A Simple Proof on the Non-existence of Shrinking Breathers for the Ricci Flow

Suppose M is a compact n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. It is known that Ricci flow is a very powerful tool in understanding the geometry and structure of manifolds. In 1982 R. Hamilton [H1] first began the study of Ricci flow on a manifold. Suppose M is a compact 3-dimensional manifold with a metric gij(x) having a strictly positive Ricci curvature. R. Hamilton proved that if the metric gij evolve by the Ricci flow ∂ ∂t gij = −2Rij (0.1) with gij(x, 0) = gij(x), then the evolving metric will converge modulo scaling to a metric of constant positive curvature. A similar result for compact 4-dimensional manifold with positive curvature operator was proved by R. Hamilton in the paper [H2]. By using a modification of the proof of Li-Yau Harnack inequality [LY] for the heat equations on manifolds R. Hamilton [H4] proved the Harnack inequality for the Ricci flow. Singularities of solutions of the Ricci flow was studied by R. Hamilton [H5] and G. Perelman [P1], [P2]. Ricci flow on non-compact manifolds was studied by W.X. Shi [S1], [S2], R. Hamilton [H3], and L.F. Wu [W1], [W2]. Existence and asymptotic behaviour of solutions of the Ricci flow equation on non-compact R was studied by S.Y. Hsu in the papers [Hs1–4]. 1991 Mathematics Subject Classification. Primary 58J35, 53C44 Secondary 58C99.