Small eigenvalues of closed Riemann surfaces for large genus

Abstract

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer k k , as the genus g g goes to infinity, the minimum of k k -th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus g g is uniformly comparable to 1 g 2 \frac {1}{g^2} in g g . And the minimum of a g ag -th eigenvalues of hyperbolic surfaces in any thick part of moduli space is bounded above by a uniform constant only depending on ε \varepsilon and a a . In the proof of the upper bound, for any constant ε > 0 \varepsilon >0 , we will construct a closed hyperbolic surface of genus g g in any ε \varepsilon -thick part of moduli space such that it admits a pants decomposition whose curves all have length equal to ε \varepsilon , and the number of separating systole curves in this surface is uniformly comparable to g g .

Large genus asymptotics for lengths of separating closed geodesics on random surfaces

“…For (3), we write $$$$\begin{equation*} X(\gamma )\setminus \gamma = (\sqcup D_i) \sqcup (\sqcup C_j), \end{equation*}$$$$where the functions of $$D_i$$ and the functions of $$C_j$$ are topological discs and cylinders, respectively, all disjoint from each other. By the classical Isoperimetric Inequality (for example, see [7, 35]), we know that $$$$\begin{equation*} \mathop {\rm Area}(D_i)\leqslant \ell (\partial D_i) \quad \text{and} \quad \mathop {\rm Area}(C_j)\leqslant \ell (\partial C_j). \end{equation*}$$$$Thus, we have …”

“…where the functions of 𝐷 𝑖 and the functions of 𝐶 𝑗 are topological discs and cylinders, respectively, all disjoint from each other. By the classical Isoperimetric Inequality (for example, see [7,35]), we know that Area(𝐷 𝑖 ) ⩽ 𝓁(𝜕𝐷 𝑖 ) and Area(𝐶 𝑗 ) ⩽ 𝓁(𝜕𝐶 𝑗 ). Proof.…”

“…Each element of $$\lbrace \xi _1,\ldots ,\xi _r\rbrace$$ appears in $$\lbrace \partial D_i\rbrace _{i\in I}$$ or $$\lbrace \partial C_j\rbrace _{j\in J}$$ or $$\lbrace \partial C^{\prime }_{p}\rbrace _{p\in P}$$ exactly once. By Isoperimetric Inequality for topological discs and cylinders on hyperbolic surfaces (for example, see [7] or [35]), we have $$$$\begin{eqnarray*} \mathop {\rm Area}(D_i) &\leqslant & \ell (\partial D_i)=\ell (\xi _i),\\ 2\mathop {\rm Area}(C_j) &=& \mathop {\rm Area}(2C_j)\leqslant \ell (\partial (2C_j)) = 2\ell (\xi _j),\\ \mathop {\rm Area}(C^{\prime }_p) &\leqslant & \ell (\partial C^{\prime }_p)=\ell (\xi _{k_p^1})+\ell (\xi _{k_p^2}), \end{eqnarray*}$$$$…”

Degenerating hyperbolic surfaces and spectral gaps for large genus

“…It was proved by Otal-Rosas [OR09] that the constant α(g) can be optimally chosen to be 1 4 . For large genus g, it was recently proved by the first named author and Xue [WX21a,WX18] that up to multiplication by a universal constant, α 1 (g) can be optimally chosen to be 1 g 2 . The other result that is relevant is [BBD88, Theorem 2.1] regarding the first eigenvalue when the limiting degenerating surface is connected:…”

Benjamini–Schramm convergence and spectra of random hyperbolic surfaces of high genus

“…In order to do so, we equip the moduli space of hyperbolic surfaces of signature (g, k) with the Weil-Petersson probability measure P g,k . This is a natural model to study typical hyperbolic surfaces, as illustrated by the rich literature that has developed in the last few years [1,6,7,8,10,11,13,20,19]. By typical, we mean that we wish to prove properties true with probability going to one in a certain asymptotic regime.…”