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Let $g_i$ be a family of smooth Riemannian metrics on the standard sphere $S^n$ ($n\geq 3$). Assume that the Gromov–Hausdorff distance satisfies $d_{GH}(g_i, g_{st}) \le 1/i$ and that $\operatorname{...
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I am interested in understanding whether mean curvature flow with Dirichlet boundary conditions can produce finite-time singularities that are localized to a neck region connecting two Euclidean ...
DimensionalBeing's user avatar
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Let $(M^{3},g(t))_{0\le t<T}$ be a Ricci flow on a closed orientable 3-manifold that performs Perelman surgery at the discrete times $0<t_{1}<\dots <t_{N}<T$. Each surgery excises one ...
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A Riemannian manifold $(M, g)$ is said to be a non-trivial gradient almost Ricci soliton if there exist smooth (and non-constant) functions $f, \lambda: M \to \mathbb{R}$ such that $$ \operatorname{...
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I'm a prospective graduate student organizing some results about Ricci soliton. P. Petersen and W. Wylie in "On the classification of gradient Ricci solitons" gave a remark on R. S. Hamilton'...
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I'm an undergraduate organizing some basic knowledge of Ricci soliton. When I read "On the classification of gradient Ricci solitons" by P. Petersen and W. Wylie, I saw the result given by R....
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I don't know if my question is too simple for this forum but let me proceed. In Ricci flow one equips a smooth manifold $M$ with a Riemannian metric $g_0$ and evolves the metric with "time": ...
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What does it mean when one says the inequality must be understood in the barrier sense, when necessary? I encountered this notion when I was reading Perelman's paper "The entropy formula for the ...
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Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("...
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I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not. So my question is if one starts with a metric that has mostly ...
Emmie's user avatar
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I have a basic question about how geometric flows such as the Ricci flow interact with the Nash embedding theorem. Say you have a 1-parameter family of Riemann metrics on a compact manifold $N$. If ...
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Let M be a (compact) smooth manifold. What kind of obstruction exist for M to admit a metric whose Ricci flow is a t-periodic flow?
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The curve-shortening flow is $$ \frac{\partial C}{\partial t} = \kappa n $$ where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...
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We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$. I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
Adrien B's user avatar
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Let $\Delta$ be a Laplacian or an elliptic operator over a manifold, can the heat equation be defined for complex time? Can we define: $$e^{-z \Delta}$$ for $Re(z)>0$ ? Also can the Ricci flow be ...
Antoine Balan's user avatar
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It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow. A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of ...
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Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $...
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I am sorry if this is a silly question, but I am new to Ricci flows. Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...
Eduardo Longa's user avatar
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In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
Matheus Andrade's user avatar
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Picture below is from Topping's Lectures on the Ricci flow. I've been stuck by the red line about two months. In fact, I asked it on ME two months ago. To describe the problem more precisely, ...
Enhao Lan's user avatar
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In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
Matheus Andrade's user avatar
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This problem comes from the book Hamilton's Ricci flow. Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...
James Chiu's user avatar
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This problem comes from the book Hamilton's Ricci flow. Given a smooth functional $f$, and following system. $$\partial_tg_{ij}=-2(R_{ij}+\nabla_i\nabla_jf)$$ If there exist a 1 parameter family of ...
James Chiu's user avatar
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Picture below is from Topping's Lectures on Ricci flow. I don't understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But ...
Enhao Lan's user avatar
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So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
James Chiu's user avatar
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Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs. There are at least two versions of Ricci curvature in the ...
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So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
James Chiu's user avatar
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29 votes
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[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.] According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of ...
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In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
Matheus Andrade's user avatar
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Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
Matheus Andrade's user avatar
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I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't ...
Gabe K's user avatar
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I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
Gabe K's user avatar
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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 ...
Naruto's user avatar
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There are two separate places where ingenious uses of gauge transformations simplify the analysis of Ricci flow considerably.  The Deturck trick is a way to break the diffeomorphism invariance of the ...
Gabe K's user avatar
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16 votes
4 answers
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Is this a theorem? Every $3$-connected planar graph $G$ may be represented as a tiling of a square by squares, one square per node of $G$, with nodes connected in $G$ corresponding to tangent squares....
Joseph O'Rourke's user avatar
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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 ...
Matheus Andrade's user avatar
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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 ...
Hollis Williams's user avatar
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13 votes
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Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
Hollis Williams's user avatar
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13 votes
3 answers
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I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics ...
Matheus Andrade's user avatar
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In a couple articles I've read lately, I've seen it mentioned that the cigar soliton has linear volume growth. What does this mean? I thought maybe, if you compute the volume of geodesic balls and ...
Matheus Andrade's user avatar
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9 votes
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I have been unable to find a reference to the following (perhaps too naive) question. Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-...
u184's user avatar
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1 answer
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All examples about geometric flow equations given in Wikipedia's Geometric flow article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order ...
Kirby's user avatar
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2 votes
1 answer
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I know that type II singularities of the Ricci flow can exist on closed 3-manifolds (e.g. on $S^3$), but on the other hand it seems to me that ODE comparison combined with Hamilton's tensor maximum ...
srp's user avatar
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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 ...
Hollis Williams's user avatar
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We read and checked the detailed proof of the Poincare conjecture. One can find the article (Ricci Flow And The Poincare Conjecture by Morgan and Tian) on arXiv. Since the proof contains some gaps and ...
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2 votes
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162 views

Let $(P,g) \to (S^2,h)$ be a Riemannian submersion. Let $g(t)$ be the Ricci flow on $P$ with initial condition $g$. Does the induced flow on $S^2$ converges to the round metric on $S^2?$ I could ...
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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 ...
Jae Ho Cho's user avatar
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Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
user162551's user avatar
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Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
Jae Ho Cho's user avatar
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I am reading Peter Topping's notes on Ricci flow: on page 99 a statement is made which is needed for his proof of a version of Perelman's no local volume collapse theorem, but I am not sure why it ...
Hollis Williams's user avatar
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