Metric Derivations on Euclidean and Non-euclidean Spaces

As introduced by Weaver, (metric) derivations extend the notion of differential operators on Euclidean spaces and provide a linear differentiation theory that is well-defined on all metric spaces with Borel measures. The main result of this paper is a rigidity theorem for measures on Euclidean spaces that support a maximal number of derivations. The proof relies substantially on ideas from geometric measure theory, such as slicing techniques. Several conjectures in the analysis on metric spaces also follow as consequences of the rigidity theorem. This includes a characterisation of top-dimensional metric currents in Euclidean spaces, as formulated by Ambrosio and Kirchheim. As another application, we study the structure of metric spaces that support generalised differentiable structures in the sense of Cheeger.