Seminar Analysis: Dominik Inauen (University of Zurich)

Flexibility and Rigidity of Isometric Embeddings

The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. C^1) than at high regularity (i.e. C^2): for example, it's possible to find C^1 isometric embeddings of the standard 2-sphere into arbitrarily small balls in R^3, and yet, in the C^2 category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture, one might ask if there is a regularity threshold in the Hölder scale which distinguishes these behaviours. In my talk I will give an overview of what is known concerning the latter question. 

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