Questions tagged [ricci-flow]

The Ricci flow on a Riemannian manifold $(M,g)$ is determined by the geometric evolution equation $\partial_t g_{ij} = -2R_{ij}$ where $R_{ij}$ is the Ricci curvature. The Ricci flow is the main ingredient in Perelman's proof of the Poincaré conjecture.

Filter by
Sorted by
Tagged with
1 vote
0 answers
17 views

Curvature estimate in Hamilton's Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold

In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
user avatar
2 votes
0 answers
33 views

Convex function on Riemannian manifold

When I read the 9.5 of Topping's Lectures on the Ricci flow, I have some problem. Assume $W$ is a vector bundle over manifold $(M,g)$, and $A$ is connection on $W$. $\{e_1,...,e_l\}$ is a frame of $...
user avatar
  • 5,140
2 votes
0 answers
36 views

How to show $A^2e_i=-\frac{1}{2} R_A(\cdot, \cdot)e_i$ in Topping's Lectures on the Ricci flow

When I read the 9.5 of Topping's Lectures on the Ricci flow, I have some problem. Assume $W$ is a vector bundle over manifold $(M,g)$, and $A$ is connection on $W$. $\{e_1,...,e_l\}$ is a frame of $...
user avatar
  • 5,140
0 votes
0 answers
23 views

Changing the system of PDE by diffeomorphism

This problem comes from the book Hamilton's Ricci flow. Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...
user avatar
1 vote
0 answers
43 views

A basic question about calculation under geodesic frame

I have been recently reading some books related to Ricci flow, and I am confused when it comes to the evolution of Levi-Citita connection. That is: If $(M^n,g)$ Is a closed Riemannian manifold, and we ...
user avatar
0 votes
0 answers
21 views

How to find evolution equation of certain geometric quantities under Ricci flow

I want to find evolution of certain geometric quantities, like Weighted Laplacian $\Delta_\phi:=\Delta-\nabla\phi\nabla$, where $\Delta$ is Laplacian operator and $\nabla$ is gradient operator, under ...
user avatar
2 votes
0 answers
65 views

Using Rauch comparison theorem to get an estimation of two metric.

Picture bottom is from the section 8.4 of Topping's Lectures on the Ricci flow. I use the following Rauch theorem From the $|Rm|\le 1$, I have sectional curavture $ K \le 1$. Therefore, I have $$ |Y|...
user avatar
  • 5,140
1 vote
0 answers
19 views

A detail of volume ratio bounds in Ricci flow

Picture bottom is from Topping's Lectures on the Ricci flow. I can't get the red line. First, I have the Bishop's theorem: where $V^\alpha(r)$ is the volume of a ball of radius $r$ in the complete ...
user avatar
  • 5,140
0 votes
0 answers
14 views

Reference about the existence of Ricci flow with surgery for all positive time

As far as I can see, the Perelman's Ricci flow with surgery on three-manifolds does not prove the existence of Ricci flow with surgery for all positive time. And it is necessary for proving ...
user avatar
  • 5,140
1 vote
0 answers
84 views

mean curvature flow vs. ricci flow

Both of these seem to describe very similar flows: curve-shortening flows of a high-dimensional hypersurface, which may form singularities, where the flow velocity at any point is normal to the ...
user avatar
  • 1,424
2 votes
0 answers
33 views

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 ...
user avatar
  • 5,140
0 votes
1 answer
55 views

How to show $\frac{n}{4}\ln [1+\int |\nabla \phi|^2 dV]\le \frac{1}{2\pi}\int |\nabla \phi|^2 dV + C(g,n)$? [closed]

Picture below is from 90th page of Topping's Lectures on the Ricci flow. The integration is take on closed smooth manifold. I can't get the red line. What I try: $$ \frac{n}{4}\ln [1+\int |\nabla \...
user avatar
  • 5,140
2 votes
0 answers
33 views

How to use delta function as initial data?

Picture below is from 74th page of Topping's Lectures on the Ricci flow. The (6.4.8) is $$ -\partial_t u -\Delta u + Ru =0 \tag{6.4.8} $$ where $R$ is scalar curvature. And the $N$ is $$ N= -\int_M u \...
user avatar
  • 5,140
0 votes
0 answers
62 views

How to show the upperbound of the Ricci tensor preserved

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
user avatar
2 votes
0 answers
33 views

Continuously extend of bounded tensor

If $g(t)$ is Ricci flow on $[0,T)$, and $$ |Rm|\le M ~~~~~~ \forall t\in[0,T), $$ By the Metric equivalence $$ e^{-2Mt} g(0) \le g(t)\le e^{2Mt} g(0), $$ I know $g(t)$ is bounded (namely, there is $A,...
user avatar
  • 5,140
1 vote
1 answer
38 views

$g=\rho^2 (d x^2 + dy^2) \Rightarrow g = ds^2 +\tanh^2 s d\theta^2$

The question is from the 10th page of Topping's Lectures on Ricci flow, the calculation about Hamilton's cigar soliton. Consider $R^2$ with metric $$ g=\rho^2 (dx^2 + dy^2),~~~\rho^2= \frac{1}{1+x^2 ...
user avatar
  • 5,140
0 votes
0 answers
24 views

Non-linearity of Ricci flow equation

I know that the Ricci flow equation, $\frac{\partial g}{\partial t} =-2 Ric $, is non-linear but I can't understand how it satisfies the non-linearity conditions. Please help me. Thanks in advance.
user avatar
4 votes
2 answers
95 views

How to show $\partial_t \hat g = \sigma'(t)\psi_t^* (g) + \sigma(t) \psi_t^*(\partial_t g) + \sigma(t) \psi_t^*(L_Xg)$?

$X(t)$ is a time dependent family of smooth vector fields on $M$, and $\psi_t$ is the local flow of $X(t)$, namely for any smooth $f:M\rightarrow R$ $$ X(\psi_t(y),t) f = \frac{\partial(f\circ \psi_t)...
user avatar
  • 5,140
0 votes
0 answers
90 views

Remark 3.1.3 of Lectures on the Ricci flow

About Theorem 3.1.1, Remark 3.1.3 state that the strong maximum principle means a better results. I find the strong maximum principle from Evans' book which is in 3th picture below. The operator $L$ ...
user avatar
  • 5,140
0 votes
0 answers
26 views

Uniformly convergence of solution in Weak maximum principle

For $T<\infty$, if $\phi: [0,T] \rightarrow R$ solves \begin{align} &\frac{d\phi}{dt} =F(\phi(t),t) \\ &\phi(0) =\alpha \end{align} where $F(x,t)$ is smooth. Then, how to show there is $...
user avatar
  • 5,140
1 vote
1 answer
49 views

How to show $|Ric|^2 \ge \frac{R^2}{n}$?

$Ric$ is Ricci curvature. Making orthogonal decomposition, there is $$ Ric =\mathring{Ric} + \frac{R}{n}g \tag{1} $$ where $R$ is scalar curvature, $g$ is Riemannian metric. I get (1) from the 42th ...
user avatar
  • 5,140
1 vote
1 answer
45 views

How to show $(\delta S)Z = \delta (S(\cdot, Z))+\frac{1}{2}\langle S, \mathcal L_Zg\rangle$?

I want to show $$ (\delta S)Z = \delta (S(\cdot, Z))+\frac{1}{2}\langle S, \mathcal L_Zg\rangle \tag{1} $$ where $\delta= -tr_{12}\nabla$ is divergence operator, $\mathcal L$ is Lie derivative, $S\...
user avatar
  • 5,140
5 votes
1 answer
67 views

How to show $\delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle $?

Picture below is from Topping's 37th page of Lectures on the Ricci flow. I try to show $$ \delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle $$ $\delta$ ...
user avatar
  • 5,140
1 vote
0 answers
30 views

How to understand $(\pm*d*\alpha) dV = \pm d(*\alpha)$ in Ricci flow

Picture below is from Topping's Lectures on the Ricci flow. First, I don't know why $$ (\pm*d*\alpha) dV = \pm d(*\alpha), $$ in fact, I don't know what is $*\alpha$ and $d*\alpha$. In my opinion, $*$ ...
user avatar
  • 5,140
2 votes
1 answer
43 views

A problem on showing $h(X, \operatorname{Ric}(W))= \operatorname{tr} h(\operatorname{Ric}(W,\cdot)\cdot, X)$

I want to show $(2.3.11)$, but in my calculation, there is no minus. What I try: from the $(2)$, I have $$ h(X,\text{Ric}(W))=h(X, \text{Ric}(W, e_i)e_i)= \text{tr}\space h( X, \text{Ric}(W, \cdot)\...
user avatar
  • 5,140
1 vote
1 answer
39 views

Show $\operatorname{tr} \nabla^2_{X,~\cdot} h(\cdot, W)= -(\nabla \delta h)(X,W)$

I want to show $$ \operatorname{tr} \nabla^2_{X,~\cdot} h(\cdot, W)= -(\nabla \delta h)(X,W) $$ where $X,W$ are tangent vector fields on Riemannian manifold. And $$ \nabla^2_{X,Y} = \nabla _X \nabla_Y ...
user avatar
  • 5,140
1 vote
0 answers
37 views

Formal adjoint of divergence which is restricted on $\Gamma(\wedge^k T^*M)$

I want to show the red line. In fact, in another problem , with many help, I get the formal adjoint of divergence without restrict. Beforehand, I think I can deal the red line if I know how to get the ...
user avatar
  • 5,140
1 vote
1 answer
59 views

Proof of Proposition 2.3.5 of Topping's Lectures on the Ricci flow

When I read the Proposition 2.3.5 of Topping's Lectures on the Ricci flow. I can't understand why the spcial vector fields which satisfy (2.3.13) do not loss generality. In fact, in my view, they ...
user avatar
  • 5,140
1 vote
1 answer
96 views

Why $(\partial_t \nabla _X \omega)Y =-\omega(\partial_t \nabla_X Y)$ means $\partial_t \nabla \omega = \omega * \nabla (\partial _t g)$?

$\omega \in \Gamma(T^*M)$, $X,Y\in \Gamma(TM) $, $(M, g(t))$ is Riemannian manifold whoes metric depends on $t$. If I have $$ (\partial_t \nabla _X \omega)Y =-\omega(\partial_t \nabla_X Y) $$ Why $\...
user avatar
  • 5,140
1 vote
0 answers
185 views

Formal adjoint of divergence operator

Picture bottom is from Topping's Lectures on the Ricci flow (author website, zbMath link). I wanto to show the formal adjoint of $\delta$ is covariant derivative. The definition of formal adjoint is ...
user avatar
  • 5,140
4 votes
0 answers
75 views

Variation on Ricci tensor about metric

Picture below is from the "Three manifolds with positive Ricci flow". I try to calculate the red line. First, in normal coordinate, I have $$ E(g_{ij})=-2R_{ij} = \frac{\partial}{\partial ...
user avatar
  • 5,140
0 votes
1 answer
46 views

Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow

In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
user avatar
3 votes
0 answers
63 views

Embedded Ricci Flow

Consider a submanifold $\mathcal{M}$ that is embedded in a higher dimensional manifold $\mathcal{N}$. Now if I infinitesimally perturb the submanifold as $$g_{\alpha\beta}\rightarrow g_{\alpha\beta}+\...
user avatar
  • 892
2 votes
0 answers
52 views

Proving singularity types of Einstein manifolds under the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
user avatar
8 votes
0 answers
121 views

Where does the $2$ in Ricci flow come from?

I started learning about Ricci flow recently, which is always given as $$ \frac{\partial g}{\partial t}=-2\textrm{Ric}. $$ It would seem more natural to me to define Ricci flow instead by the equation ...
user avatar
  • 3,785
2 votes
0 answers
21 views

Minkowski formulae for hypersurfaces in GRW spacetimes

Could someone tell me why $$\overline{\operatorname{Ric}}(U,V) = \operatorname{Ric}_M(U^* ,V^*)+ (n((\log f)')^2+(\log f)'')\langle U,V \rangle -(n-1)(\log f)'' \langle U, \partial_t \rangle \langle ...
user avatar
4 votes
1 answer
72 views

Question regarding the quadratic curvature tensor

I am studying the evolution of curvature in my study on the Ricci flow, and in The Ricci Flow in Riemannian Geometry by Hopper and Andrews, I came across the (0,4) quadratic curvature tensor defined ...
user avatar
2 votes
1 answer
126 views

Can Ricci flow be used to prove Poincaré’s conjecture for $n=2$?

This question is concerning the conjecture described in this Wikipedia article. The conjecture has been proved for dimension two ($n = 2$). For $n=3$, conjecture is proved using Ricci flow. My ...
user avatar
  • 2,886
1 vote
0 answers
45 views

Does scalar curvature is bounded by absolute value of Riemann curvature?

Is it true if the lowest eigenvalue of the Ricci tensor is negative than $Scal \le |Rm|$? What i guess is $$|Rm| \ge \frac{Scal}{n} $$ by using Cauchy-schwarz inequality. However while i read the ...
user avatar
  • 46
0 votes
0 answers
40 views

Is there any kind of "flow" that describe balloon-like expansion for 2d surfaces constructed via a 1d paths?

I've looked into Ricci Flow and mean curvature flow/curve shortening flow, but I dont think any of these describe the math behind balloon-like expansion (or even simple diffusion) well enough given ...
user avatar
3 votes
0 answers
123 views

What is missing from solution of Poincaré conjecture to Naivier-Stokes equation?

This question has arised from the links of the comments in the question What are the fundamental problem in solving the NAVIER–STOKES equation.. As I understand Ricci flow has been used in solving the ...
user avatar
  • 2,886
1 vote
1 answer
77 views

Question on Ricci flow and Einstein Tensor

My knowledge on this topic is solely based on physics books, introductory texts on differential geometry and this one paper $[1]$. One of the formulas of the paper shows what is called the Ricci ...
user avatar
  • 635
5 votes
1 answer
219 views

Ricci flow as heat flow on Riemannian manifold

I read that Ricci flow is "a nonlinear heat flow for the Riemannian metric". Can someone explain what this means? Nonlinear heat flow has a wikipedia page but I don't understand how this ...
user avatar
  • 1,424
4 votes
2 answers
192 views

Mean curvature flow vs. diffusion

Mean curvature flow (MCF) and diffusion-type flows both have smoothing effects on a curve. I can tell there is a deep connection between the two, as seen in the Merriman-Bence-Osher (MBO) numerical ...
user avatar
  • 1,424
1 vote
0 answers
57 views

Gradient of solution to heat equation under evolving metric

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
user avatar
  • 21
2 votes
0 answers
139 views

Book on the geometry of rotationally symmetric riemannian manifolds

I would like to find some references where there are specific computations and properties of rotationally symmetric riemannian manifolds, e.g. spectrum of the laplacian, schrödinger operators, ...
user avatar
2 votes
0 answers
56 views

Proving some identities about the time derivative of the k-th covariant derivatives of scalar curvature under normalized Ricci flow on surfaces

I'm trying to prove the following identities (under the normalized Ricci flow on surfaces, on which $\partial_t g = (r-R)g$ holds true, where $r$ denotes the average scalar curvature and has the same ...
user avatar
1 vote
1 answer
129 views

Motivation of Uhlenbeck’s trick without local computation(Suitable connection defined in space time manifold M $\times \mathbb{R}$)

I am a beginner to learn Ricci flow and my main study reference is Simon Brendle’s “Ricci Flow and the Sphere Theorem”. In section 2.3 of Brendle, he introduce the following: let D be the Levi-Civita ...
user avatar
1 vote
0 answers
16 views

Global Neckpinch Singularity under Ricci Flow

I am aware that in general for the Ricci flow, neckpinches are local singularities (ie. they occur on a compact subset of the manifold). The usual picture is that of a manifold shaped like a dumbbell ...
user avatar
  • 1,076
1 vote
1 answer
123 views

Curvature waves, harmonic curvature, and curvature flow

Let me first say I'm better at physics than math and have big gaps in my understanding of Riemannian geometry. I do a lot better with intuitive explanations than identities and so forth. I've been ...
user avatar

1
2 3 4 5