Inserted: 18 sep 2008
Last Updated: 7 sep 2009
Journal: Journal of Geometric Analysis
In $R^n$ equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization of when the distance from the origin in an arbitrary Carnot-Carathéodory space is viscosity infinite harmonic at a point outside the origin. We specifically show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity subsolution. We also show that at the origin, the distance function is not a viscosity supersolution.