Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence \begin{equation} 0=\lambda_0<\lambda_1\leq \lambda_2\leq ... \leq \lambda_k\nearrow\infty \end{equation} To my knowledge, a generic surface will not have eigenvalues of multiplicity $>1$ (this is a result of K.Uhlenbeck, see https://www.jstor.org/stable/2374041). On the other hand, any surface with degenerate eigenvalues (such as the Bolza surface or Klein quartic), which I am aware of, usually has a lot of symmetries (in the sense that the automorphism group of $S$ is large), which cause the high multiplicities. Are there any known inverse results of the type "If a hyperbolic surface $S$ has an eigenvalue of multiplicity >1, then it has non-trivial automorphism group". I was unable to spot any related results in the literature, so I want to to ask whether there is anything known in this direction or if there are any examples of hyperbolic surfaces violating this idea.
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3$\begingroup$ Heuristically, if you have a "generic" one-parameter family of surfaces which deforms the spectra a large amount (larger than eigenvalue gaps), you always expect to have parameter values for which the spectrum will have multiplicity. Hyperbolic surfaces of sufficient genus with trivial automorphism groups form a many-parameter family along which the spectrum is deformed by a large amount, so this should be "common." Obviously this is not a proof, and I'm not an expert on the topic; possibly there will even be explicit examples. But this is the basic heuristic from an analytic perspective. $\endgroup$– user378654Commented Jun 6, 2023 at 1:28
2 Answers
Let me extend, and correct, the argument expressed in the comment made by user378654.
Let us start with a surface $S$ for which $\Delta$ admits a double eigenvalue $\lambda$. For instance, you may choose an $S$ with a non-trivial symmetry group. Denote $V=\ker(\lambda-\Delta)$. Consider now a smooth $m$-parameter family $p\mapsto S_p$ around $S$, with say $S_0=S$. By a Lyapunov-Schmidt procedure, which essentially involves the fact that the spectrum of $\Delta$ is discrete, there exists a unique smooth map $p\mapsto V(p)$ over a neighbourhood $U$ of $0$, where $V(p)$ is a plane in $C^\infty(S_p)$, such that $V(0)=V$ and $V(p)$ is stable under $\Delta$. Choosing a smoothly varying unitary basis ${\cal B}_p$ of $V(p)$, the action of $\Delta$ over $V(p)$ is represented by a $2\times2$ symmetric real matrix $M(p)$.
Now if $m=3$ and the family is "generic", in the sense that the map $p\mapsto M(p)$ is a submersion at $p=0$, then the $3$-parameter family $S_p$ contains a $1$-parameter family of surfaces that admit a double eigenvalue.
Notice the importance of the assumption that $m\ge3$. A one-parameter family $p\mapsto S_p$ does not suffice. This phenomenon (for symmetric matrices in general) is described in a book of V. I. Arnold.
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$\begingroup$ Thank you for your answer! Could you tell me, to which book of Arnold you are referring? By the way, how do we assure that at least one surface in your 1-parameter family of surfaces with double eigenvalues has a trivial symmetry group? $\endgroup$– ClaudiusCommented Jun 6, 2023 at 12:31
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1$\begingroup$ @Claudius 1) I refer to Geometrical methods in the theory of ordinary differential equations (2nd ed.), chapter 6. 2) To ensure a trivial symmetry group (by symmetry, we mean of course isometries), just deform the surface only in a neighbourhood of a point, in such a way that the symmetry group of $S_0$ does not survive. $\endgroup$ Commented Jun 6, 2023 at 15:27
When minimizing numerically the eigenvalues of the Laplacian under area constraint in 2D it is observed that optimal shapes tend to have multiple eigenvalues. Take for example the simulations shown here: http://www.cmap.polytechnique.fr/~beniamin.bogosel/spectral_volume.html
For the minimization of $\lambda_{13}$ the observed minimizer is non-symmetric and has an eigenvalue with (numerical) multiplicity $4$.
For more details see:
