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I am currently reading Canary's "Ends of Hyperbolic 3-Manifolds". At page 17, while talking about some branched covering of a hyperbolic 3-manifold, he says that "$\hat{N}$ inherits a metric of constant negative curvature except along the branching locus, where the metric is singular and has concentrated negative curvature". $\hat{N}$ being a branched cover of a hyperbolic manifold $N$ over a collection of geodesics.

I also encountered the phrase while reading about simplicial ruled surfaces, where a global Riemannian metric is not definable, and so one defines a $CAT(-1)$ structure "with negative curvature concentrated at the vertices".

My question is: what does "concentrated negative curvature" mean? Can somebody please point me to a reference? I know that there are some Gauss-Bonnet-type theorems for these metric spaces, and I actually need them for what I am doing.

Thank you in advance.

Ted Shifrin
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Ras-Al-Shaytan
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    One could give a satisfactory and reasonably simple answer to your question in the more restrictive setting of "Gauss-Bonnet-type theorems" for metrics on 2-dimensional manifolds, depending on which theorem(s) you have in mind. There may well also be a satisfactory answer in a broader setting including Canary's paper, but it would not be as simple. – Lee Mosher Mar 15 '25 at 17:02
  • I would like to know what a metric with concentrated negative curvature is, manily. Regarding the Gauss-Bonnet thing, I need it to show that the simplicial ruled surfaces embedded in a geometrically infinite (thus simply degenerate) end of a PNC manifold have uniformly bounded volume. My ultimate goal is to show that in the case of positive injectivity radius, the positivity of the Cheeger constant is equivalent to the vanishing of the bounded fundamental class of PNC manifolds. – Ras-Al-Shaytan Mar 15 '25 at 17:31
  • Please tell me if I need to clarify some things. – Ras-Al-Shaytan Mar 15 '25 at 17:32
  • Well, let me repeat what I said in different words: explaining concentrated negative curvature *in the Gauss-Bonnet context (in 2 dimensions)* is reasonably straightforward; explaining it in the Canary setting (in 3 dimensions) is not so straightforward. So, with this in mind, if you can ask a more precise question, you should edit your post to do so. – Lee Mosher Mar 15 '25 at 20:01
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    My suggestion is to ignore this sentence: Knowing the local $CAT(-1)$ property is enough for the purpose of reading the article. If you want to have a better conceptual understanding of this situation, see my answer here and the references therein. – Moishe Kohan Mar 16 '25 at 15:33
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    Thank you Moishe, I read your answer in the linked page, and I will take a look at the references. Very helpful. – Ras-Al-Shaytan Mar 17 '25 at 07:01

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I don't think that "concentrated curvature" is a mathematical notion like "curvature" is. Instead it's the author's description of the situation.

Consider a polygon on a plane. On the sides of the polygon, the curvature vanishes, but in the corners we have a "concentrated curvature" giving a contribution to the total curvature (which is $2\pi$).

In higher dimensions you can consider a polyhedron or a cone. On the surfaces the curvature vanishes, but in the corners or a the tip of the cone, we have a "concentrated curvature" giving a contribution to the total curvature.

md2perpe
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