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}\,\...
Anton Petrunin's user avatar
8 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 ...
HJRW's user avatar
  • 23.8k
7 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. ...
ccriscitiello's user avatar
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\...
Anton Petrunin's user avatar
6 votes
Accepted

The midpoint geodesic

The answer is no. Define the manifold Let $f$ be a smooth function on $\mathbb{R}$ satisfying $f(r) = |r|$ for $|r| > 3$ $f(r) > 0$ $f''(r) \geq 0$. The corresponding warped product metric ...
Willie Wong's user avatar
  • 36.5k
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|...
Robert Bryant's user avatar
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 ...
Joel Hass's user avatar
  • 871
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 ...
Logan Fox's user avatar
  • 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.
Anton Petrunin's user avatar
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 ...
HenrikRüping's user avatar
3 votes

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\ \...
Anton Petrunin's user avatar
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 $\Lambda$ ...
AGenevois's user avatar
  • 7,471
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 $$ \int_{B_R(0)}\|u\|^pd\mathbb{P}(u) > \frac{1}{2}\int_{\mathbb{R}^n}\|u\|^pd\mathbb{P}(u). $$ ...
hthi's user avatar
  • 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 ...
Moishe Kohan's user avatar
  • 9,624

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