Uniform Comparison of Hyperbolic Ball Volumes on the Universal Cover

Abstract

Let $\Vert M{\Vert }_{\Delta }$ denote the simplicial volume of $M$, $V_r(X,h)={\sup }_{x\in X}{\mathrm{Vol}}_h(B_h(x,r))$, and ${\mathbb{H}}^n$ denotes hyperbolic $n$-space. We prove that, if a closed oriented $n$-manifold $M$ admits a hyperbolic metric, then there is a dimensional constant ${\delta }_n>0$ such that every Riemannian metric $g$ on $M$ with \[ \frac{\operatorname{Vol}_g(M)}{\|M\|_Δ}<δ_n \] satisfies \[ V_r(\widetilde M,\widetilde g)\ge V_r(\mathbb{H}^n) \quad\text{for every }r\ge 1. \]

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