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 ${ℂ}^2$. 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.