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I am reading a book on Ricci flow and at one point there is a computation $\Box^* v$ where $v=[\tau(2\Delta f - |\nabla f|^2 + R) + f -n]u $ and $\Box^* = -\partial_t - \Delta + R.$ To compute $\B …

I am currently reading 'Lectures on Ricci Flow' by Peter Topping and I have got to Chapter 3 where he states the 'weak maximum principle for scalars'. Suppose for $t \in [0,T]$ for finite $T$ that $g …

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 …

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 …

Is there some result on existence of marginally trapped surfaces in spacetime 4-manifolds? Am I right in saying that a marginal surface (like a trapped surface in general) is a compact spacelike 2- …

I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system i …

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 ex …

I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, …

Not a survey per se, but a good article on the theorem is the one below which gives a quick, clear sketch of the second part of Schoen and Yau's proof of the positive mass theorem. It might be helpful …

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 …

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 …

I have noticed that in the literature on causality in general relativity one sees apparent counterexamples to the cosmic censorship hypothesis (somehow you have models for gravitational collapse which …

Having thought about this more and discussed it with others, the answer seems to be that there are likely no counterexamples to the Penrose inequality, even if one allows for unphysical violations. Fo …

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 …

It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves). I …