Sur la croissance du volume dans une classe conforme. (On the volume growth in a conformal class) (Q1117472)

The growth function of a Riemannian manifold pointed at x assigns to each positive R the volume of the geodesic ball of radius R with center x. Perform a conformal change of metric. To what extent does the growth function change? The answer is a dichotomy: either every increasing function can be obtained after a conformal change (``parabolic'' case, illustrated by Euclidean space) or exactly those which arise as growth functions of metrics of revolution conformal to a Euclidean disk (``hyperbolic'' case). The result is sharp only for a class of weakly regular metrics. The dichotomy is shown to be quasiconformally invariant. The main technical point in the paper is the investigation of weakly regular conformal metrics. A conformal factor in \(L^ n\) still defines a metric space with approximately differentiable distance function.