Counter-intuitive existence and approximation results

29.11.2018, 16:15  –  Raum 0.14
Forschungsseminar Differentialgeometrie

Christian Bär

We will discuss a general approximation theorem which allows to solve overdetermined partial differential relations on an open dense subset of the domain. Let K be a real number. Applications will show that

• Every C¹-function can be uniformly approximated by Lipschitz functions with derivative = K on an open dense subset;
• Every embedding of a surface in R³ can be C¹-approximated by embeddings with Gauss curvature = K on an open dense subset;
• Every manifold of dimension at least 2 has a C^1,1-metric which is smooth and has constant sectional curvature K on an open dense subset.

This is joint work with Bernhard Hanke (Augsburg).