Variational Methods in Geometry Seminar

Back in the 70s, Gromov started to study the relationship between the Lipschitz constant of a map (also called the dilation) and its topology. The Lipschitz constant describes the local geometric features of the map, and the problem is to understand how it relates to the global geometric features of the map -- a bit like trying to understand the relationship between the curvature of a Riemannian manifold and its topology. For example, for maps from the unit 3-sphere to the unit 2-sphere, Gromov proved that there are about L^4 different homotopy classes of maps that can be realized by maps with Lipschitz constant at most L.

The k-dilation is a generalization of the Lipschitz constant that measures how much a map stretches k-dimensional areas. We say that the k-dilation of a map f is at most D if, for any k-dim surface S in the domain Vol_k ( f(S) ) <= D Vol_k (S).

We study the following problem. Suppose that f is a map from the unit m-sphere to the unit n-sphere with tiny k-dilation. Does the map f have to be contractible? If f maps S^m to S^{m-1}, we get sharp results which show the following threshold. If k > (m+1)/2 then there are non-contractible maps with arbitrarily small k-dilation, but if k <= (m+1)/2 then every non-contractible map has k-dilation at least some positive constant. When k > (m+1)/2, the maps with small k-dilation are a little in the spirit of the h-principle. When k <= (m+1)/2, the proof involves two steps. First, we have to understand how to detect when a map is topologically non-trivial. This is a classical topic in algebraic topology, but we have to describe it in as geometric a way as possible so that we can relate it to the k-dilation. In the second step, we bound the relevant topological invariant using isoperimetric-type inequalities. This second step is related to the 2T problem in geometric measure theory.