Christian Bär - Counter-intuitive approximations

The Nash-Kuiper embedding theorem is a prototypical example
of a counter-intuitive approximation result: any short embedding of a
Riemannian manifold into Euclidean space can be approximated by
*isometric* ones. As a consequence, any surface can be isometrically
C1C^1C1-embedded into an arbitrarily small ball in R3\mathbb{R}^3R3.
For C2C^2C2-embeddings this is impossible due to curvature
restrictions.

We will present a general result which will allow for approximations
by functions satisfying strongly overdetermined equations on open
dense subsets. This will be illustrated by three examples: real
functions, embeddings of surfaces, and abstract Riemannian metrics on
manifolds.

Our method is based on "weak flexibility", a concept introduced by
Gromov in 1986. This is joint work with Bernhard Hanke (Augsburg).