Controlled-norm extensions with singularities and non-rectifiable Sobolev spaces

A classical way to detect "holes" in a compact manifold M is to check whether every continuous function u from a k-dimensional sphere S^k to M extends to a continuous function from the (k+1)-dimensional ball to M. The obstruction to this is the k-th fundamental group of M.

The question I consider in this talk is the natural analogue of this qualitative question when we replace the qualitative property of being a continuous function by the quantitative property of having L^p-integrable gradient. After formulating it, I will describe the following features of this Sobolev extension problem:

1. the p corresponding the conformally invariant energy ||Du||_{L^k(S^k)} is the critical one, providing the most intriguing behaviors
2. extensions will tend to create singularities, a big difference to the C⁰ case
3. the setting is complementary to that of VMO functions and to so-called epsilon-regularity results
4. points 2, 3 are symptoms of the fact that our function spaces are "not rectifiable"
5. * uniformization using hyperbolic geometry provides short and elegant extension constructions

The talk is based on a recent result in collaboration with Jean Van Schaftingen.

(*: if time constraints will allow it)