Eternal solutions to the Ricci flow

Ancient Solutions to Kähler-ricci Flow

In this paper, we prove that any non-flat ancient solution to KählerRicci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results then are applied to study the geometry ...

Limits of Solutions to the K Ahler - Ricci Flow

Uniqueness of Solutions of Ricci Flow on Complete Noncompact Manifolds

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures. Recently there is a lot of study on the Ricci flow on manifolds by R. Hamilton [H1–6], S.Y. Hsu [Hs...

On the Conditions to Extend Ricci Flow

Consider {(M, g(t)), 0 ≤ t < T < ∞} as an unnormalized Ricci flow solution: dgij dt = −2Rij for t ∈ [0, T ). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T ) then the solution can be extended over T . Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, the...

Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

Consider a sequence of pointed n–dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T ] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n–dimensional solution to the Ricci...