Backwards Uniqueness for the Ricci Flow

Backwards Uniqueness of the Mean Curvature Flow

In this note we prove the backwards uniqueness of the mean curvature flow in codimension one case. More precisely,let Ft, e Ft : M → M n+1 be two complete solutions of the mean curvature flow on M×[0, T ] with bounded second fundamental form in a complete ambient manifold with bounded geometry. Suppose FT = e FT , then Ft = e Ft on M n × [0, T ]. This is an analog of a recent result of Kotschwa...

Strong Uniqueness of the Ricci Flow

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

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...

The Ricci Flow for Nilmanifolds

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product with respect to time and the evolution of structure constants with respect to time, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.’s for some of these flows and des...

Ricci Lower Bound for Kähler–ricci Flow