Stability of compact Ricci solitons under Ricci flow

Abstract

In this paper we establish stability results for Ricci solitons under the Ricci flow, i.e. small perturbations of the Ricci soliton result in small variations in the solution under Ricci flow.

Warped product spaces with Ricci conditions

“…It is well known that when ρ ≤ 0 all compact solitons are necessarily Einstein [6], and a Ricci soliton on a compact manifold has a constant curvature in 2 dimensions [7] as well as in 3 dimensions [8]. Moreover, a Ricci soliton on a compact manifold is a gradient Ricci soliton [9,14] and Vaghef and Razavi [15] studied the stability of compact Ricci solitons under Ricci flow. * Correspondence: bhkim@khu.ac.kr 2010 AMS Mathematics Subject Classification: Primary 53C25; Secondary 53B21…”

“…Vaghef and Razavi [15] In [12], we studied the gradient Ricci soliton in the warped product space and obtained the criterion for the base space of the warped product space with a gradient Ricci soliton to be a gradient Ricci soliton or an Einstein space by taking the derivative twice of the warping function. However, there are no such criteria or methods of construction of a Ricci soliton for the Riemannian products or warped products as far as we know.…”