The Ricci Flow Series

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).