Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface M of genus γ>1 (I am using the more usual letter “g” to denote metric), and the normalized Ricci flow on it. It is known that at time t, the scalar curvature R satisfies |R−r|<Cert, where r=∫MRdμ∫Mdμ is the average scalar curvature of M, and C is a constant depending only … Read more

Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the evolution equation $$\frac{\partial}{\partial t}(\nabla \text{Rm})=\Delta(\nabla \text{Rm})+\text{Rm} \ast \nabla \text{Rm}$$ and the uniqueness of the solution due to the compactness of … Read more

Is there a version of Ricci Flow for Pseudo-Riemannian Metrics?

The Ricci flow deforms a Riemannian metric. I was wondering if there was something very similar which deforms a pseudo-Riemannian metric or if not, is there reason why such a geometric flow cannot exist? Also, why in particular does the Ricci flow only work with a Riemannian metric? What would fail in particular for the … Read more

Classifying singularities of the Ricci flow

Context: A solution (Mn,g(t)) of the Ricci flow is said to encounter a Type III Singularity if g(t) is defined for all t≥0 and: supMn×[0,∞)‖ Similarly, if g(t) is defined for all t \geq 0 but: \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\operatorname{Rm}(\cdot, t) \| t= \infty the solution is said to encounter a Type IIb singularity. … Read more

Low boundary of $\mathcal W$ function

Picture below is from Topping’s Lectures on Ricci flow. I don’t understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But how to get $\mathcal W$ has uniform low boundary on $(0,\tau_0]$ ? PS: I asked the problem in ME without answer. Therefore, … Read more

Neckpinch singularity of Ricci flow

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the sense that it occurs on a compact subset of a manifold. The classic picture is that of a … Read more

Energy functional

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman’s works $\mathcal{F}(g,f)=\int_M(R+|\nabla f|^2)e^{-f}d\mu$ is introduced as an energy functuional, where $M$ is a closed manifold, $g$ is Riemannian metric, $R$ is Ricci scalar, and $f$ is any function that … Read more

Curvature blow up along Ricci flow

In the book on Ricci flow by Andrews and Hopper, it has been proved that if Ricci flow on M has a finite time singularity at time T then lim. I am wondering whether the following assertions are false or unknown (I know that the following assertions are true if \lim is replaced by \lim … Read more

Prerequisites for reading Gregory Perelman’s work

What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture? I am referring to the last three papers here. Answer AttributionSource : Link , Question Author : Alan , Answer Author : Community