# Questions tagged [non-positive-curvature]

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21
questions

**8**

votes

**0**answers

118 views

### 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) ...

**2**

votes

**1**answer

53 views

### 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 ...

**0**

votes

**0**answers

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 ...

**-1**

votes

**1**answer

59 views

### 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_{\...

**4**

votes

**0**answers

62 views

### 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 ...

**6**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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 ...

**9**

votes

**0**answers

366 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $...

**3**

votes

**1**answer

97 views

### Billera Tree Space

I am studying the tree space of Billera and I do not really understand why it is an Hadamard Space. I have already read L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, ...

**5**

votes

**1**answer

112 views

### Convex embedding with a positivity condition

We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...

**6**

votes

**1**answer

217 views

### Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...

**1**

vote

**1**answer

127 views

### Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...

**9**

votes

**1**answer

368 views

### Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given $...

**6**

votes

**0**answers

103 views

### 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 ...

**10**

votes

**0**answers

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 ...

**12**

votes

**2**answers

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 $\mathcal{F}$ ...

**3**

votes

**1**answer

141 views

### Geodesic comparison in Hadamard space

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-...

**5**

votes

**0**answers

135 views

### Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?
Examples for what I have in mind:
Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and $\...

**15**

votes

**3**answers

587 views

### Embedding expanders in CAT(0) spaces

It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.
Can anyone provide a reference (...

**6**

votes

**0**answers

240 views

### Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...

**4**

votes

**1**answer

437 views

### Volume of a geodesic simplex on a manifold of non-positive curvature.

Let $M$ be a simply connected manifold that admits a metric of non-positive curvature. For example take $\mathbb{R}^k\times \mathbb{H}^n$. Take $m+1$ points $x_0$, $x_1, \ldots$ $x_m$, $m>k+1$ and ...