佩雷尔曼关于庞加莱猜想的论文0211159.pdf

The Ricci flow equation, introduced by Richard Hamilton [H 1], is the
evolution equation d gij(t) = −2Rij for a riemannian metric gij(t). In his dt
seminal paper, Hamilton proved that this equation has a unique solution for a short time for an arbitrary (smooth) metric on a closed manifold. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric|2, so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched point- wisely as the curvature is getting large. This observation allowed him to prove the convergence results: the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator