Title
On metric currents and their connection to differentiability and PDEs
Abstract
A metric k-current is an appropriate generalisation of an oriented k dimensional manifold, which makes sense in general metric spaces. The way this is done is by axiomatizing properties of the action of the k-current T on k+1 tuples of Lipschitz functions (f,π1,...,πk) which intuitively correspond to differential k-forms:
(f,π1,...,πk) = f∧dπ1∧...∧dπk.
The remarkable property of the axiomatisation is that it leads to very fruitful theory, allowing one to use currents as suitable tools for a number of geometric problems, yet the structure of general metric k-currents is not well understood even in Rd.
The problem lies in the fact that one of the axioms, joint continuity, is very difficult to understand. In 2016, DePhilippis and Rindler used PDE and Harmonic Analysis techniques to give a complete description of d-currents in Rd, giving first indication of the deep connection to PDEs. More recently, a number of papers studying different connections of metric currents to PDEs appeared providing further insight into the problem. I will discuss the results of these papers.
