Abstract
Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdor sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 487-510 |
| Number of pages | 24 |
| Journal | Geometry and Topology |
| Volume | 7 |
| State | Published - 2003 |
| Externally published | Yes |
Fingerprint
Keywords
- Gromov-hausdor convergence
- Ricci flow
ASJC Scopus subject areas
- Geometry and Topology
Cite this
Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates. / Glickenstein, David A.
In: Geometry and Topology, Vol. 7, 2003, p. 487-510.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates
AU - Glickenstein, David A
PY - 2003
Y1 - 2003
N2 - Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdor sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.
AB - Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdor sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.
KW - Gromov-hausdor convergence
KW - Ricci flow
UR - http://www.scopus.com/inward/record.url?scp=4243148519&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=4243148519&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:4243148519
VL - 7
SP - 487
EP - 510
JO - Geometry and Topology
JF - Geometry and Topology
SN - 1465-3060
ER -