Uniform measures of dimension 1

Início: 11/06/2020 17:00

Término: 11/06/2020 18:00

Palestrante: Mircea Petrache (Pontificia Universidad Católica de Chile)

E-mail do Palestrante:

Resumo: In his fundamental 1987 paper on the geometry of measures, Preiss posed the problem of classifying uniform measures in d-dimensional Euclidean space, a question at the interface of measure theory and differential geometry. A uniform measure is a positive measure such that for all $r>0$, all balls of radius $r$ with center in the support of the measure, are given equal masses. It was proved by Kirchheim-Preiss that a uniform measure in $mathbb{R}^d$ is a multiple of the k-dimensional Hausdorff measure restricted to a k-dimensional analytic variety. This establishes the link to differential geometry. An important class of uniform measures are G-invariant measures, for G any subgroup of isometries of Euclidean space. These are called homogeneous measures. Intriguing examples of non-homogeneous uniform measures do exist (the surface area of the 3D cone $x^2=y^2+w^2+z^2$ in $mathbb{R}^4$ is one), but they are not well understood, making Preiss' classification question is still widely open. After a historical survey, I will describe a recent joint paper with Paul Laurain, about uniform measures of dimension 1 in d-dimensional Euclidean space: we prove by a direct approach that these are all given by at most countable unions of congruent helices or of congruent toric knots. In particular, 1-dimensional uniform measures with connected support are homogeneous.