In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written: Fix a closed manifold $M$ with a probability measure $m$, and …

I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, …

I want to study gradient Ricci solitons. Can anyone help me to find a simple and good reference about solitons and their applications? Thanks

In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then $\frac{d …

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow ca …

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 $ …

In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile …

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vani …

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free …

This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I fo …

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, …

We know Laplace equation (elliptic equations) $ Δ u = 0$ Heat equation (parabolic equations) $u_t − Δu = 0$ Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$ we have - H …

Reading some of my old notes, I came across a remark, I don't understand. Summarized: Let $(M^n,g)$ be an open riemannian manifold (in that case it came with sectional curvature $ …

I just started to learn about the Ricci flow and try to understand the Ricci flow evolution equation. It states that a one-parameter family $g_t$, $t\in[0,T)$ of Riemannian metrics …

The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing. …