On topological upper-bounds on the number of small cuspidal eigenvalues

TL;DR

This paper investigates the behavior of small cuspidal eigenvalues of the Laplace operator on noncompact hyperbolic surfaces, establishing topological bounds and continuity properties as surfaces vary in moduli space, supporting Otal-Rosas conjecture.

Contribution

It describes the limiting behavior of small cuspidal eigenpairs on hyperbolic surfaces and proves openness of certain eigenvalue sets in moduli space, advancing understanding of eigenvalue distributions.

Findings

Sets of surfaces with eigenvalues greater than 1/4 are open and contain neighborhoods of specific moduli spaces.

Certain eigenvalue sets include neighborhoods of moduli spaces of lower genus surfaces.

Results support the Otal-Rosas conjecture regarding eigenvalues on hyperbolic surfaces.

Abstract

Let $S$ be a noncompact, finite area hyperbolic surface of type $(g, n)$. Let $\Delta_S$ denote the Laplace operator on $S$. As $S$ varies over the {\it moduli space} ${\mathcal{M}_{g, n}}$ of finite area hyperbolic surfaces of type $(g, n)$, we study, adapting methods of Lizhen Ji \cite{Ji} and Scott Wolpert \cite{Wo}, the behavior of {\it small cuspidal eigenpairs} of $\Delta_S$. In Theorem 2 we describe limiting behavior of these eigenpairs on surfaces ${S_m} \in {\mathcal{M}_{g, n}}$ when $({S_m})$ converges to a point in $\overline{\mathcal{M}_{g, n}}$. Then we consider the $i$-th {\it cuspidal eigenvalue}, ${\lambda^c_i}(S)$, of $S \in {\mathcal{M}_{g, n}}$. Since {\it non-cuspidal} eigenfunctions ({\it residual eigenfunctions} or {\it generalized eigenfunctions}) may converge to cuspidal eigenfunctions, it is not known if ${\lambda^c_i}(S)$ is a continuous function. However, applying Theorem 2 we prove that, for all $k \geq 2g-2$, the sets $${{\mathcal{C}_{g, n}^{\frac{1}{4}}}}(k)= \{ S \in {\mathcal{M}_{g, n}}: {\lambda_k^c}(S) > \frac{1}{4} \}$$ are open and contain a neighborhood of ${\cup_{i=1}^n}{\mathcal{M}_{0, 3}} \cup {\mathcal{M}_{g-1, 2}}$ in $\overline{\mathcal{M}_{g, n}}$. Moreover, using topological properties of nodal sets of {\it small eigenfunctions} from \cite{O}, we show that ${{\mathcal{C}_{g, n}^{\frac{1}{4}}}}(2g-1)$ contains a neighborhood of ${\mathcal{M}_{0, n+1}} \cup {\mathcal{M}_{g, 1}}$ in $\overline{\mathcal{M}_{g, n}}$. These results provide evidence in support of a conjecture of Otal-Rosas \cite{O-R}.