This thesis studies three analytic and quantitative questions on doubling metric (measure) spaces. These results are largely independent and will be presented in separate chapters.
The first question concerns representing metric spaces arising from complete Riemannian manifolds in Euclidean space. More precisely, we find bi-Lipschitz embeddings ƒ for subsets A of complete Riemannian manifolds M of dimension n, where N could depend on a bound on the curvature and diameter of A. The main difficulty here is to control the distortion of such embeddings in terms of the curvature of the manifold. In constructing the embeddings, we will study the collapsing theory of manifolds in detail and at multiple scales. Similar techniques give embeddings for subsets of complete Riemannian orbifolds and quotient metric spaces.
The second part of the thesis answers a question about finding quantitative and weak conditions that ensure large families of rectifiable curves connecting pairs of points. These families of rectifiable curves are quantified in terms of Poincaré inequalities. We identify a new quantitative connectivity condition in terms of curve fragments, which is equivalent to possessing a Poincaré inequality with some exponent. The connectivity condition arises naturally in three different contexts, and we present methods to find Poincaré inequalities for the spaces involved. In particular, we prove such inequalities for spaces with weak curvature bounds and thus resolve a question of Tapio Rajala.
In the final part of the thesis we study the local geometry of spaces admitting differentiation of Lipschitz functions with certain Banach space targets. The main result shows that such spaces can be characterized in terms of Poincaré inequalities and doubling conditions. In fact, such spaces can be covered by countably many pieces, each of which is an isometric subset of a doubling metric measure space admitting a Poincaré inequality. In proving this, we will find a new way to use hyperbolic fillings to enlarge certain sub-sets into spaces admitting Poincaré inequalities.
|Commitee:||Austin, Tim, Cheeger, Jeff, Kohn, Robert V., Young, Robert|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 79/01(E), Dissertation Abstracts International|
|Keywords:||Analysis, Differentiability, Embedding, Lipschitz, Metric spaces, Poincare|
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