# Questions tagged [non-positive-curvature]

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### Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) ...
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### Tangent cone of metric graph

I am starting to study some lecture notes about metric geometry and I would appreciate it if someone could some questions regarding the notion of the tangent cone. Consider 3 half lines joined by ...
33 views

### how to normalize curvature to be between -1 and -1/4

On paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one", lemma 8.1, when the author tried to construct finite energy map from Quaternion hyperbolic ...
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### Horospherical distance in CAT($-1$) space

In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then d_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d_{\...
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### Proximal isometries in CAT($-1$) metric space

Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
178 views

### Is the completion of a CAT(0) open ball a closed ball?

It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball! But CAT(0) spaces are ...
390 views

### The midpoint geodesic

Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment. Pick three distinct ...
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### Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
139 views

### What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible ...
506 views

### Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
Let $X$ be a hadamard space and $\gamma_1, \gamma_2 \colon \mathbb{R}\rightarrow X$ be two geodesics. Part 2 of Coroallary 2.5 in http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-...