Some comparison theorems in Finsler-Hadamard manifolds

TL;DR

This paper establishes bounds on volume-to-sphere ratios and volume growth entropy in Finsler-Hadamard manifolds with pinched S-curvature, generalizing classical Riemannian results.

Contribution

It provides new upper and lower bounds for volume and surface area ratios in Finsler-Hadamard manifolds, extending known Riemannian theorems to Finsler geometry.

Findings

Bounds for volume-to-sphere ratio in Finsler-Hadamard manifolds

Limit of the ratio at infinity established

Volume growth entropy estimates derived

Abstract

We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian geometry. We also estimate the volume growth entropy for the balls in such manifolds.