Classification of uniformly distributed measures of dimension 1 in general codimension

2019/05/23 by Laurain, Paul, Petrache, Mircea
#28C10 #49Q15 #51F20 #53A04 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1905.09601

Abstract

Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in \mathbb R^d has remained open, except for d=1 and for compactly supported measures in d=2, and for codimension 1. In this paper we study 1-dimensional measures in \mathbb R^d for all d and classify uniform measures with connected 1-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension 1 in the absence of the connected support hypothesis.

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