Abstract. In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

IntroductionIn the recent work [2] Hamilton proved that for any compact 3-manifold with positive Ricci curvature one can deform the initial metric along the heat flow defined by the equation: ~gis 2 c3 t = -2Rii + ~ rglj We consider the complex version of Hamilton's equation of the following type:

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.Combining Theorem 1.1 and Theorem 1.2, we also have the following consequence, which was obtained previously in [10] and [15] respectively. Corollary 1.1. Let (M n , g ij , f ) be a complete noncompact gradient shrinking Ricci soliton. Then we have M |u|e −f dV < +∞ for any function u on M with |u(x)| ≤ Ae αr 2 (x) , 0 ≤ α < 1 4 and A > 0. In particular, the weighted volume of M is finite, M e −f dV < +∞.

In this paper, we classify n-dimensional (n ≥ 3) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

Abstract. In this paper, we classify n-dimensional (n ≥ 4) complete Bachflat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R 4 or the round cylinder S 3 ×R. More generally, for n ≥ 5, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton R n or the product N n−1 × R, where N n−1 is Einstein. The resultsA complete Riemannian manifold (M n , g ij ) is called a gradient Ricci soliton if there exists a smooth function f on M n such that the Ricci tensor R ij of the metric g ij satisfies the equation R ij + ∇ i ∇ j f = ρg ij for some constant ρ. For ρ = 0 the Ricci soliton is steady, for ρ > 0 it is shrinking and for ρ < 0 expanding. The function f is called a potential function of the gradient Ricci soliton. Clearly, when f is a constant the gradient Ricci soliton is simply an Einstein manifold. Thus Ricci solitons are natural extensions of Einstein metrics. Gradient Ricci solitons play an important role in Hamilton's Ricci flow as they correspond to self-similar solutions, and often arise as singularity models. Therefore it is important to classify gradient Ricci solitons or understand their geometry.In this paper we shall focus our attention on gradient shrinking Ricci solitons, which are possible Type I singularity models in the Ricci flow. We normalize the constant ρ = 1/2 so that the shrinking soliton equation is given byIn recent years, inspired by Perelman's work [22,23], much efforts have been devoted to study the geometry and classifications of gradient shrinking Ricci solitons. We refer the reader to the survey papers [4,5] by the first author and the references therein for recent progress on the subject. In particular, it is known (cf. [23,21,7]) that any complete 3-dimensional gradient shrinking Ricci soliton is a finite quotient of either the round sphere S 3 , or the Gaussian shrinking soliton R 3 , or the round cylinder S 2 × R. For higher dimensions, it has been proven that complete locally conformally flat gradient shrinking Ricci solitons are finite quotients of either the round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n−1 × R (first due to Z. H. Zhang that n-dimensional complete gradient shrinking solitons with harmonic Weyl tensor are rigid in the sense that they are finite quotients of the product of an Einstein manifold N k with the Gaussian shrinking soliton R n−k . Our aim in this paper is to investigate an interesting class of complete gradient shrinking Ricci solitons: those with vanishing Bach tensor. This well-known tensor was introduced by R. Bach [1] in early 1920s' to study conformal relativity. On any n-dimensional manifold (M n , g ij ) (n ≥ 4), the Bach tensor is defined byHere W ikjl is the Weyl tensor. It is easy to see that if (M n , g ij ) is either locally conformally flat (i.e., W ikjl = 0) or...