delta_g)$?

Although the answers to Q2 and Q3 seem to be "obviously yes", I have not been able to find a rigorous argument to prove that. I have tried to argue by contradiction, to show that if $Spec(\Delta_g)$ contains infinitely many natural numbers (or all of them), then we would somehow violate Weyl's formula (see this post) which says that $\lambda_k\sim \frac{k}{gen(\Sigma)-1}$ as $k\to+\infty$, but again without success (even in the case $gen(\Sigma)=2$).
 A: I was informed by Sugata Mondal at the MPI that Scott Wolpert proved the following result in his 1994 Annals paper Disappearance of cusp forms in special families:

Theorem 5.14. The eigenvalues of the Laplacian above $\tfrac14$ on a closed hyperbolic surface vary nontrivially under analytic deformations.

That is, if $g_t$ is a real-analytic path of hyperbolic metrics on $\Sigma$, then no real-analytic branch $\lambda_k(t)$ of an eigenvalue of the Laplacian $\Delta_{g_t}$ can be constant. Recall that by the Kato Selection Theorem, up to relabeling the indices $k$'s, the functions $\lambda_k(t)$ are real-analytic.
I also contacted Scott Wolpert, who confirmed that this indeed answers all my above questions. In particular, he had called Q1 the "middle C embarrassment question", since prior to the above result it could have been that middle C on the piano is a frequency for every hyperbolic Laplacian. Finally, note that his result also clearly allows to prescribe the genus of the closed hyperbolic surface $\Sigma$ at the same time as avoiding any given real number $\lambda>\tfrac14$ in its spectrum.
