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

We consider the rate of volume growth of large Carnot–Carathéodory metric balls on a class of unbounded model hypersurfaces in 2. When the hypersurface has a uniform global structure, we show that a metric ball of radius δ 1 either has volume on the order of δ3 or δ4. We also give necessary and sufficient conditions on the hypersurface to display either behavior.

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Carnot–Carathéodory metric, global behavior, volume growth
Mathematical Subject Classification 2010
Primary: 53C17
Secondary: 32V15, 43A85
Received: 5 August 2016
Revised: 11 December 2016
Accepted: 3 January 2017
Published: 17 July 2017

Communicated by Michael Dorff
Ethan Dlugie
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
United States
Aaron Peterson
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
United States