An almost isometric sphere theorem and weak strainers on Alexandrov spaces

2017 ◽  
Vol 66 (4) ◽  
pp. 1267-1286
Author(s):  
Qingsong Cai
Keyword(s):  
Sphere Theorem ◽  
Alexandrov Spaces

Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1

2017 ◽  
Vol 19 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Xiaole Su ◽  
Hongwei Sun ◽  
Yusheng Wang
Keyword(s):  
Upper Bound ◽  
Geometric Structure ◽  
Alexandrov Space ◽  
Sphere Theorem ◽  
Alexandrov Spaces ◽  
Riemannian Case ◽  
Packing Radius ◽  
Distance Formula

In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

scholarly journals The Riemannian structure of Alexandrov spaces

1994 ◽  
Vol 39 (3) ◽  
pp. 629-658 ◽  
Author(s):  
Yukio Otsu ◽  
Takashi Shioya
Keyword(s):  
Riemannian Structure ◽  
Alexandrov Spaces

scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

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