A Basmajian-type inequality for Riemannian surfaces

TL;DR

The paper extends Basmajian-type relations from hyperbolic surfaces to general Riemannian surfaces with one geodesic boundary by incorporating the volume entropy of the doubled surface $S'$. It first proves a Basmajian-type inequality for a class of metric graphs $\Gamma$ formed from a boundary circle and orthogeodesic chords, establishing the inequality $\tanh\left(\frac{h(\Gamma) L}{2}\right) < 2\sum_{i=1}^n \frac{1}{1+e^{h(\Gamma)\ell_i}} < \sinh\left(\frac{h(\Gamma) L}{2}\right)$, where $h(\Gamma)$ is the graph's volume entropy. The authors then transfer this graph result to surfaces by constructing a sequence $\Gamma_n$ from the boundary data of $S$ and showing, via a ping-pong argument, that $h(\Gamma_n) \le h(S')$ for all $n$. Consequently, taking $n\to\infty$ yields the main inequality $\ell(\partial S) \ge \frac{2}{h(S')} \mathrm{arcsinh}\left(\sum_{\ell\in\mathcal{O}(S)} \frac{1}{1+e^{h(S')\ell}}\right)$, providing a curvature-free analogue of Basmajian's identity. The work also discusses optimality, limitations, and open questions regarding extending the inequality to multiple boundary components.

Abstract

We explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of the orthospectrum when fixing a metric invariant of the surface related to the classical notion of volume entropy. This inequality can be thought of as a Riemannian analog of Basmajian's identity for hyperbolic surfaces.

Figures (3)

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Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['th:graph']}
• Proposition 2.1
• proof
• Example 2.2
• Remark 2.3: Non-sharpness of the upper bound
• Proposition 3.1
• proof
• Lemma 3.2