A Family of Expanding Ricci Solitons

Abstract

The Ricci flow is a natural evolution equation for Riemannian metrics,introduced by Richard S. Hamilton in 1982. A family \({\left( {g\left( {t, \cdot } \right)} \right)_{t \in I}}\) of metrics on a Riemannian manifold M, depending on a time parameter \(t \in I \subseteq \mathbb{R} \) is a solution to the Ricci flow if it solves the equation $$\frac{\partial }{{\partial t}}g\left( {t,\cdot} \right) = - 2Ricg\left( {t,\cdot} \right),$$ where Ric g(t, •) is the Ricci tensor associated with the evolving metric g(t, •). In general, a solution of the Ricci flow starting with smooth initial data will not possess a smooth continuation for all time. The formation of singularities has been discussed extensively in Hamilton’s article [H], and there is a particular type of solutions which is expected (and in some cases known) to appear as parabolic blowup limit of Ricci flows around a singularity, namely Ricci solitons. Ricci solitons are solutions to the Ricci flow for which there exist scalars σ(t) and diffeomorphisms Ψ t :M → M such that $$g\left( {t,\cdot} \right) = \sigma \left( t \right)\Psi _t^* g\left( {T,\cdot} \right){\text{ for all t}} \in {\text{I and some fixed T}} \in {\text{I}}{\text{.}}$$ (1)