Geometric bounds for low Steklov eigenvalues of finite volume hyperbolic surfaces

TL;DR

The paper addresses geometric lower bounds for the low Steklov eigenvalues on finite-volume hyperbolic surfaces with geodesic boundary. It develops an adapted thick-thin decomposition and leverages DR86 energy-oscillation estimates for Steklov eigenfunctions, together with a topological lemma controlling the length of the shortest separating multi-geodesics $\ell_k$, to relate $\sigma_k$ to $\ell_k$ and boundary geometry. The main result provides a sharp bound $\sigma_k(\Sigma) \ge \frac{C}{b|\chi|^3}\min\{\frac{1}{(1+\beta)^2 e^{\beta}}, \frac{\ell_k}{\beta}\}$ for $0<k\le K$ and a uniform bound for $\sigma_{K+1}$, with extensions to curvature-pinched and noncompact finite-volume cases and connections to the Schoen–Wolpert–Yau inequality for Laplacians. The work improves previous compact-case results, broadens them to noncompact and pinched settings, and provides a framework for analyzing the dependence of the Steklov spectrum on the geometric data via $\ell_k$ and boundary length $\beta$.

Abstract

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the {corresponding Steklov} eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.

Figures (4)

• Figure 1: $\Sigma_\epsilon$ is obtained as a union of two discs connected by a thin neck of length $\epsilon$ and width $\epsilon^3$, and removing a small disc around the centre of one of the discs, then performing a connected sum with a surface of genus at least $1$.
• Figure 2: On the left, the grey parts show $(\mathcal{D}\Sigma)_{{\text{thin}}}$ where $\Sigma$ is a hyperbolic surface with 3 boundary components $B_1, B_2$ and $B_3$, and on the right, the thin part $\Sigma_{{\text{thin}}}^{{\varepsilon_\circ}}$ of $\Sigma$. Note that since ${\varepsilon_\circ} \leq \mathop{\mathrm{arsinh}}\nolimits(1)$, some of the original thin tubes are no longer in the thin part. Furthermore, by definition, the half-collar of each boundary component is part of $\Sigma_{{\text{thin}}}^{{\varepsilon_\circ}}$.
• Figure 4: A surface with 6 boundary components and genus 0 and how to obtain 4 disjoint components each containing part of the boundary by considering its double with respect to two of the boundary components.
• Figure 5: Example of surface with $6$ boundary components and a large $\ell_3$.

Theorems & Definitions (17)

• Theorem 1.1
• Lemma 2.1: DR86
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof : Proof of Proposition \ref{['lem:ell_1']}
• Remark 3.4