The Ricci flow equation is the evolution equation

for a Riemannian metric , where is the Ricci curvature tensor. Hamilton (1982) showed that there is a unique solution to this equation for an arbitrary smooth metric on a closed manifold over a sufficiently short time. Hamilton (1982, 1986) also showed that Ricci flow preserves positivity of the Ricci curvature tensor in three dimensions and the curvature operator in all dimensions (Perelman 2002).

Collins, G. P. "The Shapes of Space." Sci. Amer. 291, 94-103, July 2004.

Hamilton, R. S. "Three Manifolds with Positive Ricci Curvature." J. Diff. Geom. 17, 255-306, 1982.

Hamilton, R. S. "Four Manifolds with Positive Curvature Operator." J. Diff. Geom. 24, 153-179, 1986.

Kleiner, B. and Lott, J. "Notes and Commentary on Perelman's Ricci Flow Papers." http://www.math.lsa.umich.edu/research/ricciflow/perelman.html.

Perelman, G. "The Entropy Formula for the Ricci Flow and Its Geometric Application" 11 Nov 2002. http://arxiv.org/abs/math.DG/0211159/.

Robinson, S. "Russian Reports He Has Solved a Celebrated Math Problem." The New York Times, p. D3, April 15, 2003.

CITE THIS AS:

Weisstein, Eric W. "Ricci Flow." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RicciFlow.html