On Uniform Large-Scale Volume Growth for the Carnot-Carath\'eodory Metric on Unbounded Model Hypersurfaces in $\mathbb{C}^2$

Abstract

We consider the rate of volume growth of large Carnot-Carath\'eodory metric balls on a class of unbounded model hypersurfaces in $\mathbb{C}^2$. When the hypersurface has a uniform global structure, we show that a metric ball of radius $\delta \gg 1$ either has volume on the order of $\delta^3$ or $\delta^4$. We also give necessary and sufficient conditions on the hypersurface to display either behavior.