In this paper we establish stability results for Ricci solitons under the Ricci flow, . small perturbations of the Ricci soliton result in small variations in the solution under Ricci flow. | Turk J Math (2015) 39: 490 – 500 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Stability of compact Ricci solitons under Ricci flow Mina VAGHEF, Asadollah RAZAVI∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran Received: • Accepted/Published Online: • Printed: Abstract: In this paper we establish stability results for Ricci solitons under the Ricci flow, . small perturbations of the Ricci soliton result in small variations in the solution under Ricci flow. Key words: Stability, Ricci flow, Ricci soliton, compact manifolds 1. Introduction and preliminaries Differential equations are interesting mathematical topics employed throughout the sciences for modeling dynamic processes. When differential equations are difficult to be solved, we try to obtain qualitative information about the long-term or asymptotic behavior of solutions. Ricci flow is a partial differential equation that evolves a Riemannian metric g¯ on a manifold M under the following equation: ∂ g(t) = −2Ric(g(t)), ∂t g(0) = g¯. () It was introduced by Hamilton in his seminal paper [14] in order to study the geometry and topology of manifolds. Ricci flow has been developed in the past several decades. In addition to being applied as a useful tool in geometry, it also has some applications in other fields such as computer science [28] and physics [18]. Therefore, it is important to study the equation of Ricci flow. There are some interesting questions about this equation, such as stability. The term “stable” means that a stated property is not destroyed when certain perturbations are made. The stability of solutions of differential equations is a quite difficult property to determine. Even though various kinds of stability may be discussed, the one we study here is dynamical stability. g˜ is dynamically stable if for g¯ belonging to a .
