Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Area growth rate of the level surface of the potential function on the 3-dimensional steady gradient Ricci soliton


Author: Hongxin Guo
Journal: Proc. Amer. Math. Soc. 137 (2009), 2093-2097
MSC (2000): Primary 53C44
Published electronically: January 29, 2009
MathSciNet review: 2480291
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short note we show that on a 3-dimensional steady gradient Ricci soliton with positive curvature and which is $ \kappa$-noncollapsed on all scales, the scalar curvature and the mean curvature of the level surface of the potential function both decay linearly. Consequently we prove that the area of the level surface grows linearly.


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

  • 1. Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. Geometric aspects. MR 2302600 (2008f:53088)
  • 2. Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006. MR 2274812 (2008a:53068)
  • 3. Sun-Chin Chu, Geometry of 3-dimensional gradient Ricci solitons with positive curvature, Comm. Anal. Geom. 13 (2005), no. 1, 129–150. MR 2154669 (2006h:53060)
  • 4. Chu, Sun-Chin. Personal communications.
  • 5. Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol.\ II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255 (97e:53075)
  • 6. Lott, John. Dimensional reduction and the long-time behavior of Ricci flow. arXiv: 0711.4063.
  • 7. Naber, Aaron. Noncompact shrinking $ 4$-solitons with nonnegative curvature. arXiv: 0710.5579.
  • 8. Ni, Lei; Wallach, Nolan. On a classification of the gradient shrinking solitons. Math. Res. Lett. 15 (2008), no. 5, 941-955.
  • 9. Ni, Lei; Wallach, Nolan. On $ 4$-dimensional gradient shrinking solitons. Int. Math. Res. Notices (2008), vol. 2008, article ID rnm152.
  • 10. Perelman, Grisha. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159.
  • 11. Perelman, Grisha. Ricci flow with surgery on three-manifolds. arXiv:math.DG/ 0303109.
  • 12. Petersen, Peter; Wylie, William. Rigidity of gradient Ricci solitons. arXiv:0710.3174.
  • 13. Petersen, Peter; Wylie, William. On the classification of gradient Ricci solitons. arXiv:0712.1298.

Similar Articles

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

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


Additional Information

Hongxin Guo
Affiliation: School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang, 325035 People’s Republic of China
Email: hguo2006@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09792-5
PII: S 0002-9939(09)09792-5
Received by editor(s): May 30, 2008
Received by editor(s) in revised form: September 26, 2008
Published electronically: January 29, 2009
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.