Three hyperbolic metrics

Up to a constant factor, there is just one conformally invariant Riemannian metric {\rho} on the disk {\mathbb{D}=\{z\in {\mathbb C}\colon |z|<1\}}. Indeed, on the tangent space at {0} the metric must be a multiple of the Euclidean one, due to rotational invariance. Normalizing so that at {0} both metrics coincide, we can use the invariance under conformal automorphisms (Möbius transformations)

\displaystyle \psi_a(z) = \frac{z+a}{1+\bar a z},\quad |a|<1

to find that on the tangent space at {a} the metric {\rho} differs from Euclidean only by the factor of {|\psi'_a(a)| (1-|a|^2)^{-1}}. This can be written as {d\rho(z) = (1-|z|^2)^{-1}|dz|}, indicating what we integrate to find the length of curves with respect to {\rho}. This is the Poincaré disk model of the hyperbolic plane. The geodesics of {\rho} are precisely the circles orthogonal to {\partial\mathbb{D}}, and diameters.

Hyperbolic geodesics
Hyperbolic geodesics

We could model the hyperbolic plane on any other proper simply-connected domain {\Omega\subset {\mathbb C}}, just by pulling back {\rho} under the Riemann map {\phi\colon\Omega\rightarrow\mathbb{D}}. Explicitly, {d\rho_{\Omega}(z)=(1-|\phi(z)|^{-2})\,|\phi'(z)|\,|dz|}. But is this really explicit? We have no closed form for {\phi} except for very special domains {\Omega}. The density {\rho_\Omega} can be quite tricky: see this MathOverflow question.

On the disk {\mathbb{D}}, the density of {\rho} is roughly the reciprocal of the distance to the boundary. The Koebe distortion theorem yields the same for all simply connected domains:

\displaystyle \mathrm{dist}\,(z,\partial \Omega)^{-1} \le \frac{|d\rho_\Omega(z)|}{|dz|}\le 4\, \mathrm{dist}\,(z,\partial \Omega)^{-1}

Disk and slit plane realize 1 and 4, respectively.
Disk and slit plane realize 1 and 4, respectively.

This suggests replacing the hyperbolic metric {\rho_\Omega} with the quasihyperbolic metric {d\delta_\Omega(z)=\mathrm{dist}\,(z,\partial \Omega)^{-1}\,|dz|}. The density of {\delta_\Omega} is about as simple as we could wish for, but the shape of its geodesics is not as obvious; indeed, a number of papers were written on this subject in the last 35 years.

Can we have a hyperbolic-type metric with explicit geodesics and explicit density? The Hilbert metric delivers both, at least in bounded convex domains. The idea is simple: instead of dividing the length of each tangent vector {v\in T_a} by {\mathrm{dist}\,(a,\partial \Omega)}, we divide it by {\mathrm{dist}_v\,(a,\partial \Omega)}, which is the distance measured in the direction of {v}. In other words, this is how long you could walk in the direction {v} before hitting the boundary. It is clear that the length of any curve going to the boundary is infinite due to the divergence of {\int_{0^+} \frac{dx}{x}}, same as for hyperbolic and quasihyperbolic metrics. Since it is awkward to have a “metric” that is not symmetric (the lengths of {v} and {-v} are not the same), we symmetrize:

\displaystyle \|v\|_{\Omega,a} =\frac12 \left(\frac{1}{\mathrm{dist}_v\,(a,\partial \Omega)} +    \frac{1}{\mathrm{dist}_{-v}\,(a,\partial \Omega)} \right) |v|_{{\mathbb R}^2}

Directional distances to the boundary
Directional distances to the boundary

Now the distances between points are defined in the usual way, as the infimum of lengths of connecting curves {\gamma \colon [a,b]\rightarrow\Omega},

\displaystyle L(\gamma) = \int_a^b \|\gamma'(t)\|_{\Omega,\gamma(t)}\,dt

This is the Hilbert metric {d_\Omega}. Despite all its non-Euclideanness, the geodesics of {d_\Omega} are line segments. This (nontrivial) fact makes it easy to calculate the distance between two points {a,b\in\Omega}: besides the points themselves, one only needs to consider the pair of points {a',b'} where the line through {a,b} meets {\partial \Omega}. The integral ends up being (half of) the logarithm of the cross-ratio of these four points. Assuming {a',a,b,b'} are situated in the listed order, the relevant cross-ratio is

\displaystyle \frac{|a'-b|\, |a-b'|}{|a'-a|\,|b-b'|}

which is greater than {1}, making the logarithm positive.

I personally find the integral of reciprocals of distances more intuitively accessible than the logarithm of cross-ratio. Either version of the definition makes it clear that the Hilbert metric is invariant under projective transformations, a property not shared by the metrics {\rho} and {\delta}.

It turns out that {d_{\mathbb D}} is also a model of the hyperbolic plane, but with geodesics being line segments rather than circular arcs. Along each diameter {d_{\mathbb D}} coincides with {\rho}, because at Euclidean distance {r} from the center it scales vectors by

\displaystyle    \frac{1}{2}\left(\frac{1}{1-r}+\frac{1}{1+r}\right) = \frac{1}{1-r^2}

Behold the magic of partial fractions!

Quasi-isometries and stability of quasi-geodesics

Continuation of expository series on Gromov hyperbolicity. Recall that a map {f\colon X\rightarrow Y} is a quasi-isometry if there are constants {L,M} such that {L^{-1}|x_1x_2|-M\le |f(x_1)f(x_2)|\le L|x_1x_2|+M} for all {x_1,x_2\in X}. This is a coarse version of the bi-Lipschitz condition. Surprisingly, Gromov hyperbolicity is preserved under quasi-isometries of geodesic spaces. The surprising part is that the multiplicative constant {L} does not kill the additive constant {\delta}.

Theorem. Suppose {X} and {Y} are geodesic metric spaces, and {Y} is Gromov hyperbolic. If there exists a quasi-isometry {f\colon X\rightarrow Y}, then {X} is also Gromov hyperbolic.

Proof goes like this. Assuming that {X} contains a fat geodesic triangle {a,b,c}, we consider the geodesic triangle in {Y} with vertices {f(a),f(b),f(c)}, and want to prove that it is also fat. Since {f} is a quasi-isometry, it follows that the images of geodesics {ab}, {bc} and {ac} form a roughly-triangular shape which has the fatness property: there is a point on one of the sides that is far away from the other two sides. The problem reduces to showing that this roughly-triangular shape lies within a certain distance {R} (independent of {a,b,c}) from the actual geodesic triangle with vertices {f(a),f(b),f(c)}. This is known as stability of quasi-geodesics. A quasi-geodesic is a quasi-isometric image of a line segment, similar to how a geodesic is a (locally) isometric image of a segment.

By the way, quasi-geodesic stability fails in {\mathbb R^2}. We can connect the points {(-n,0)} and {(n,0)} by the quasi-geodesic {y=n-|x|}, which is at distance {n} from the true geodesic between these points.

Quasi-geodesics are not stable in R^2

I’ll prove a more specialized and weaker statement, which however contains the essence of the full result. Namely, let \mathbb H^2 denote the hyperbolic plane and assume that {f\colon [a,b]\rightarrow \mathbb H^2} is bi-Lipschitz: {L^{-1}|x_1-x_2|\le |f(x_1)f(x_2)|\le L|x_1-x_2|} for all {x_1,x_2\in [a,b]}. The claim is that the image of {f} lies in the {R}-neighborhood of the geodesic through {f(a)} and {f(b)}, where {R} depends only on {L}.

There are three standard models of hyperbolic plane: disk, halfplane and infinite strip. I’ll use the last one, because it’s the only model in which a geodesic is represented by Euclidean line. Specifically, {\mathbb H^2} is identified with the infinite strip {\{x+iy\in\mathbb C\colon |y|<\pi/2\}} equipped with the metric {|dz|/\cos y}. (To see where the metric comes from, apply z\mapsto i e^{iz} to map the strip onto upper halfplane and pull back the hyperbolic metric |dw|/\mathrm{Im}\,w.)

The hyperbolic and Euclidean metrics coincide on the real line, which is where we place {f(a)} and {f(b)} with the help of some hyperbolic isometry. Let {\Gamma=f[a,b]} be our quasi-geodesic. Being a bi-Lipschitz image of a line segment, {\Gamma} satisfies the chord-arc condition: the length of any subarc of {\Gamma} does not exceed {L^2} times the distance between its endpoints. Pick {y_0\in (0,\pi/2)} such that {1/\cos y_0=2L^2}. Let {D} be the hyperbolic distance between the lines {y=y_0} and {y=0}. This distance could be calculated as {\int_0^{y_0}\sec y\,dy}, but I’d rather keep this integral as an exquisite Calculus II torture device.

The problem facing us is that quasigeodesic may be L^2 times longer than the distance between its endpoints, which seems to allows it to wander far off the straight path. However, it turns out there is a uniform bound on the length of any subarc {\Gamma'} of {\Gamma} that lies within the substrip {\{y \ge y_0\}}. We lose no generality in assuming that the endpoints of \Gamma' are on the line y=y_0; they will be denoted {x_j+iy_0}, {j=1,2}. The key point is that connecting these two points within \{y\ge y_0\} is rather inefficient, and such inefficiency is controlled by the chord-arc property.

Quasi-geodesic and its inefficient subarc

The hyperbolic distance between {x_j+iy_0} is at most |x_1-x_2|+2D, because we can go from {x_1+iy_0} to {x_1} (distance {D}), then from {x_1} to {x_2} (distance {|x_1-x_2|}), and finally from {x_2} to {x_2+iy_0} (distance {D}). On the other hand, the length of {\Gamma'} is at least {|x_1-x_2|/\cos y_0 = 2L^2|x_1-x_2|} because the density of hyperbolic metric is at least 1/\cos y_0 where \Gamma' lives. The chord-arc property yields {2L^2 |x_1-x_2| \le L^2 (|x_1-x_2|+2D)}, which simplifies to {|x_1-x_2| \le 2D}. Hence, the distance between the endpoints of {\Gamma'} is at most {4D}, and another application of the chord-arc property bounds the length of {\Gamma'} by {4DL^2}.

In conclusion, the claimed stability result holds with {R= D+2DL^2}.

Complete proofs can be found in many books, for example Metric Spaces of Non-Positive Curvature by Bridson and Haefliger or Elements of Asymptotic Geometry by Buyalo and Schroeder. I used Schroeder’s lecture notes An introduction to asymptotic geometry.