Counterintuitive approximations

The Nash-Kuiper embedding theorem is a prototypical example of a counterintuitive 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 C^1-embedded into an arbitrarily small ball in R^3. For C^2-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.