Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Differential Geometry

arXiv:0710.1276 (math)
[Submitted on 5 Oct 2007 (v1), last revised 8 Oct 2007 (this version, v2)]

Title:Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows

Authors:David Glickenstein
View PDF
Abstract: In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross curvature flow of SL(2,R) limits to Sol.
Subjects: Differential Geometry (math.DG)
MSC classes: 53C44, 53C21
Cite as: arXiv:0710.1276 [math.DG]
  (or arXiv:0710.1276v2 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.0710.1276
Journal reference: Int. Math. Res. Not. IMRN 2008, no. 12, Art. ID rnn034, 49 pp

Submission history

From: David Glickenstein [view email]
[v1] Fri, 5 Oct 2007 17:42:07 UTC (38 KB)
[v2] Mon, 8 Oct 2007 17:23:24 UTC (38 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math.DG
< prev   |   next >
new | recent | 2007-10
Change to browse by:
math

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)