Abstract. In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman’s no non-trivial expanding breat...

By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the two-dimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert...

It tries to distribute the curvature evenly around a manifold. In doing so, the Ricci flow preserves the symmetries of the space, and, in the limit, can increase the symmetries of the space. It has been extensively studied in [Ham82], [Pera], [Perc], and [Perb] and it was applied most famously to solve the Poincare conjecture. In general, the Ricci flow is a PDE, however on a homogeneous space ...

We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n = 2, we prove an optimal dimension reduction theorem for complete translating KählerRicci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the Kähler-Ricci flow with nonneg...

As a step toward understanding the analytic behavior of TypeIII Ricci flow singularities, i.e. immortal solutions that exhibit |Rm | ≤ C/t curvature decay, we examine the linearization of an equivalent flow at certain fixed points discovered recently by Baird–Danielo and Lott: non-gradient homogeneous Ricci solitons on nilpotent Lie groups. We show that the linearized operators at these fixed p...

Yau’s uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to C. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kähler-Ricci flow with focus on its application to...

∣ 2 u dV. This implies in particular that d dt μ(g(t), τ(t)) ≥ 0 with equality exactly for homothetically shrinking solutions of Ricci flow. An important consequence of this entropy formula is a lower volume ratio bound for solutions of Ricci flow on a closed manifold for a finite time interval [0, T ) asserting the existence of a constant κ > 0, only depending on n, T and g(0), such that the i...

An invariant cone in the space of curvature operators is one that is preserved by a flow. For Ricci flow, the condition R ≥ 0 is preserved in all dimensions, while the conditionR ≤ 0 is preserved only in real dimension two. Positive curvature operator is preserved in all dimensions [11], but positive sectional curvature is not preserved in dimensions four and above. The known counterexamples, c...

This paper concerns conditions related to the first finite singularity time of a Ricci flow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality method, suggested by N. Le [13] and F. He [11]. We also display a close connection between this method and time slice analysis as in [23]. As an application, we prove several inequalities for a Ric...