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 C^1-embedded into an arbitrarily small ball in R^3. For C^2-embeddings this is impossible because of curvature restrictions. I will present a general result which allows for approximations by functions satisfying strongly overdetermined equations on open dense subsets. This will be illustrated by three examples: Lipschitz functions with surprising derivative, surfaces in R^3 with unexpected curvature properties, and a similar statement for abstract Riemannian metrics on manifolds. Our method is based on ‘cut-off homotopy’, a concept introduced by Gromov in 1986. Joint work with Bernhard Hanke.