Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

Abstract

In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a “dual” problem of building Lipschitz maps from our metric space into a sphere with good topological properties. These families of curves can be used to control the values of a function in terms of its gradient (suitably interpreted on a general metric space), and to derive Sobolev and Poincaré inequalities.