Lipschitz Maps in Metric Spaces
In this dissertation we study Lipschitz and bi-Lipschitz mappings on abstract, non-smooth metric measure spaces. The dissertation consists of two separate parts.
The first part considers a well-known class of questions that ask the following: If X and Y are metric measure spaces and f is a Lipschitz mapping from X to Y whose image has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors regular topological manifolds. In general, these manifolds need not admit bi-Lipschitz embeddings into any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds.
In the second part, we study the class of ``Lipschitz differentiability spaces'' introduced by Cheeger. These are spaces on which an appropriate version of Rademacher's theorem holds. We show that if an Ahlfors regular Lipschitz differentiability space has a differentiable structure of maximal dimension, then at almost every point all its tangents are uniformly rectifiable. In particular, it admits Euclidean tangents at almost every point. Conversely, we show that if the dimension of the differentiable structure is not extremal, then the space is strongly unrectifiable, in the sense of Ambrosio-Kirchheim. In proving these results, we generalize some results of Cheeger from the setting of doubling spaces with Poincare inequalities (PI spaces) to general doubling Lipschitz differentiability spaces. The starting point is a result of Bate on the local structure of these spaces.