11
votes
Accepted
Is the completion of a CAT(0) open ball a closed ball?
The answer is "no".
Let $\Sigma$ be the suspension over Poincaré homology sphere.
It admits a polyhedral $\mathrm{CAT}[1]$-metric.
Let $B$ be the unit ball in the Euclidean cone $\mathrm{Cone}\,\...
- 45k
9
votes
Amalgamated product acting on CAT(0) cube complex
To extend the gluing result from Bridson--Haefliger to non-positively curved cube complexes, it is important to work in the correct category.
If we want the result to also be a non-positively curved ...
- 25k
8
votes
Sectional curvature of the manifold of symmetric positive definite matrices
Dolcetti and Pertici give an explicit expression for the Riemannian curvature tensor on the symmetric space $\mathcal{P}_n$ of $n \times n$ positive definite matrices with affine-invariant metric. ...
- 637
6
votes
Accepted
What are the extremal CAT(0) metrics?
Let me describe a 6-point counterexample.
Let $K$ be a 2-dimensional cone with total angle $\theta=2{\cdot}\pi+\varepsilon$, where $\varepsilon$ is small and positive (any $0<\varepsilon<\tfrac\...
- 45k
4
votes
Accepted
Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?
The answer is "no" even for 3-dimensional Hadamard manifolds.
Moreover, implication
$$\measuredangle[a\,^b_c]<\tfrac\pi2 \ \&\ \measuredangle[a\,^b_d]<\tfrac\pi2\ \Rightarrow\ \...
- 45k
4
votes
Accepted
Upper bound on volume growth of area minimizers
Consider the complex curve $w = z^k$ in $\mathbb{C}^2\simeq\mathbb{R}^4$, which is calibrated and therefore area-minimizing. The area of the part of this curve that lies inside the polydisk $\max\{|z|...
- 108k
4
votes
Convexity of Isoperimetric Domains
Using the definition of a Cartan Hadamard as a complete, simply connected manifold having non-positive curvature, a region in a Cartan Hadamard manifold realizing the optimal isoperimetric ratio need ...
- 881
3
votes
Amalgamated product acting on CAT(0) cube complex
I assume that isomorphic embedding really meant isometric embedding in the assumptions and thus $Y$ is really a convex subset of $X_i$ for all $i$.
Then the CAT(0)-space on which the amalgamated ...
- 11.1k
3
votes
Tangent cone of metric graph
Loosely speaking, if you are standing at the origin then there are only three directions you can travel. So the space of directions $S_o$ is only 3 points. Taking the product of the space of ...
- 267
3
votes
Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
It meant to be an answer to another question, https://mathoverflow.net/a/406833/, but it is an answer to this question as well.
- 45k
3
votes
Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?
From Lemma 5.2 in Hamenstädt's article Rank-one isometries of proper CAT(0) spaces:
Let $X$ be a proper CAT(0) space and $G \leq \mathrm{Isom}(X)$ a non-elementary group with limit set $\Lambda$ ...
- 8,401
2
votes
Accepted
$L^p$-barycenters via continuous selectors
There exists a minimizer for all $p\in [1,\infty[$. To see that, let $R > 0$ be big enough so that
$$
\int_{B_R(0)}\|u\|^pd\mathbb{P}(u) > \frac{1}{2}\int_{\mathbb{R}^n}\|u\|^pd\mathbb{P}(u).
$$
...
- 415
2
votes
Convex hulls of quasi-convex sets in proper CAT(0) spaces
There are counter-examples to this even among 3-dimensional Hadamard Riemannian manifolds of curvature $\le -1$. They are due to Ancona:
Ancona, Alano, Convexity at infinity and Brownian motion on ...
- 12.3k
Only top scored, non community-wiki answers of a minimum length are eligible