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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnowitt S., Deser S. and Misner C. (1961). Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122: 997–1006
Bartnik R. (1986). The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39: 661–693
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
Chen B. and Zhu X. (2000). Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140(2): 423–452
Ecker K. and Huisken G. (1991). Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105: 547–569
Gutperle M., Headrick M., Minwalla S. and Schomerus V. (2003). Spacetime Energy Decreases under World-sheet RG Flow. JHEP 0301: 073
Greene R., Petersen P. and Zhu S. (1994). Riemannian manifolds of faster-than-quadratic curvature decay. Internat. Math. Res. Notices 9: 363–377
Hamilton R. (1995). The formation of Singularities in the Ricci flow. Surveys in Diff. Geom. 2: 7–136
Hamilton R. (1993). The Harnack estimate for the Ricci flow. J. Diff. Geom. 37: 225–243
Kapovitch V. (2005). Curvature bounds via Ricci smoothing. Illinois J. Math. 49(1): 259–263
Lee J.M. and Parker T. (1987). The Yamabe problem. Bull. AMS 17: 37–91
Li P. and Yau S.T. (1986). On the parabolic kernel of the Schrodinger operators. Acta. Math. 158: 153–201
Oliynyk, T., Woolgar, E.: Asymptotically Flat Ricci Flows. http://arxiv.org/list/math.DG/0607438, 2006
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http:// arxiv.org/list/math.DG/math.DG/0211159, 2002
Shi W.X. (1989). Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Diff. Geom. 30: 303–394
Shi W.X. (1989). Deforming the metric on complete Riemannian manifolds. J. Diff. Geom. 30: 223–301
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, 1989
Schoen R. and Yau S.T. (1994). Lectures on Differential Geometry. International Press, Cambridge, MA
Schoen R. and Yau S.T. (1979). On the proof the positive mass conjecture in general relativity. Commun. Math. Phys. 65: 45–76
Witten E. (1981). A new proof of the positive energy theorem. Commun. Math. Phys. 80: 381–402
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G.W. Gibbons
Partially supported by NSF and NSFC.
The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.
Rights and permissions
About this article
Cite this article
Dai, X., Ma, L. Mass Under the Ricci Flow. Commun. Math. Phys. 274, 65–80 (2007). https://doi.org/10.1007/s00220-007-0275-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0275-6