Hyperbolic volume of n-manifolds with geodesic boundary and orthospectra

TL;DR

This paper introduces a function linking the volume of hyperbolic n-manifolds with geodesic boundary to their orthospectra, providing an integral formula and bounds based on boundary area.

Contribution

It defines a new function relating volume to orthospectrum and derives an integral formula for it, offering bounds based on boundary area.

Findings

Established a function F_n for hyperbolic n-manifolds with boundary

Derived an integral formula for F_n in elementary functions

Provided a lower volume bound based on boundary area

Abstract

In this paper we describe a function $F_n:{\bf R}_+ \to {\bf R}_{+}$ such that for any hyperbolic n-manifold $M$ with totally geodesic boundary $\partial M \neq \emptyset$, the volume of $M$ is equal to the sum of the values of $F_n$ on the {\em orthospectrum} of $M$. We derive an integral formula for $F_n$ in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.