Perturbed Metrics in the Poincaré Ball: Volume Bounds, Matrix Jacobi Fields, and Stability of Finite-Scale Geometric Indices
Abstract
We study the geometry of the Poincaré ball equipped with a rotationally symmetric metric \(g_\eta\) conformally close to the hyperbolic metric \(g_{\mathbb{H}}\), with curvature pinched in \([-1-\eta, -1+\eta]\). We work with the matrix Jacobi field \(A_\eta\) and the shape operator \(S_\eta = A_\eta' A_\eta^{-1}\), whose trace \(\sigma_\eta = \operatorname{tr} S_\eta\) satisfies a Riccati equation. For rotationally symmetric non-homogeneous metrics and centres \(z \neq 0\), the geodesic spheres need not be isotropic: the volume element in geodesic polar coordinates centred at \(z\) is generally \(\det A_\eta(t, \omega, z)\,dt\,d\omega\), not a power of a scalar radial profile. Five results are established under the stated assumptions. First, using Günther and Bishop--Gromov comparison, we prove two-sided model-volume bounds \[ V_{-(1-\eta)}^{\mathrm{model}}(\delta) \le V_\eta(\delta, z) \le V_{-(1+\eta)}^{\mathrm{model}}(\delta) \] uniformly in \(z\). Second, we prove an exact weighted-average identity for \(\Sigma_\eta(\delta, z) = \partial_\delta\log \partial_\delta V_\eta(\delta, z)\), in which the anisotropy of the geodesic spheres enters as a positive variance term. Third, we prove a monotonicity-transfer lemma: if \(q_\eta(\delta, z) = \delta\,\Sigma_\eta(\delta, z)\) is strictly increasing in \(\delta\), then the volume indicator \(\phi_\eta(\delta, z) = \delta\,\partial_\delta V_\eta/V_\eta\) is strictly increasing. Fourth, under two explicit assumptions on the…
Read the paper
Related studies
- eggNOG 5.0: a hierarchical, functionally and phylogenetically annotated orthology resource based on 5090 organisms and 2502 viruses
- Online Survival Analysis Software to Assess the Prognostic Value of Biomarkers Using Transcriptomic Data in Non-Small-Cell Lung Cancer
- OptiType: precision HLA typing from next-generation sequencing data
- Nicotinic Cholinergic System and COVID-19: In Silico Identification of an Interaction between SARS-CoV-2 and Nicotinic Receptors with Potential Therapeutic Targeting Implications
- Saccharomyces cerevisiae cells have three Omega class glutathione S-transferases acting as 1-Cys thiol transferases
- Hemolysis at low blood flow rates: in-vitro and in-silico evaluation of a centrifugal blood pump
Go deeper in Evidence Hub
Evidence Hub adds a study data sheet (design, sample size, evidence level, key numbers), an AI summary and a chat grounded in the paper's text.