The eternal solution to the cross curvature flow exists in 3-manifolds of negative sectional curvature

Given a closed 3-manifold $M^3$ endowed with a radial symmetric metric of negative sectional curvature, wedefine the cross curvature flow on $M^3$; using the maximum principle theorem, we demonstrated that the solution to thecross curvature flow exists for all time and converges pointwise to a hyperbolic metric.

PDF

___

[1] Buckland J. Short-time existence of solutions to the cross curvature flow on 3-manifolds. Proceedings of the American Mathematical Society 2006; 134 (6): 1803-1807.
[2] Chow B, Hamilton RS. The cross curvature flow of 3-manifolds with negative sectional curvature. Turkish Journal of Mathematics 2004; 28 (1): 1-10.
[3] Hamilton RS. The formation of singularities in the Ricci flow. In: Proceedings of the Conference on Geometry and Topology; University of Cambridge, MA, USA; 1993. International Press, 1995. pp. 7-136.
[4] Knopf D, Young A. Asymptotic stability of the cross curvature flow at a hyperbolic metric. Proceedings of the American Mathematical Society 2009; 137 (2): 699-709.
[5] Ma L, Chen DZ. Examples for cross curvature flow on 3-manifolds. Calculus of Variations and Partial Differential Equations 2006; 26 (2): 227-243.
[6] Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv preprint math/0211159, 2002.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv