Cyclic Projections in Hadamard Spaces

Abstract

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.

Appendix: Three Disks

While the cyclic product of projections P constructed in Sect. 2 is not asymptotically regular, its square $P^2$ is the identity on $C_1$; in particular, $P^2$ is asymptotically regular. The construction in Sect. 2 produces a Möbius band B divided into three rectangles and a map from B to a Hadamard space that is distance-preserving on each rectangle.

In this appendix, we produce a Hadamard space that contains an embedding of a twisted solid torus with arbitrary twisting angle, such that the solid torus consists of 3 isometrically embedded flat cylinders. In this case, we obtain again 3 projections onto convex sets, each of them isometric to a Euclidean disk, the bases of the cylinders. Then the cyclic product of these projections is the rotation of a disk by the prescribed twisting angle $\alpha$. In particular, if $\tfrac{\alpha }{\pi }$ is irrational, then any power of this cyclic product of projections may not be asymptotically regular.

Theorem A.1

There is a cyclic projection P as in Theorem 1.1 such that any of its powers $P^m$ is not asymptotically regular.

Proof of A.1

Fix an angle $\alpha$ and a small $\epsilon >0$. Consider the closed $\epsilon$-neighborhood A of a closed geodesic $\gamma$ in the unit sphere ${\mathbb {S}}^3$. Note that the boundary of A is a saddle surface in ${\mathbb {S}}^3$; hence, it has curvature bounded from above by 1. Thus, A is a compact Riemannian manifold with boundary, such that the curvature of the interior and of the boundary is bounded from above by 1. Therefore, by the result of Stephanie Alexander, David Berg and Richard Bishop [1], A equipped with the induced intrinsic metric is locally CAT(1). The universal cover ${\tilde{A}}$ of A with its induced metric is locally CAT(1) as well. Since ${\tilde{A}}$ does not contain closed geodesics, it is CAT(1), by the generalized Hadamard–Cartan theorem [3, 8.13.3], [10, 6.8+6.9], [13].

Denote by E the inverse image of $\gamma$ in ${\tilde{A}}$. The isometry group of ${\tilde{A}}$ contains the group of translations along E and the rotations that fix E. Let T be the composition of translation along E of length $2\cdot \pi +10\cdot \epsilon$ and the rotation by angle $\alpha$. The element T generates a discrete subgroup $\Gamma$ in the group of isometries of ${\tilde{A}}$ that acts freely and discretely on $\tilde{A}$.

Set $Y ={\tilde{A}}/\Gamma$. Since $\varepsilon$ is small, any non-trivial element of $\Gamma$ moves every point of ${\tilde{A}}$ by more than $2\cdot \pi$. Therefore, Y is a compact locally CAT(1) space that does not contain closed geodesics of length less than $2\cdot \pi$. Hence, by the generalized Hadamard–Cartan theorem [3], Y is CAT(1). By construction, Y is locally isometric to ${\mathbb {S}}^3$ outside its boundary B. The projection of E to Y is a closed geodesic G of length $2\cdot \pi +10\cdot \epsilon$.

Denote by X the Euclidean cone over Y; since Y is CAT(1), we get that X is CAT(0); see [3]. Moreover, X is locally Euclidean outside its boundary — the cone over B.

The cone Z over the closed geodesic G is the Euclidean cone over a circle of length $2\cdot \pi +10\cdot \epsilon$. By construction, Z is a locally convex subset of X. Hence, Z is a convex subset of X [2, 2.2.12]. Let us consider a geodesic triangle $[q_1q_2q_3]$ in Z that surrounds the origin o of the cone Z.

By construction, the sides of the triangle $[q_1q_2q_3]$ lie in the flat part of X. Thus, we can find a small $r>0$ such that the $2\cdot r$-neighborhood $U_1$ of the geodesic $[q_1q_2]$ is isometric to a convex subset of the Euclidean space. We can assume that $2\cdot r$-neighborhoods $U_2$ of $[q_2q_3]$ and $U_3$ of $[q_3q_1]$ have the same property.

Denote by $C_i$ the disk of radius r centered at $q_i$ and being orthogonal to Z. By construction, $C_i$ and $C_{i+1}$, for $i=1,2,3\pmod 3$ are contained in $U_i$. Since Z is convex, $C_i$ and $C_{i+1}$ are parallel inside $U_i$, thus their convex hull $Q_i$ is isometric to the cylinder $C_i \times [q_i,q_{i+1}]$ with bottom and top $C_i$ and $C_{i+1}$. In particular, the projection $P_i$ defines an isometry $C_{i+1}\rightarrow C_{i}$.

By construction, the composition $P=P_1\circ P_2\circ P_3:C_1\rightarrow C_1$ rotates $C_1$ by angle $\alpha$. If $\tfrac{\alpha }{\pi }$ is irrational, then P, as well as all its powers, are not asymptotically regular.

As before, setting $C_3=\cdots =C_k$ defines examples for any $k\ge 3$. $\square$