CONTROLLED GEOMETRY VIA VOLUMES ON ALEXANDROV SPACES Controlled Geometry via Volumes on Alexandrov Spaces

Nan Li, Nan Li
2010 unpublished
In this thesis we recall the basic definitions and properties for Alexandrov space and describe two geometry phenomenons controlled via volume (Hausdorff measure or rough volume) conditions. (1) For a path in X ∈ Alex n (κ) (the compact n-dimensional Alexandrov spaces with curvature ≥ κ.), the sum of the length and the turning angle is bounded from below in terms of κ, n, diameter and volume of X. This generalizes a basic estimate by Cheeger on the length of a closed geodesic in closed
more » ... n manifold ([Ch]). (2) Let Σ p be the space of directions at p ∈ X and the pointed radius R = inf{r : X ⊂ B r (p)}. If X ∈ Alex n (κ), then vol(X) ≤ vol(C R κ (Σ p)), where C R κ (Σ p) is the metric R-ball at the vertex in the κ-suspension C κ (Σ p). We give an isometric classification of X ∈ Alex n (κ) whose volume achieves the maximal possible value vol(C R κ (Σ p)). We also determine homeomorphic types of such X when X is a topological manifold. These results are natural extension of K. Grove and P. Petersen's work in 1992 ([GP 92]). ii