TY - JOUR
TI - Controlled geometry via volumes on Alexandrov spaces
DO - https://doi.org/doi:10.7282/T37944FC
PY - 2010
AB - 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 [Greek letter epsilon] 2 Alex [superscript n] (Greek letter kappa) (the compact n-dimensional Alexandrov spaces with curvature [greater than or equal to kappa].), the sum of the length and the turning angle is bounded from below in terms of [kappa], n, diameter and volume of X. This generalizes a basic estimate by Cheeger on the length of a closed geodesic in closed Riemannian manifold ([Ch]). (2) Let [epsilon]p be the space of directions at p 2 [epsilon] X and the pointed radius R = inf{r : X C B[subscript r](p)}. If X [epsilon] Alex[superscript n](kappa), then vol(X) [less than or equal to symbol] vol(CR/k (Epsilon subscript p). where (CR/k (Epsilon subscript p) is the metric R-ball at the vertex in the [kappa]-suspension (CR/k (Epsilon subscript p). We give an isometric classification of X X [epsilon] Alex[superscript n](kappa) whose volume achieves the maximal possible value (CR/k (Epsilon subscript 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]).
KW - Mathematics
KW - Geodesics (Mathematics)
KW - Geometry, Differential
KW - Rigidity (Geometry)
LA - eng
ER -