Finding Curves on General Spaces through Quantitative Topology, with Applicationsfor Sobolev and Poincare Inequalities

摘要

In many metric spaces one can connect an arbitrary pair of points with a curve offinite 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 persent paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by solving first a dual problem of building Lipschitz maps from our metric space into a sphere with good 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 Poincare inequalities.