On the classification of gradient Ricci solitons

Abstract

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton-Ivey-Perelman classification of 3-dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of53C25

The Evolution of the Weyl Tensor under the Ricci Flow

“…In the shrinking case, the analysis of Kotschwar [23] of rotationally invariant shrinking, gradient Ricci solitons gives the following classification where the Gaussian soliton is defined as the flat R n with a potential function f = α|x| 2 /2n, for a constant α ∈ R. This classification of shrinking, gradient, LCF Ricci solitons was obtained in the paper of P. Petersen and W. Wylie [29] (using also results of Z.-H. Zhang [33]). Many other authors contributed to the subject, including X. Cao, B. Wang and Z. Zhang [7], B.-L. Chen [8], M. Fernández-López and E. García-Río [14], L. Ni and N. Wallach [27], O. Munteanu and N. Sesum [25] and again P. Petersen and W. Wilye [30].…”

“…If n 4, any n-dimensional, LCF Ricci soliton with constant scalar curvature is either a quotient of R n , S n and H n with their canonical metrics or a quotient of R × S n−1 and R × H n−1 (see also [29]). …”

On the classification of warped product Einstein metrics

“…When there is a critical point at t = 0, g S must be the round sphere to obtain a smooth metric, if there is no critical point, then g S is the metric of a level set of w. The result follows easily from these equations. For more details see, [7,11,20,35]. …”

Area growth rate of the level surface of the potential function on the 3-dimensional steady gradient Ricci soliton

“…See Hamilton's important work [5]. Recently, quite a few results on the classification and rigidity of the gradient solitons have appeared; see [6], [7], [8], [9], [12], and [13], for example.…”