Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

Estimating the trace-free Ricci tensor in Ricci flow


Author: Dan Knopf
Journal: Proc. Amer. Math. Soc. 137 (2009), 3099-3103
MSC (2000): Primary 53C44, 58J35
DOI: https://doi.org/10.1090/S0002-9939-09-09940-7
Published electronically: April 30, 2009
MathSciNet review: 2506468
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An important and natural question in the analysis of Ricci flow behavior in all dimensions $ n\geq4$ is this: What are the weakest conditions that guarantee that a solution remains smooth? In other words, what are the weakest conditions that provide control of the norm of the full Riemann curvature tensor? In this short paper, we show that the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, for all compact solutions in all dimensions $ n\geq3$, without any hypotheses on the initial data.


References [Enhancements On Off] (What's this?)

  • 1. Angenent, Sigurd B.; Knopf, Dan. Precise asymptotics of the Ricci flow neckpinch. Comm. Anal. Geom. 15 (2007), no. 4, 773-844. MR 2395258 (2009b:53106)
  • 2. Böhm, Christoph; Wilking, Burkhard. Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geom. Funct. Anal. 17 (2007), no. 3, 665-681. MR 2346271 (2008h:53050)
  • 3. Böhm, Christoph; Wilking, Burkhard. On Ricci flow in high dimensions. In preparation.
  • 4. Demailly, Jean-Pierre; Paun, Mihai. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math. (2) 159 (2004), no. 3, 1247-1274. MR 2113021 (2005i:32020)
  • 5. Hamilton, Richard S. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497 (84a:53050)
  • 6. Hamilton, Richard S. The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136, Internat. Press, Cambridge, MA, 1995. MR 1375255 (97e:53075)
  • 7. Ivey, Thomas. Ricci solitons on compact three-manifolds. Differential Geom. Appl. 3 (1993), no. 4, 301-307. MR 1249376 (94j:53048)
  • 8. Knopf, Dan. Positivity of Ricci curvature under the Kähler-Ricci flow. Commun. Contemp. Math. 8 (2006), no. 1, 123-133. MR 2208813 (2006k:53114)
  • 9. Šešum, Nataša. Curvature tensor under the Ricci flow. Amer. J. Math. 127 (2005), no. 6, 1315-1324. MR 2183526 (2006f:53097)
  • 10. Simon, Miles. Deformation $ C^{0}$ Riemannian metrics in the direction of their Ricci curvature. Comm. Anal. Geom.  10 (2002), no. 5, 1033-1074. MR 1957662 (2003j:53107)
  • 11. Tian, Gang; Zhang, Zhou. On the Kähler-Ricci flow on projective manifolds of general type. Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179-192. MR 2243679 (2007c:32029)
  • 12. Wang, Bing. On the conditions to extend Ricci flow. Int. Math. Res. Not. (2008), Article ID rnn012, 30 pp. MR 2428146

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C44, 58J35

Retrieve articles in all journals with MSC (2000): 53C44, 58J35


Additional Information

Dan Knopf
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-0257
Email: danknopf@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09940-7
Received by editor(s): July 14, 2008
Published electronically: April 30, 2009
Additional Notes: The author acknowledges NSF support in the form of grants DMS-0545984 and DMS-0505920.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 by the author

American Mathematical Society