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.

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Positive scalar curvature Einstein manifolds are noncollapsed

I am currently working through some of the exercises in Ricci Solitons in Low Dimensions by Bennett Chow, and I've been stuck on Exercise 1.23. It asks you to prove that positive scalar curvature ...
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Smoothness of Ricci flow solution on a closed interval

In the paper "Deforming the metric on complete Riemannian manifolds" by Wan-Xiong Shi, the author proves the following theorem which I copy verbatim below: Theorem 1.1. Let $(M, g_{ij}(x)$ ...
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A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
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Definition of Ricci flow

My undergraduate thesis is related to the Ricci flow, and I have a number of basic questions. Let $M$ be a smooth manifold. At the start of Chapter 2.3 of Peter Topping's Lectures on the Ricci flow, ...
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Unbounded diameter for the blow up on Ricci Flow

Let $(M,g_0)$ be a compact n-dimensional Riemannian Manifold. Suppose that $g(t)$ is an smooth solution to the Ricci flow $\partial_tg(t)=-2Rc(g_t)$ on a maximal time interval $[0,T)$ of existence, ...
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Why does bounded curvature mean that ricci flow has solution?

Picture below is from Hamilton's The Ricci flow on surfaces. In this paper, the author consider the equation $$\partial_t g_{ij}=(r-R)g_{ij} \tag{1}$$ where $r$ is the average of $R$. From the 4.7 ...
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Problem calculating simmetrized product of Ricci tensor

I was reading the following article, about an example of neckpinching for Ricci flow on $S^{n+1}$, with $n\geq 2$, but I have a problem in the proof of Proposition 5.7. The setting is the following. ...
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