Summary: Digital Object Identifier (DOI) 10.1007/s00220-007-0275-6
Commun. Math. Phys. 274, 6580 (2007)
Mass Under the Ricci Flow
Xianzhe Dai1,2, , Li Ma3,
1 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA.
2 Chern Institute of Mathematics, Tianjin, China
3 Department of Mathematical Science, Tsinghua University, Peking 100084, People's Republic of China.
Received: 25 January 2006 / Accepted: 25 January 2007
Published online: 6 June 2007 © Springer-Verlag 2007
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.