On complete gradient shrinking Ricci solitons

Abstract

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 < +∞.

On rigidity of gradient Kähler-Ricci solitons with harmonic Bochner tensor

“…Recently, a rigidity result for Bach flat gradient shrinkers has also been obtained by H.-D. Cao and the first author [9]. We refer the reader to a recent survey paper of H.-D. Cao [5] on Ricci shrinkers.…”

A necessary and sufficient condition for some steady Ricci solitons to have positive asymptotic volume ratio

“…In particular, more recently, observing the results in [1], [2], [4] and [8], B. Chow, P. Lu and B. Yang [3] derived a necessary and sufficient condition for noncompact shrinking Ricci soliton to have positive AVR as follows:…”

Rigid properties of quasi-Einstein metrics

“…This fact is very useful in studying the geometry of complete gradient shrinking Ricci solitons [10,11,12]. The authors of [5] obtained some estimates of the scalar curvature for an m−dimensional quasi-Einstein metric on closed manifolds.…”

“…The authors of [5] obtained some estimates of the scalar curvature for an m−dimensional quasi-Einstein metric on closed manifolds. In [10,11], the authors considered the estimate of scalar curvature for a gradient Ricci soliton on noncompact manifolds. In Section 4, we get the lower estimate of the scalar curvature for expanding a quasi-Einstein metric on a noncompact manifold.…”