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

TL;DR

This paper investigates the volume growth rates of large Carnot-Carathéodory metric balls on unbounded model hypersurfaces in c2b2, identifying conditions for cubic or quartic volume growth.

Contribution

It characterizes the volume growth behavior of large metric balls on unbounded hypersurfaces and provides necessary and sufficient conditions for different growth regimes.

Findings

Volume of large balls is either on the order of c2b3 or c2b4.

Conditions are established for when each growth behavior occurs.

The results depend on the global structure of the hypersurface.

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.