0
votes
0answers
78 views

What are the most important papers to read about Ricci flow? [duplicate]

I would like to know, which papers are important to read, if one wants to work about Ricci flow. What papers have to be known by every person conducting research in this field? Can you provide me with ...
2
votes
1answer
188 views

Optimal, conformal diffeomorphisms between two surfaces in 3D

Let $S_1$ and $S_2$ be two smooth, closed surfaces embedded in $\mathbb{R}^3$. Q. Is there a natural definition of the optimal, conformal diffeomorphism between $S_1$ and $S_2$? I am imagining ...
6
votes
1answer
511 views

Yang-Mills flow, Ricci flow and the holonomy

Is the holonomy group (based at some point) preserved along the Yang-Mills flow/ Ricci flow? (1) For Yang-Mills case, we know that the centralizer of the holonomy $H_x$ is the isotropy group of the ...
-1
votes
2answers
168 views

On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows: Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...
3
votes
1answer
331 views

Reverse Ricci Flow and Longtime Existence

The usual Ricci flow and normalized Ricci flow for surfaces are $$ \partial_t g = -2Kg $$ and $$ \partial_t g = -2Kg + 2sg,$$ where $K$ is the Gaussian curvature and $s$ is its average. The latter ...
3
votes
2answers
272 views

Bryant Soliton is asymptotically cylindrical?

This is my first question in mathoverflow. I'm now reading Brendle's paper http://arxiv.org/pdf/1203.0270.pdf. I'm confused about how to check Condition (ii) of asymptotically cylindrical condition ...
1
vote
1answer
198 views

On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have ...
8
votes
1answer
381 views

How fast does Ricci flow converge on the three-sphere?

Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the ...
1
vote
1answer
378 views

Ricci flow as a gradient flow and its Lyapunov function

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 ...
3
votes
1answer
228 views

What is visualization of gradient flow of a functional?

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 can be written as a ...
3
votes
2answers
494 views

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 ...
0
votes
0answers
134 views

Isoperimetric profile

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 is defined by ...
0
votes
0answers
185 views

A question from Hamilton's Ricci Flow book by bennett chow

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 condition results ...
3
votes
1answer
251 views

geometric meaning of Ricci-flatness

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 vanished at a point, ...
5
votes
1answer
394 views

How submanifolds evolve under Ricci flow?

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 found online... ...
4
votes
1answer
301 views

Rigorous solution to Ricci Flow on dumbbell $S^3$

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, so I am wondering ...
1
vote
1answer
349 views

Ricci flow descending from an universal cover

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 $K\ge0$, but I don't ...
6
votes
2answers
419 views

the left hand side of the Ricci flow equation at the initial value

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 on a smooth closed ...
1
vote
2answers
477 views

Does normalized Ricci flow on surfaces yield a bundle?

As is well known, the normalized Ricci flow is defined for all $t>0$ on compact surfaces, and every metric on a compact surfaces converges to a metric constant curvature if $X \neq S^2$ (at least I ...
8
votes
3answers
1k views

Ricci flow with surgery in dimension 2

Is it possible to define the Ricci flow with surgery in dimension 2 and use it to classify the surfaces? I know this is overkill, there are simpler ways to classify surfaces, but I would like to ...