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Results tagged with riemannian-geometry
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user 119114
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3
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Accepted
Does the Cheeger constant satisfy a heat-type equation?
Answer to (2) is negative. For a counterexample, consider a $2$-sphere and blow up the radius. The isoperimetric ratio $C_H$ remains the same, but the Cheeger constant $h$ shrinks, essentially becau …
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14
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1
answer
395
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Does the Cheeger constant satisfy a heat-type equation?
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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1
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0
answers
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Relation between Sormani-Wenger and Gromov-Hausdorff Convergence
I am reading a bit about Sormani-Wenger intrinsic flat distance between compact oriented Riemannian manifolds out of curiosity.
There are some settings which it can be shown that Gromov-Hausdorff conv …
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0
votes
Rigorous solution to Ricci Flow on dumbbell $S^3$
Yes, these pictures have now been made rigorous. Another paper which you might be interested in on this topic is
Simon, M. (2000). A class of Riemannian manifolds that pinch when evolved by Ricci flo …
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1
vote
What are the tricks for computing/estimating Gromov-Hausdorff distance?
Not sure if this is a helpful example but if $Y$ is the space of a single point, then
$d_{GH}(X,Y) \leq \text{rad} \: X = \inf_{y \in X} \sup_{x \in X} d(x,y),$
which is just the radius of the smalles …
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1
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0
answers
135
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Perturbation of a spacetime in general relativity
In general relativity one has the Schwarzchild metric for a non-rotating black hole
$g_{SC} = -\phi^2 \: dt^2 + \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $
and from this one has the spacelike Schwarzc …
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13
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2
answers
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Is there a solution of the Yamabe problem using Ricci flow?
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 pa …
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4
votes
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answer
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Example of a Manifold which has One Non-zero Component of Ric corresponding to Scalar Curvature
I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $R$, I can find a manifold such that the Ricci tensor has all zero components except for one …
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0
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Does the Volume Ratio of a Geodesic Ball for a Complete Riemannian Manifold tend to the volu...
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 hol …
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1
vote
0
answers
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Estimate on Covariant Derivatives of Coordinate Derivatives
I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that
$\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\partia …
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2
votes
1
answer
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Principal Symbol for the Ricci-DeTurck Flow
I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 …
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