Rigidity of gradient Ricci solitons

Abstract

We define a gradient Ricci soliton to be rigid if it is a flat bundle N × ‫ޒ‬ k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

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.…”

Ricci solitons on almost Kenmotsu 3-manifolds

“…4/ R is a rigid gradient Ricci soliton (see [20,21]). For other results on existence of quasi-Einstein metrics on this product we refer the reader to [22].…”

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

“…Note that one can always find p, q i being solutions of (24). A straightforward calculation from (22) shows that (M, g, X) is a Ricci soliton.…”

“…A gradient Ricci soliton (M, g, f ) is said to be rigid if (M, g) is isometric to a quotient of N × R k , where N is an Einstein manifold and the potential function f is defined on the Euclidean factor as f = λ 2 x 2 (thus generalizing the Gaussian soliton) [24]. Although rigidity is a rather restrictive condition, rigid Ricci solitons are the only solitons in many important situations as shown in [23], where it is proven that any homogeneous gradient Ricci soliton is rigid if the metric is positive definite.…”