Asian Journal of Mathematics

Volume 13 (2009)

Number 4

Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)

Pages: 421 – 458

DOI: http://dx.doi.org/10.4310/AJM.2009.v13.n4.a1

Authors

Xiaodong Cao

Laurent Saloff-Coste

Abstract

In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.

Keywords

Cross Curvature Flow (XCF); locally homogeneous 3-manifold

2010 Mathematics Subject Classification

Primary 53C44. Secondary 58J35.

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