Communications in Mathematical Physics

, Volume 274, Issue 1, pp 65–80 | Cite as

Mass Under the Ricci Flow

Article

Abstract

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n = 3. If m(g) ≠ 0, then the Ricci flow starting at g can not have Euclidean space as its (uniform) limit.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnowitt S., Deser S. and Misner C. (1961). Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122: 997–1006 MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Bartnik R. (1986). The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39: 661–693 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bando S., Kasue A. and Nakajima H. (1989). On a construction of coordiantes at infinity on manifold with fast curvature decay and maximal volume growth. Invent. Math. 97: 313–349 MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Chen B. and Zhu X. (2000). Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140(2): 423–452 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ecker K. and Huisken G. (1991). Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105: 547–569 MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Gutperle M., Headrick M., Minwalla S. and Schomerus V. (2003). Spacetime Energy Decreases under World-sheet RG Flow. JHEP 0301: 073 CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Greene R., Petersen P. and Zhu S. (1994). Riemannian manifolds of faster-than-quadratic curvature decay. Internat. Math. Res. Notices 9: 363–377 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hamilton R. (1995). The formation of Singularities in the Ricci flow. Surveys in Diff. Geom. 2: 7–136 MathSciNetGoogle Scholar
  9. 9.
    Hamilton R. (1993). The Harnack estimate for the Ricci flow. J. Diff. Geom. 37: 225–243 MATHMathSciNetGoogle Scholar
  10. 10.
    Kapovitch V. (2005). Curvature bounds via Ricci smoothing. Illinois J. Math. 49(1): 259–263 MATHMathSciNetGoogle Scholar
  11. 11.
    Lee J.M. and Parker T. (1987). The Yamabe problem. Bull. AMS 17: 37–91 MATHMathSciNetGoogle Scholar
  12. 12.
    Li P. and Yau S.T. (1986). On the parabolic kernel of the Schrodinger operators. Acta. Math. 158: 153–201 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Oliynyk, T., Woolgar, E.: Asymptotically Flat Ricci Flows. http://arxiv.org/list/math.DG/0607438, 2006Google Scholar
  14. 14.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http:// arxiv.org/list/math.DG/math.DG/0211159, 2002Google Scholar
  15. 15.
    Shi W.X. (1989). Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Diff. Geom. 30: 303–394 MATHGoogle Scholar
  16. 16.
    Shi W.X. (1989). Deforming the metric on complete Riemannian manifolds. J. Diff. Geom. 30: 223–301 MATHGoogle Scholar
  17. 17.
    Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of Variations, LNM 1365, Berlin-Heidelberg-New York: Springer-Verlag, 1989Google Scholar
  18. 18.
    Schoen R. and Yau S.T. (1994). Lectures on Differential Geometry. International Press, Cambridge, MA MATHGoogle Scholar
  19. 19.
    Schoen R. and Yau S.T. (1979). On the proof the positive mass conjecture in general relativity. Commun. Math. Phys. 65: 45–76 MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Witten E. (1981). A new proof of the positive energy theorem. Commun. Math. Phys. 80: 381–402CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Chern Institute of MathematicsTianjinChina
  3. 3.Department of Mathematical ScienceTsinghua UniversityPekingPeople’s Republic of China

Personalised recommendations