Kolloquium: Christian Bär (Universität Potsdam): “Counterintuitive Approximations”

Counterintuitive Approximations – Vortragender: Christian Bär, Universität Potsdam – Einladender: H. Schulz-Baldes

Abstract:
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.