Universität Augsburg
Institut für Mathematik

The Nash-Kuiper embedding theorem is a prototypical example of a counterintuitive approximation result: any short (but highly non-isometric) embedding of a Riemannian manifold into Euclidean space can be approximated by isometric C¹-embeddings. As a consequence, any surface can be isometrically C¹-embedded into an arbitrarily small ball in ℝ³. For C²-embeddings this is impossible due to 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 3-space 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.

This is based on joint work with Bernhard Hanke.