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

Abstract

We consider the rate of volume growth of large Carnot-Carathéodory 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.

14 pages