Abstract.

Doubling metric measure spaces supporting a Poincar\'e inequality constitute a good environment to carry out analysis even without the presence of a smooth structure. It is therefore of interest to find characterizations for the validity of such an inequality. I will show that a complete doubling metric measure space $X$ supports a weak $1$-Poincar\'e inequality if and only if it admits generalized pencils of curves (GPCs) joining any pair of distinct points in $X$. We define GPCs in terms of normal $1$-currents and they turn out to act as a relaxed version of Semmes' notion of "pencil of curves". The construction of GPCs is based on the max flow - min cut theorem in graph theory. This is joint work with Tuomas Orponen.