Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues

Question

Hello,

Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-manifolds with sequence of genus $g_n$ and varying hyperbolic metrics $d_n$ such that its ($2g_n - 3$)rd eigenvalue $\lambda_n$ is of the order $o(g_n ^3)?$

Any answer or any reference would be appreciated.

P.S. : in a paper by Schoen-Wolpert-Yau, they stated a relation between $\lambda_n$ and $l_n,$ of the form $c(g) l_n \leq \lambda_n \leq d(g) l_n,$ which I do not want to use, since I do not understand fully the proof of $\lambda_n \geq c(g)l_n\ldots$.I might ask a question about it later. The quantity l_n is defined below : consider all the family F of sets of disjoint closed geodesics on the compact hyperbolic surface M such that they divide M into ( n+1 ) components. Then l_n = infimum of sum of all the closed such geodesics in all such sets in the family F.

Answer 1

Dear SPal,

while I didn't try to answer your main question, let me make a few comments around the first eigenvalue. In general, there is a nice trick (of taking cyclic coverings) in order to get $\lambda_1$ arbitrarily small. More precisely, given $\varepsilon>0$ and any compact hyperbolic surface $S$, there is a finite (cyclic) covering $S'$ of $S$ such that $0<\lambda_1(S')\leq\varepsilon$.

A (quick) proof can be found in the excellent book (in French) of Nicolas Bergeron (see this link here, Corollaire 3.39)

Just as a side remark, this construction goes back to Selberg (which obtained this result in connection with his lower bounds on the first eigenvalue of arithmetic hyperbolic surfaces). Another observation is that one can use the Cheeger-Buser inequalities to give lower and upper bounds on the first eigenvalues in terms of appropriate isoperimetric inequalities (see Bergeron's book).

Furthermore, the problem of getting large eigenvalues was discussed here .

In any case, I believe that there are similar results for higher eigenvalues, but I don't remember any references right now.

Best,

Matheus