Geometric embeddings and rigidity

Existence of Bi-Lipschitz embeddings can be a useful tool in studying the geometry of the space. In the first paper on quantitative bi-Lipschitz embeddings I gave a positive answer employing a detailed study of collapsing theory on the space. We were able to construct bi-Lipschitz embeddings for subsets of manifold, orbifolds and group quotients and bound their distortion and target dimension using diameter and a both-sided bound on sectional curvature. 

On the other hand, occasionally the geometry of the space and rigidity prevent such embeddings. This was pursued jointly with Guy C. David. We showed that a certain slit carpet fails to embed into any uniformly convex Banach space. Here, the novelty was to prove a rigidity for a space with many curves but without a Poincaré inequality, as conventionally would be required. This gave an example of a space which admits a Lipschitz regular map to Euclidean space but fails to admit a bi-Lipschitz embedding. This was an old question of Heinonen and Semmes. The proof was an application of an idea from a different type of context: that of classifying Jacobians of Lipschitz regular mappings and the proof of Burago and Kleiner. Indeed, the framework is more general than the particular example.


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