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 $C^\infty(\Sigma)$.



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



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> **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 $n\in\mathbb N$?
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Or even more,



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> **Question 3:** Does there exist a hyperbolic metric $g\in\mathcal M(\Sigma)$ such that $\mathbb N\not\subset Spec(\Delta\_g)$?
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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](https://mathoverflow.net/questions/91397/exotic-spectrum-of-laplace-operator)) which says that $\lambda\_k\sim \frac{k}{gen(\Sigma)-1}$ as $k\to+\infty$, but again without success (even in the case $gen(\Sigma)=2$).



Avoiding integers in the spectrum of the Laplacian of a Riemann surface
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*:



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


