Counter-intuitive approximations

Abstract: 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 $\mathbb{R}^3$. For $C^2$-embeddings this is impossible due to curvature restrictions.

We will present a general result which will allow for approximations by functions satisfying strongly overdetermined equations on open dense subsets. This will be illustrated by three examples: real functions, embeddings of surfaces, and abstract Riemannian metrics on manifolds.

Our method is based on "weak flexibility", a concept introduced by Gromov in 1986. This is joint work with Bernhard Hanke (Augsburg).

Mathematics

Audience: researchers in the topic

( paper )

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Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

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