Large-Scale Nonpositive Curvature

• Large-scale nonpositive curvature is a framework defining metric spaces and manifolds with curvature constraints, blending hyperbolic and flat geometry behaviors.
• It employs synthetic methods like consistent geodesic bicombings to extend classical results such as the Flat Torus Theorem and Gromov’s hyperbolicity criterion.
• Applications include analyzing volume growth, entropy in Riemannian manifolds, and obstructing coarse embeddings through sharp convexity and martingale methods.

Large-scale nonpositive curvature concerns global geometric and analytic properties arising in metric spaces and manifolds whose local curvature is constrained to be nonpositive—typically in the sense of Alexandrov, Busemann, or via synthetic convexity conditions. At large scale, these spaces exhibit a dichotomy between hyperbolicity (thin triangle behavior, no flat subspaces) and the existence of isometrically embedded normed (flat) spaces; these phenomena pervade the study of geometric group actions, ergodic theory, volume growth, and embedding theory. Recent advances have unified and extended classical results—such as the Flat Torus Theorem and Gromov's hyperbolicity criterion—beyond spaces with unique geodesics, establishing robust frameworks rooted in geodesic bicombings and sharp convexity inequalities, and posing significant constraints on metric embedding universality.

1. Synthetic Nonpositive Curvature: Foundational Notions

A geodesic metric space $(X,d_X)$ is said to have nonpositive curvature in the sense of Alexandrov (a $\mathrm{CAT}(0)$ space) if for every quadruple $x,y,z,w\in X$ with $w$ the midpoint of $x,y$, the following four-point condition is satisfied: $$d_X(z,w)^2 + \frac{1}{4}d_X(x,y)^2 \leq \frac{1}{2}d_X(z,x)^2 + \frac{1}{2}d_X(z,y)^2.$$ This characterizes spaces where triangles are at least as thin as in Euclidean geometry and allows a synthetic (metric rather than differential) definition applicable even in the absence of a manifold structure (Eskenazis et al., 2018).

Convexity can be further abstracted via geodesic bicombings. A consistent geodesic bicombing on $(X,d)$ is a map

$$\sigma:X\times X\times[0,1]\rightarrow X, \quad \sigma_{xy}(t),$$

subject to geodesicity-reversibility, conicality, and a consistency property. Such structures permit the analysis of large-scale curvature-type phenomena without enforcing unique geodesics or even full Busemann convexity (Descombes et al., 2015).

2. Convex Geodesic Bicombings and Large-Scale Dichotomies

The Descombes–Lang framework extends core nonpositive curvature results to spaces admitting a consistent geodesic bicombing (Descombes et al., 2015). In this setting, the following are central:

• Flat-Plane Criterion (Gromov's Hyperbolicity Test): For a proper metric space with a consistent bicombing and cocompact isometry group, the space is (Gromov-)hyperbolic if and only if it contains no isometrically embedded normed plane. Formally,

$\text{No embedded normed plane} \iff \text{space is %%%%6%%%%-hyperbolic.}$

The proof employs a slim-triangle lemma adapted to $\sigma$-geodesics and constructs arbitrarily fat triangles to produce ruled half-planes; gluing such half-planes yields full normed planes.

• Flat-Torus Theorem: Under proper, cocompact group actions, the existence of a free abelian subgroup $A\cong \mathbb{Z}^n$ acting by isometries forces the presence of an isometrically embedded $n$-dimensional normed space invariant under $A$ [Theorem 1.2 in (Descombes et al., 2015)].
• Recovery of Large-Scale Geometry: The axioms of conicality and consistency suffice to derive Flat-Strip and Half-Plane Theorems, and to recover the classical "no flat $\iff$ hyperbolic" dichotomy at the coarse scale.

This synthetic approach broadens classical Busemann- and CAT(0)-based rigidity theories; injective metric spaces, $\ell^1$- and $\ell^\infty$-normed spaces, and certain Finsler symmetric spaces all admit natural consistent bicombings, thus realizing large-scale NPC phenomena without requiring strict convexity or unique geodesics.

3. Volume Asymptotics and Entropy in Riemannian NPC Manifolds

For closed, rank-one, $C^\infty$ Riemannian manifolds $(M,g)$ with nonpositive sectional curvature and universal cover $X$, a central analytic feature is the asymptotic behavior of ball volumes in $X$: $$\lim_{t \to \infty} \frac{b_t(x)}{e^{h t}/h} = c(x),$$ where $b_t(x)$ is the volume of the ball of radius $t$ about $x\in X$, $h$ is the topological entropy of the geodesic flow, and $c:X\to(0,\infty)$ is a positive, continuous, $\Gamma$-invariant function (the Margulis function) (Wu, 2021).

The function $c(x)$ possesses $C^1$ regularity, arising from an integral with respect to a Patterson–Sullivan measure on the ideal boundary: $$c(x) = \int_{\partial X} e^{-h\beta_\xi(x,p)}\, d\mu_p(\xi),$$ where $\beta_\xi(x,p)$ is the Busemann function. Rigidity phenomena are observed:

• On surfaces without flat strips, $c(x)$ is constant if and only if the curvature is constant negative;
• Local symmetry is characterized by flip invariance of the boundary measure under reversal of geodesics.

These results interlace ergodic-theoretic mixing, geometric group actions, and Busemann boundary theory, highlighting the fine structure of large-scale entropy, rigidity, and geometric invariants (Wu, 2021).

4. Metric Embeddings and Coarse Universality

A salient question in large-scale geometry is the existence of coarse (large-scale) embeddings into model spaces of NPC. A map $f:Y\to X$ between metric spaces is a coarse embedding if there are non-decreasing control moduli $\omega, S$ with $\omega(t)\to\infty$ as $t\to\infty$, such that for all $y,y'\in Y$,

$$\omega(d_Y(y,y')) \leq d_X(f(y),f(y')) \leq S(d_Y(y,y')).$$

While Alexandrov spaces of nonnegative curvature are coarsely universal for metric spaces, Alexandrov spaces of nonpositive curvature (i.e., $\mathrm{CAT}(0)$ spaces) are not [Theorem 2.1, (Eskenazis et al., 2018)]. Specifically, $\ell_p$ for $p>2$ cannot coarsely embed into any $\mathrm{CAT}(0)$ space.

Underlying this is the sharp metric cotype property, derived via nonlinear (metric space-valued) martingale inequalities: any $q$-barycentric space (including $\mathrm{CAT}(0)$, which are $2$-barycentric) enjoys sharp metric cotype $q$; this places strong obstructions on coarse embedding into NPC spaces (Eskenazis et al., 2018).

5. Barycentric Convexity and Martingale Methods

A metric space $(X,d_X)$ is $q$-barycentric with constant $\beta$ if for every finitely supported probability measure $\mu$ on $X$ and every $x\in X$,

$$d_X(\mathfrak{B}(\mu),x)^q + \frac{1}{\beta^q} \int_X d_X(\mathfrak{B}(\mu),y)^q d\mu(y) \leq \int_X d_X(x, y)^q d\mu(y),$$

where $\mathfrak{B}$ is a barycenter map. This property is a nonlinear analog of uniform convexity. The barycentric framework enables powerful martingale and cotype inequalities, yielding exact distortion estimates for bi-Lipschitz embeddings of grids into $\ell_q$, and imposing rigidity on possible embeddings into NPC targets (Eskenazis et al., 2018).

For instance, for $q>2$ and the $n$-dimensional $\ell_\infty$ grid $[m]^n_\infty$, the distortion of embedding into $\ell_q$ is

$$c_{\ell_q}([m]^n_\infty) \asymp \max\{1,\, m^{1-2/q}\, n^{1/q}\},$$

with no intermediate distortion possible between the trivial identity and forgetful embeddings. This provides a sharp quantitative obstruction to coarse universality.

6. Implications, Generalizations, and Open Directions

The large-scale framework based on convex bicombings generalizes classical results predicated on uniqueness and strong convexity. The unification encompasses CAT(0), Busemann, Banach, and injective spaces under the umbrella of convexity along bicombing geodesics, unifying diverse metric theories (Descombes et al., 2015).

A key implication is that large-scale nonpositive curvature is not, in general, a “benign” property: the existence of flats, growth rates, and the fine structure of measure-theoretic invariants (such as the Margulis function) drive significant dichotomies and obstructions in geometry and group theory. Open problems include the existence of bounded-geometry metric spaces or finitely generated groups that cannot be coarsely embedded into any $\mathrm{CAT}(0)$ space—a direction tied to the theory of expanders and rigidity (Eskenazis et al., 2018).

In summary, large-scale nonpositive curvature furnishes both a robust geometric framework for understanding global phenomena—such as hyperbolicity, rigidity, and periodic flat subspaces—and sharp analytic and combinatorial tools that delimit the possible embeddings and universality properties of these spaces.