Geometric Structures of Collapsing Riemannian Manifolds I
| dc.creator | Naber, Aaron | |
| dc.creator | Tian, Gang | |
| dc.date | 2008-04-14 | |
| dc.date | 2008-06-18 | |
| dc.date.accessioned | 2026-07-07T09:44:57Z | |
| dc.date.available | 2026-07-07T09:44:57Z | |
| dc.description | Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded Alexandroff curvature. In particular, if n \leq 4 then B is empty and (X,d) is a Riemannian orbifold. Our main application is to prove that a collapsed limit of Einstein four manifolds has a smooth Riemannian orbifold structure away from a finite number of points, and that near these points the curvatures has a -dist^{-2} lower bound. | |
| dc.description | 35 Pages | |
| dc.identifier | https://arxiv.org/abs/0804.2275 | |
| dc.identifier | http://arxiv.org/abs/0804.2275 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163051 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53B20 | |
| dc.title | Geometric Structures of Collapsing Riemannian Manifolds I | |
| dc.type | text |