#### Vol. 11, No. 1, 2018

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On uniform large-scale volume growth for the Carnot–Carathéodory metric on unbounded model hypersurfaces in $\mathbb{C}^2$

### Ethan Dlugie and Aaron Peterson

Vol. 11 (2018), No. 1, 103–118
DOI: 10.2140/involve.2018.11.103
##### 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.

##### Keywords
Carnot–Carathéodory metric, global behavior, volume growth
##### Mathematical Subject Classification 2010
Primary: 53C17
Secondary: 32V15, 43A85