Title

Counterintuitive approximations

Abstract

One of the oldest questions in differential geometry is whether every curved space arises as a subspace of Euclidean space ℝⁿ. More precisely, does every Riemannian manifold admit an isometric (i.e. length preserving) smooth embedding into ℝⁿ? Nash showed that the answer is yes but the dimension n of the Euclidean space may be much larger than that of the manifold. Often there are non-isometric embeddings of much lower codimension.

The Nash-Kuiper embedding theorem is a prototypical example of a counterintuitive approximation result in geometry: any short (but highly non-isometric) embedding of a Riemannian manifold into Euclidean space can be approximated by isometric embeddings. They are generally not smooth but of regularity C¹. This implies that any surface with a given geometry can be isometrically C¹-embedded into an arbitrarily small ball in ℝ³. For C²-embeddings this completely false due to curvature restrictions.

After explaining this historical and conceptual context, I will present a general result which ensures approximations by maps satisfying strongly overdetermined equations (such as being isometric) 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.