Busemann function

In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".

Definition and elementary properties

Let $(X,d)$ be a metric space. A geodesic ray is a path $\gamma :[0,\mathrm{\infty })\to X$ which minimizes distance everywhere along its length. i.e., for all $t,t^{\prime }\in [0,\mathrm{\infty })$, $$d(\gamma (t),\gamma (t^{\prime }))=|t-t^{\prime }|.$$ Equivalently, a ray is an isometry from the "canonical ray" (the set $[0,\mathrm{\infty })$ equipped with the Euclidean metric) into the metric space X.

Given a ray γ, the Busemann function $B_{\gamma }:X\to ℝ$ is defined by

$$B_{\gamma }(x)=\underset{t\to \mathrm{\infty }}{\lim }\left(d\right(\gamma (t),x)-t)$$

Thus, when t is very large, the distance $d(\gamma (t),x)$ is approximately equal to $B_{\gamma }(x)+t$. Given a ray γ, its Busemann function is always well-defined: indeed the right hand side above $F_t(x)\stackrel{\text{def}}{=}d(\gamma (t),x)-t$, tends pointwise to the left hand side on compacta, since $t-d(\gamma (t),x)=d(\gamma (t),\gamma (0))-d(\gamma (t),x)$ is bounded above by $d(\gamma (0),x)$ and non-increasing since, if $s\le t$,

$$t-s+d(x,\gamma (s))-d(x,\gamma (t))\ge t-s-d(\gamma (s),\gamma (t))=0.$$

It is immediate from the triangle inequality that

$$|B_{\gamma }(x)-B_{\gamma }(y)|\le d(x,y),$$

so that $B_{\gamma }$ is uniformly continuous. More specifically, the above estimate above shows that

• Busemann functions are Lipschitz functions with constant 1.

By Dini's theorem, the functions $F_t(x)=d(x,\gamma (t))-t$ tend to $B_{\gamma }(x)$ uniformly on compact sets as t tends to infinity.

Example: Poincaré disk

Let $D$ be the unit disk in the complex plane with the Poincaré metric

$$d{s}^2=\frac{4|dz{|}^2}{(1-|z{|}^2{)}^2}.$$

Then, for $|z|<1$ and $|\zeta |=1$, the Busemann function is given by

$$B_{\zeta }(z)=-\log \left(\frac{1-|z{|}^2}{|z-\zeta {|}^2}\right),$$

where the term in brackets on the right hand side is the Poisson kernel for the unit disk and $\zeta$ corresponds to the radial geodesic $\gamma$ from the origin towards $\zeta$, $\gamma (t)=\zeta \tanh (t/2)$. The computation of $d(x,y)$ can be reduced to that of $d(z,0)=d(|z|,0)=2\mathrm{artanh}(|z|)=\log \left(\frac{1+|z|}{1-|z|}\right)$, since the metric is invariant under Möbius transformations in $SU(1,1)$; the geodesics through $0$ have the form $\zeta g_t(0)$ where $g_t$ is the 1-parameter subgroup of $SU(1,1)$,

$$g_t=\left(\begin{array}{cc}\cosh (t/2)& \sinh (t/2)\\ \sinh (t/2)& \cosh (t/2)\end{array}\right)$$

The formula above also completely determines the Busemann function by Möbius invariance.

Busemann functions on a Hadamard space

In a Hadamard space, where any two points are joined by a unique geodesic segment, the function $F=F_t$ is convex, i.e. convex on geodesic segments $[x,y]$. Explicitly this means that if $z(s)$ is the point which divides $[x,y]$ in the ratio Template:Math, then $F(z(s))\le sF(x)+(1-s)F(y)$. For fixed $a$ the function $d(x,a)$ is convex and hence so are its translates; in particular, if $\gamma$ is a geodesic ray in $X$, then $F_t$ is convex. Since the Busemann function $B_{\gamma }$ is the pointwise limit of $F_t$,

• Busemann functions are convex on Hadamard spaces.
• On a Hadamard space, the functions $F_t(y)=d(y,\gamma (t))-t$ converge uniformly to $B_{\gamma }$ uniformly on any bounded subset of $X$.

Let Template:Math. Since $\gamma (t)$ is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function Template:Math is convex. Hence for Template:Math

$$g(s)\le (1-\frac{s}{t})g(0)+\frac{s}{t}g(t).$$

Thus

$$2sh(s)\le (h(s)+s{)}^2-s^2=g(s)\le (1-\frac{s}{t})d(x,y{)}^2+\frac{s}{t}(2th(t)+h(t{)}^2)\le d(x,y{)}^2+2sh(t)+\frac{s}{t}d(x,y{)}^2,$$

so that

$$|F_s(y)-F_t(y)|=|h(s)-h(t)|\le \frac{1}{2}(s^{-1}+t^{-1})d(x,y{)}^2.$$

Letting t tend to ∞, it follows that

$$|F_s(y)-B_{\gamma }(y)|\le \frac{d(x,y{)}^2}{2s},$$

so convergence is uniform on bounded sets.

Note that the inequality above for $|F_s(y)-F_t(y)|$ (together with its proof) also holds for geodesic segments: if Template:Math is a geodesic segment starting at Template:Math and parametrised by arclength then

$$|d(y,\mathrm{\Gamma }(s))-s-d(y,\mathrm{\Gamma }(t))+t|\le (s^{-1}+t^{-1})d(x,y{)}^2.$$

Next suppose that Template:Math are points in a Hadamard space, and let Template:Math be the geodesic through Template:Math with Template:Math and Template:Math, where Template:Math. This geodesic cuts the boundary of the closed ball Template:Math at the point Template:Math. Thus if Template:Math, there is a point Template:Math with Template:Math such that Template:Math.

This condition persists for Busemann functions. The statement and proof of the property for Busemann functions relies on a fundamental theorem on closed convex subsets of a Hadamard space, which generalises orthogonal projection in a Hilbert space: if Template:Math is a closed convex set in a Hadamard space Template:Math, then every point Template:Math in Template:Math has a unique closest point Template:Math in Template:Math and Template:Math; moreover Template:Math is uniquely determined by the property that, for Template:Math in Template:Math,

$$d(x,y{)}^2\ge d(x,a{)}^2+d(a,y{)}^2,$$

so that the angle at Template:Math in the Euclidean comparison triangle for Template:Math is greater than or equal to Template:Math.

• If Template:Math is a Busemann function on a Hadamard space, then, given Template:Math in Template:Math and Template:Math, there is a unique point Template:Math with Template:Math such that Template:Math. For fixed Template:Math, the point Template:Math is the closest point of Template:Math to the closed convex Template:Math set of points Template:Math such that Template:Math and therefore depends continuously on Template:Math.

Let Template:Math be the closest point to Template:Math in Template:Math. Then Template:Math and so Template:Math is minimised by Template:Math in Template:Math where Template:Math is the unique point where Template:Math is minimised. By the Lipschitz condition Template:Math. To prove the assertion, it suffices to show that Template:Math, i.e. Template:Math. On the other hand, Template:Math is the uniform limit on any closed ball of functions Template:Math. On Template:Math, these are minimised by points Template:Math with Template:Math. Hence the infimum of Template:Math on Template:Math is Template:Math and Template:Math tends to Template:Math. Thus Template:Math with Template:Math and Template:Math tending towards Template:Math. Let Template:Math be the closest point to Template:Math with Template:Math. Let Template:Math. Then Template:Math, and, by the Lipschitz condition on Template:Math, Template:Math. In particular Template:Math tends to Template:Math. Passing to a subsequence if necessary it can be assumed that Template:Math and Template:Math are both increasing (to Template:Math). The inequality for convex optimisation implies that for Template:Math.

$$d(u_n,u_m{)}^2\le R_n^2-R_m^2\le 2r|R_n-R_m|,$$

so that Template:Math is a Cauchy sequence. If Template:Math is its limit, then Template:Math and Template:Math. By uniqueness it follows that Template:Math and hence Template:Math, as required.

Uniform limits. The above argument proves more generally that if Template:Math tends to infinity and the functions Template:Math tend uniformly on bounded sets to Template:Math, then Template:Math is convex, Lipschitz with Lipschitz constant 1 and, given Template:Math in Template:Math and Template:Math, there is a unique point Template:Math with Template:Math such that Template:Math. If on the other hand the sequence Template:Math is bounded, then the terms all lie in some closed ball and uniform convergence there implies that Template:Math is a Cauchy sequence so converges to some Template:Math in Template:Math. So Template:Math tends uniformly to Template:Math, a function of the same form. The same argument also shows that the class of functions which satisfy the same three conditions (being convex, Lipschitz and having minima on closed balls) is closed under taking uniform limits on bounded sets.

Comment. Note that, since any closed convex subset of a Hadamard subset of a Hadamard space is also a Hadamard space, any closed ball in a Hadamard space is a Hadamard space. In particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of Template:Math or even a semi-infinite interval Template:Math. The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space.

• If Template:Math is a convex function, Lipschitz with constant 1 and Template:Math assumes its minimum on any closed ball centred on Template:Math with radius Template:Math at a unique point Template:Math on the boundary with Template:Math, then for each Template:Math in Template:Math there is a unique geodesic ray Template:Math such that Template:Math and Template:Math cuts each closed convex set Template:Math with Template:Math at Template:Math, so that Template:Math. In particular this holds for each Busemann function.

The third condition implies that Template:Math is the closest point to Template:Math in the closed convex set Template:Math of points $u$ such that Template:Math. Let Template:Math for Template:Math be the geodesic joining Template:Math to Template:Math. Then Template:Math is a convex Lipschitz function on Template:Math with Lipschitz constant 1 satisfying Template:Math and Template:Math and Template:Math. So Template:Math vanishes everywhere, since if Template:Math and Template:Math. Hence Template:Math. By uniqueness it follows that Template:Math is the closest point to Template:Math in Template:Math and that it is the unique point minimising Template:Math in Template:Math. Uniqueness implies that these geodesics segments coincide for arbitrary Template:Math and therefore that Template:Math extends to a geodesic ray with the stated property.

• If Template:Math, then the geodesic ray Template:Math starting at Template:Math satisfies $\sup d(\gamma (t),\delta (t))<\mathrm{\infty }$. If Template:Math is another ray starting at Template:Math with $\sup d(\delta (t),{\delta }_1(t))<\mathrm{\infty }$ then Template:Math.

To prove the first assertion, it is enough to check this for Template:Math sufficiently large. In that case Template:Math and Template:Math are the projections of Template:Math and Template:Math onto the closed convex set Template:Math. Therefore, Template:Math. Hence Template:Math. The second assertion follows because Template:Math is convex and bounded on Template:Math, so, if it vanishes at Template:Math, must vanish everywhere.

• Suppose that Template:Math is a continuous convex function and for each Template:Math in Template:Math there is a unique geodesic ray Template:Math such that Template:Math and Template:Math cuts each closed convex set Template:Math at Template:Math, so that Template:Math; then Template:Math is a Busemann function. Template:Math is a constant function.

Let Template:Math be the closed convex set of points Template:Math with Template:Math. Since Template:Math is a Hadamard space for every point Template:Math in Template:Math there is a unique closest point Template:Math to Template:Math in Template:Math. It depends continuously on Template:Math and if Template:Math lies outside Template:Math, then Template:Math lies on the hypersurface Template:Math—the boundary ∂Template:Math of Template:Math—and Template:Math satisfies the inequality of convex optimisation. Let Template:Math be the geodesic ray starting at Template:Math.

Fix Template:Math in Template:Math. Let Template:Math be the geodesic ray starting at Template:Math. Let Template:Math, the Busemann function for Template:Math with base point Template:Math. In particular Template:Math. It suffices to show that Template:Math. Now take Template:Math with Template:Math and let Template:Math be the geodesic ray starting at Template:Math corresponding to Template:Math. Then

$$d(x,y)\ge d(\gamma (t),\delta (t)),d(x,\delta (t){)}^2\ge d(x,\gamma (t){)}^2+d(\gamma (t),\delta (t){)}^2,d(y,\gamma (t){)}^2\ge d(y,\delta (t){)}^2+d(\gamma (t),\delta (t){)}^2.$$

On the other hand, for any four points Template:Math, Template:Math, Template:Math, Template:Math in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds:

$$|d(a,c{)}^2+d(b,d{)}^2-d(a,d{)}^2-d(b,d{)}^2|\le 2d(a,b)d(c,d).$$

Setting Template:Math, Template:Math, Template:Math, Template:Math, it follows that

$$|d(y,\gamma (t){)}^2-d(x,\delta (t){)}^2|\le 2d(x,y{)}^2,$$

so that

$$|d(y,\gamma (t))-d(x,\gamma (t))|\le 2\frac{d(x,y{)}^2}{d(y,\gamma (t))+d(x,\gamma (t))}\le \frac{d(x,y{)}^2}{t}.$$

Hence Template:Math. Similarly Template:Math. Hence Template:Math on the level surface of Template:Math containing Template:Math. Now for Template:Math and Template:Math in Template:Math, let Template:Math the geodesic ray starting at Template:Math. Then Template:Math and Template:Math. Moreover, by boundedness, Template:Math. The flow Template:Math can be used to transport this result to all the level surfaces of Template:Math. For general Template:Math, if Template:Math, take Template:Math such that Template:Math and set Template:Math. Then Template:Math, where Template:Math. But then Template:Math, so that Template:Math. Hence Template:Math, as required. Similarly if Template:Math, take Template:Math such that Template:Math. Let Template:Math. Then Template:Math, so Template:Math. Hence Template:Math, as required.

Finally there are necessary and sufficient conditions for two geodesics to define the same Busemann function up to constant:

• On a Hadamard space, the Busemann functions of two geodesic rays ${\gamma }_1$ and ${\gamma }_2$ differ by a constant if and only if $\underset{t\ge 0}{\sup }d({\gamma }_1(t),{\gamma }_2(t))<\mathrm{\infty }$.

Suppose firstly that Template:Math and Template:Math are two geodesic rays with Busemann functions differing by a constant. Shifting the argument of one of the geodesics by a constant, it may be assumed that Template:Math, say. Let Template:Math be the closed convex set on which Template:Math. Then Template:Math and similarly Template:Math. Then for Template:Math, the points Template:Math and Template:Math have closest points Template:Math and Template:Math in Template:Math, so that Template:Math. Hence Template:Math.

Now suppose that Template:Math. Let Template:Math be the geodesic ray starting at Template:Math associated with Template:Math. Then Template:Math. Hence Template:Math. Since Template:Math and Template:Math both start at Template:Math, it follows that Template:Math. By the previous result Template:Math and Template:Math differ by a constant; so Template:Math and Template:Math differ by a constant.

To summarise, the above results give the following characterisation of Busemann functions on a Hadamard space:

THEOREM. On a Hadamard space, the following conditions on a function Template:Math are equivalent:

• Template:Math is a Busemann function.
• Template:Math is a convex function, Lipschitz with constant Template:Math and Template:Math assumes its minimum on any closed ball centred on Template:Math with radius Template:Math at a unique point Template:Math on the boundary with Template:Math.
• Template:Math is a continuous convex function and for each Template:Math in Template:Math there is a unique geodesic ray Template:Math such that Template:Math and, for any Template:Math, the ray Template:Math cuts each closed convex set Template:Math at Template:Math.

Bordification of a Hadamard space

In the previous section it was shown that if Template:Math is a Hadamard space and Template:Math is a fixed point in Template:Math then the union of the space of Busemann functions vanishing at Template:Math and the space of functions Template:Math is closed under taking uniform limits on bounded sets. This result can be formalised in the notion of bordification of Template:Math. In this topology, the points Template:Math tend to a geodesic ray Template:Math starting at Template:Math if and only if Template:Math tends to Template:Math and for Template:Math arbitrarily large the sequence obtained by taking the point on each segment Template:Math at a distance Template:Math from Template:Math tends to Template:Math.

If Template:Math is a metric space, Gromov's bordification can be defined as follows. Fix a point Template:Math in Template:Math and let Template:Math. Let Template:Math be the space of Lipschitz continuous functions on Template:Math, i.e. those for which Template:Math for some constant Template:Math. The space Template:Math can be topologised by the seminorms Template:Math, the topology of uniform convergence on bounded sets. The seminorms are finite by the Lipschitz conditions. This is the topology induced by the natural map of Template:Math into the direct product of the Banach spaces Template:Math of continuous bounded functions on Template:Math. It is give by the metric Template:Math.

The space Template:Math is embedded into Template:Math by sending Template:Math to the function Template:Math. Let Template:Math be the closure of Template:Math in Template:Math. Then Template:Math is metrisable, since Template:Math is, and contains Template:Math as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic. Let Template:Math. Then Template:Math lies in Template:Math. It is non-zero on Template:Math and vanishes only at Template:Math. Hence it extends to a continuous function on Template:Math with zero set Template:Math. It follows that Template:Math is closed in Template:Math, as required. To check that Template:Math is independent of the basepoint, it suffices to show that Template:Math extends to a continuous function on Template:Math. But Template:Math, so, for Template:Math in Template:Math, Template:Math. Hence the correspondence between the compactifications for Template:Math and Template:Math is given by sending Template:Math in Template:Math to Template:Math in Template:Math.

When Template:Math is a Hadamard space, Gromov's ideal boundary Template:Math can be realised explicitly as "asymptotic limits" of geodesic rays using Busemann functions. If Template:Math is an unbounded sequence in Template:Math with Template:Math tending to Template:Math in Template:Math, then Template:Math vanishes at Template:Math, is convex, Lipschitz with Lipschitz constant Template:Math and has minimum Template:Math on any closed ball Template:Math. Hence Template:Math is a Busemann function Template:Math corresponding to a unique geodesic ray Template:Math starting at Template:Math.

On the other hand, Template:Math tends to Template:Math uniformly on bounded sets if and only if Template:Math tends to Template:Math and for Template:Math arbitrarily large the sequence obtained by taking the point on each segment Template:Math at a distance Template:Math from Template:Math tends to Template:Math. For Template:Math, let Template:Math be the point in Template:Math with Template:Math. Suppose first that Template:Math tends to Template:Math uniformly on Template:Math. Then for Template:Math, Template:Math. This is a convex function. It vanishes as Template:Math and hence is increasing. So it is maximised at Template:Math. So for each Template:Math, Template:Math tends towards 0. Let Template:Math, Template:Math and Template:Math. Then Template:Math is close to Template:Math with Template:Math large. Hence in the Euclidean comparison triangle Template:Math is close to Template:Math with Template:Math large. So the angle at Template:Math is small. So the point Template:Math on Template:Math at the same distance as Template:Math lies close to Template:Math. Hence, by the first comparison theorem for geodesic triangles, Template:Math is small. Conversely suppose that for fixed Template:Math and Template:Math sufficiently large Template:Math tends to 0. Then from the above Template:Math satisfies

$$|F_s(y)-B_{\gamma }(y)|\le \frac{d(x_0,y{)}^2}{2s},$$

so it suffices show that on any bounded set Template:Math is uniformly close to Template:Math for Template:Math sufficiently large.

For a fixed ball Template:Math, fix Template:Math so that Template:Math. The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since

$$|d(y,x_n)-d(y,x_0)-d(y,x_n(s))+s|\le \frac{d(x_0,y{)}^2}{s}\le \epsilon .$$

Hence, if Template:Math in Template:Math and Template:Math is sufficiently large that Template:Math, then

$$|h_n(y)-B_{\gamma }(y)|=|d(y,x_n)-d(y,x_0)-B_{\gamma }(y)|\le |d(y,x_n)-d(y,x_0)-d(y,x_n(s))+s|+d(x_n(s),\gamma (s))+|F_s(y)-B_{\gamma }(y)|\le 3\epsilon .$$

Busemann functions on a Hadamard manifold

Suppose that Template:Math are points in a Hadamard manifold and let Template:Math be the geodesic through Template:Math with Template:Math. This geodesic cuts the boundary of the closed ball Template:Math at the two points Template:Math. Thus if Template:Math, there are points Template:Math with Template:Math such that Template:Math. By continuity this condition persists for Busemann functions:

• If Template:Math is a Busemann function on a Hadamard manifold, then, given Template:Math in Template:Math and Template:Math, there are unique points Template:Math, Template:Math with Template:Math such that Template:Math and Template:Math. For fixed Template:Math, the points Template:Math and Template:Math depend continuously on Template:Math.

Taking a sequence Template:Math tending to Template:Math and Template:Math, there are points Template:Math and Template:Math which satisfy these conditions for Template:Math for Template:Math sufficiently large. Passing to a subsequence if necessary, it can be assumed that Template:Math and Template:Math tend to Template:Math and Template:Math. By continuity these points satisfy the conditions for Template:Math. To prove uniqueness, note that by compactness Template:Math assumes its maximum and minimum on Template:Math. The Lipschitz condition shows that the values of Template:Math there differ by at most Template:Math. Hence Template:Math is minimized at Template:Math and maximized at Template:Math. On the other hand, Template:Math and for Template:Math and Template:Math the points Template:Math and Template:Math are the unique points in Template:Math maximizing this distance. The Lipschitz condition on Template:Math then immediately implies Template:Math and Template:Math must be the unique points in Template:Math maximizing and minimizing Template:Math. Now suppose that Template:Math tends to Template:Math. Then the corresponding points Template:Math and Template:Math lie in a closed ball so admit convergent subsequences. But by uniqueness of Template:Math and Template:Math any such subsequences must tend to Template:Math and Template:Math, so that Template:Math and Template:Math must tend to Template:Math and Template:Math, establishing continuity.

The above result holds more generally in a Hadamard space.

• If Template:Math is a Busemann function on a Hadamard manifold, then Template:Math is continuously differentiable with Template:Math for all Template:Math.

From the previous properties of Template:Math, for each Template:Math there is a unique geodesic γ(t) parametrised by arclength with Template:Math such that Template:Math. It has the property that it cuts Template:Math at Template:Math: in the previous notation Template:Math and Template:Math. The vector field Template:Math defined by the unit vector $\dot{\gamma }(0)$ at Template:Math is continuous, because Template:Math is a continuous function of Template:Math and the map sending Template:Math to Template:Math is a diffeomorphism from Template:Math onto Template:Math by the Cartan-Hadamard theorem. Let Template:Math be another geodesic parametrised by arclength through Template:Math with Template:Math. Then Template:Math $(\dot{\delta }(0),\dot{\gamma }(0))$. Indeed, let Template:Math, so that Template:Math. Then

$$|H(\delta (s))-H(x)|\le d(\delta (s),x).$$

Applying this with Template:Math and Template:Math, it follows that for Template:Math

$$(r-d(\delta (s),u))/s\le (h(\delta (s))-h(y))/s\le (d(\delta (s),v)-r)/s.$$

The outer terms tend to $(\dot{\delta }(0),\dot{\gamma }(0))$ as Template:Math tends to 0, so the middle term has the same limit, as claimed. A similar argument applies for Template:Math.

The assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows. Let $a=\dot{\delta }(0)$ and $b=\dot{\gamma }(0)$, both unit vectors. Then for tangent vectors Template:Math and Template:Math at Template:Math in the unit ball

$$d({\exp }_{y}p,{\exp }_{y}q)=\Vert p-q\Vert +\epsilon \max \Vert p{\Vert }^2,\Vert q{\Vert }^2$$

with Template:Math uniformly bounded. Let Template:Math and Template:Math. Then

$$(d(\delta (s),v)-r)/s=(d({\exp }_y(t^{3}a),{\exp }_y(-t^{2}b))-t^2)/t^3=(\Vert t^{3}a+t^{2}b\Vert -t^2)/t^3+\epsilon |t|=(\Vert ta+b\Vert -1)/t+\epsilon |t|.$$

The right hand side here tends to Template:Math as Template:Math tends to 0 since

$$\frac{d}{dt}\Vert b+ta\Vert {|}_{t=0}=\frac{1}{2}\frac{d}{dt}\Vert b+ta{\Vert }^2{|}_{t=0}=(a,b).$$

The same method works for the other terms.

Hence it follows that Template:Math is a Template:Math function with Template:Math dual to the vector field Template:Math, so that Template:Math. The vector field Template:Math is thus the gradient vector field for Template:Math. The geodesics through any point are the flow lines for the flow Template:Math for Template:Math, so that Template:Math is the gradient flow for Template:Math.

THEOREM. On a Hadamard manifold Template:Math the following conditions on a continuous function Template:Math are equivalent:

1. Template:Math is a Busemann function.
2. Template:Math is a convex, Lipschitz function with constant 1, and for each Template:Math in Template:Math there are points Template:Math at a distance Template:Math from Template:Math such that Template:Math.
3. Template:Math is a convex Template:Math function with Template:Math.

It has already been proved that (1) implies (2).

The arguments above show mutatis mutandi that (2) implies (3).

It therefore remains to show that (3) implies (1). Fix Template:Math in Template:Math. Let Template:Math be the gradient flow for Template:Math. It follows that Template:Math and that Template:Math is a geodesic through Template:Math parametrised by arclength with Template:Math. Indeed, if Template:Math, then

$$|s-t|=|h({\alpha }_s(x))-h({\alpha }_t(x))|\le d({\alpha }_s(x),{\alpha }_t(x))\le \underset{s}{\overset{t}{\int }}\Vert d{\alpha }_{\tau }(x)/d\tau \Vert d\tau =\underset{s}{\overset{t}{\int }}\Vert dh({\alpha }_{\tau }(x))\Vert d\tau =|s-t|,$$

so that Template:Math. Let Template:Math, the Busemann function for Template:Math with base point Template:Math. In particular Template:Math. To prove (1), it suffices to show that Template:Math.

Let Template:Math be the closed convex set of points Template:Math with Template:Math. Since Template:Math is a Hadamard space for every point Template:Math in Template:Math there is a unique closest point Template:Math to Template:Math in Template:Math. It depends continuously on Template:Math and if Template:Math lies outside Template:Math, then Template:Math lies on the hypersurface Template:Math—the boundary Template:Math of Template:Math—and the geodesic from Template:Math to Template:Math is orthogonal to Template:Math. In this case the geodesic is just Template:Math. Indeed, the fact that Template:Math is the gradient flow of Template:Math and the conditions Template:Math imply that the flow lines Template:Math are geodesics parametrised by arclength and cut the level curves of Template:Math orthogonally. Taking Template:Math with Template:Math and Template:Math,

$$d(x,y)\ge d({\alpha }_t(x),{\alpha }_t(y)),d(x,{\alpha }_t(y){)}^2\ge d(x,{\alpha }_t(x){)}^2+d({\alpha }_t(x),{\alpha }_t(y){)}^2,d(y,{\alpha }_t(x){)}^2\ge d(y,{\alpha }_t(y){)}^2+d({\alpha }_t(x),{\alpha }_t(y){)}^2.$$

On the other hand, for any four points Template:Math, Template:Math, Template:Math, Template:Math in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds:

$$|d(a,c{)}^2+d(b,d{)}^2-d(a,d{)}^2-d(b,d{)}^2|\le 2d(a,b)d(c,d).$$

Setting Template:Math, Template:Math, Template:Math, Template:Math, it follows that

$$|d(y,{\alpha }_t(x){)}^2-d(x,{\alpha }_t(x){)}^2|\le 2d(x,y{)}^2,$$

so that

$$|d(y,{\alpha }_t(x))-d(x,{\alpha }_t(x))|\le 2\frac{d(x,y{)}^2}{d(y,{\alpha }_t(x))+d(x,{\alpha }_t(x))}\le \frac{d(x,y{)}^2}{t}.$$

Hence Template:Math on the level surface of Template:Math containing Template:Math. The flow Template:Math can be used to transport this result to all the level surfaces of Template:Math. For general Template:Math take Template:Math such that Template:Math and set Template:Math. Then Template:Math, where Template:Math. But then Template:Math, so that Template:Math. Hence Template:Math, as required.

Note that this argument could be shortened using the fact that two Busemann functions Template:Math and Template:Math differ by a constant if and only if the corresponding geodesic rays satisfy Template:Math. Indeed, all the geodesics defined by the flow Template:Math satisfy the latter condition, so differ by constants. Since along any of these geodesics Template:Math is linear with derivative 1, Template:Math must differ from these Busemann functions by constants.

Compactification of a proper Hadamard space

Template:Harvtxt defined a compactification of a Hadamard manifold Template:Math which uses Busemann functions. Their construction, which can be extended more generally to proper (i.e. locally compact) Hadamard spaces, gives an explicit geometric realisation of a compactification defined by Gromov—by adding an "ideal boundary"—for the more general class of proper metric spaces Template:Math, those for which every closed ball is compact. Note that, since any Cauchy sequence is contained in a closed ball, any proper metric space is automatically complete. The ideal boundary is a special case of the ideal boundary for a metric space. In the case of Hadamard spaces, this agrees with the space of geodesic rays emanating from any fixed point described using Busemann functions in the bordification of the space.

If Template:Math is a proper metric space, Gromov's compactification can be defined as follows. Fix a point Template:Math in Template:Math and let Template:Math. Let Template:Math be the space of Lipschitz continuous functions on Template:Math, .e. those for which Template:Math for some constant Template:Math. The space Template:Math can be topologised by the seminorms Template:Math, the topology of uniform convergence on compacta. This is the topology induced by the natural map of C(X) into the direct product of the Banach spaces Template:Math. It is give by the metric Template:Math.

The space Template:Math is embedded into Template:Math by sending Template:Math to the function Template:Math. Let Template:Math be the closure of Template:Math in Template:Math. Then Template:Math is compact (metrisable) and contains Template:Math as an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic. Compactness follows from the Arzelà–Ascoli theorem since the image in Template:Math is equicontinuous and uniformly bounded in norm by Template:Math. Let Template:Math be a sequence in Template:Math tending to Template:Math in Template:Math. Then all but finitely many terms must lie outside Template:Math since Template:Math is compact, so that any subsequence would converge to a point in Template:Math; so the sequence Template:Math must be unbounded in Template:Math. Let Template:Math. Then Template:Math lies in Template:Math. It is non-zero on Template:Math and vanishes only at Template:Math. Hence it extends to a continuous function on Template:Math with zero set Template:Math. It follows that Template:Math is closed in Template:Math, as required. To check that the compactification Template:Math is independent of the basepoint, it suffices to show that Template:Math extends to a continuous function on Template:Math. But Template:Math, so, for Template:Math in Template:Math, Template:Math. Hence the correspondence between the compactifications for Template:Math and Template:Math is given by sending Template:Math in Template:Math to Template:Math in Template:Math.

When Template:Math is a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary Template:Math can be realised explicitly as "asymptotic limits" of geodesics by using Busemann functions. Fixing a base point Template:Math, there is a unique geodesic Template:Math parametrised by arclength such that Template:Math and $\dot{\gamma }(0)$ is a given unit vector. If Template:Math is the corresponding Busemann function, then Template:Math lies in Template:Math and induces a homeomorphism of the unit Template:Math-sphere onto Template:Math, sending $\dot{\gamma }(0)$ to Template:Math.

Quasigeodesics in the Poincaré disk, CAT(-1) and hyperbolic spaces

Morse–Mostow lemma

In the case of spaces of negative curvature, such as the Poincaré disk, CAT(-1) and hyperbolic spaces, there is a metric structure on their Gromov boundary. This structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays. Quasigeodesics were first studied for negatively curved surfaces—in particular the hyperbolic upper halfplane and unit disk—by Morse and generalised to negatively curved symmetric spaces by Mostow, for his work on the rigidity of discrete groups. The basic result is the Morse–Mostow lemma on the stability of geodesics.

By definition a quasigeodesic Γ defined on an interval Template:Math with Template:Math is a map Template:Math into a metric space, not necessarily continuous, for which there are constants Template:Math and Template:Math such that for all Template:Math and Template:Math:

$${\lambda }^{-1}|s-t|-\epsilon \le d(\mathrm{\Gamma }(s),\mathrm{\Gamma }(t))\le \lambda |s-t|+\epsilon .$$

The following result is essentially due to Marston Morse (1924).

Morse's lemma on stability of geodesics. In the hyperbolic disk there is a constant Template:Math depending on Template:Math and Template:Math such that any quasigeodesic segment Template:Math defined on a finite interval Template:Math is within a Hausdorff distance Template:Math of the geodesic segment Template:Math.

Classical proof for Poincaré disk

The classical proof of Morse's lemma for the Poincaré unit disk or upper halfplane proceeds more directly by using orthogonal projection onto the geodesic segment.

• It can be assumed that Γ satisfies the stronger "pseudo-geodesic" condition:

$${\lambda }^{-1}|s-t|-\epsilon \le d(\mathrm{\Gamma }(s),\mathrm{\Gamma }(t))\le \lambda |s-t|.$$

Template:Math can be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from Template:Math less than Template:Math: break up the interval on which Template:Math is defined into equal subintervals of length Template:Math and take the geodesics between the images under Template:Math of the endpoints of the subintervals. Since Template:Math is piecewise geodesic, Template:Math is Lipschitz continuous with constant Template:Math, Template:Math, where Template:Math. The lower bound is automatic at the endpoints of intervals. By construction the other values differ from these by a uniformly bounded depending only on Template:Math and Template:Math; the lower bound inequality holds by increasing ε by adding on twice this uniform bound.

• If Template:Math is a piecewise smooth curve segment lying outside an Template:Math-neighbourhood of a geodesic line and Template:Math is the orthogonal projection onto the geodesic line then:

$$\ell (P\circ \gamma )\le \frac{\ell (\gamma )}{\cosh s}.$$

Applying an isometry in the upper half plane, it may be assumed that the geodesic line is the positive imaginary axis in which case the orthogonal projection onto it is given by Template:Math and Template:Math. Hence the hypothesis implies Template:Math, so that

$$\ell (P\circ \gamma )=\underset{a}{\overset{b}{\int }}\frac{|d\gamma |}{|\gamma |}\le \underset{a}{\overset{b}{\int }}\frac{|d\gamma |}{\cosh (s)\mathrm{\Im }\gamma }=\frac{\ell (\gamma )}{\cosh (s)}.$$

• There is a constant Template:Math depending only on Template:Math and Template:Math such that Template:Math lies within an Template:Math-neighbourhood of the geodesic segment Template:Math.

Let Template:Math be the geodesic line containing the geodesic segment Template:Math. Then there is a constant Template:Math depending only on Template:Math and Template:Math such that Template:Math-neighbourhood Template:Math lies within an Template:Math-neighbourhood of Template:Math. Indeed for any Template:Math, the subset of Template:Math for which Template:Math lies outside the closure of the Template:Math-neighbourhood of Template:Math is open, so a countable union of open intervals Template:Math. Then

$$\ell (\mathrm{\Gamma }{|}_{[c,d]})\le s_1\equiv {\lambda }^2(2s+\epsilon )(1-\frac{{\lambda }^2}{\cosh (s)}),$$

since the left hand side is less than or equal to Template:Math and

$$\frac{|c-d|}{\lambda }-\epsilon \le d(\mathrm{\Gamma }(c),\mathrm{\Gamma }(d))\le 2s+d(P\circ \mathrm{\Gamma }{|}_{[c,d]})\le 2s+\frac{\lambda |c-d|}{\cosh (s)}.$$

Hence every point lies at a distance less than or equal to Template:Math of Template:Math. To deduce the assertion, note that the subset of Template:Math for which Template:Math lies outside the closure of the Template:Math-neighbourhood of Template:Math is open, so a union of intervals Template:Math with Template:Math and Template:Math both at a distance Template:Math from either Template:Math or Template:Math. Then

$$\ell (\mathrm{\Gamma }{|}_{[c,d]})\le s_2\equiv {\lambda }^2(2(s+s_1)+\epsilon ),$$

since

$$\frac{|c-d|}{\lambda }-\epsilon \le d(\mathrm{\Gamma }(c),\mathrm{\Gamma }(d))\le 2(s+s_1).$$

Hence the assertion follows taking any Template:Math greater than Template:Math.

• There is a constant Template:Math depending only on Template:Math and Template:Math such that the geodesic segment Template:Math lies within an Template:Math-neighbourhood of Template:Math.

Every point of Template:Math lies within a distance Template:Math of Template:Math. Thus orthogonal projection Template:Math carries each point of Template:Math onto a point in the closed convex set Template:Math at a distance less than Template:Math. Since Template:Math is continuous and Template:Math connected, the map Template:Math must be onto since the image contains the endpoints of Template:Math. But then every point of Template:Math is within a distance Template:Math of a point of Template:Math.

Gromov's proof for Poincaré disk

The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation of the classical proof. There is also a generalisation for the more general class of hyperbolic metric spaces due to Gromov. Gromov's proof is given below for the Poincaré unit disk; the properties of hyperbolic metric spaces are developed in the course of the proof, so that it applies mutatis mutandi to CAT(-1) or hyperbolic metric spaces.

Since this is a large-scale phenomenon, it is enough to check that any maps Template:Math from Template:Math for any Template:Math to the disk satisfying the inequalities is within a Hausdorff distance Template:Math of the geodesic segment Template:Math. For then translating it may be assumed without loss of generality Template:Math is defined on Template:Math with Template:Math and then, taking Template:Math (the integer part of Template:Math), the result can be applied to Template:Math defined by Template:Math. The Hausdorff distance between the images of Template:Math and Template:Math is evidently bounded by a constant Template:Math depending only on Template:Math and Template:Math.

Now the incircle of a geodesic triangle has diameter less than Template:Math where Template:Math; indeed it is strictly maximised by that of an ideal triangle where it equals Template:Math. In particular, since the incircle breaks the triangle breaks the triangle into three isosceles triangles with the third side opposite the vertex of the original triangle having length less than Template:Math, it follows that every side of a geodesic triangle is contained in a Template:Math-neighbourhood of the other two sides. A simple induction argument shows that a geodesic polygon with Template:Math vertices for Template:Math has each side within a Template:Math neighbourhood of the other sides (such a polygon is made by combining two geodesic polygons with Template:Math sides along a common side). Hence if Template:Math, the same estimate holds for a polygon with Template:Math sides.

For Template:Math let Template:Math, the largest radius for a closed ball centred on Template:Math which contains no Template:Math in its interior. This is a continuous function non-zero on Template:Math so attains its maximum Template:Math at some point Template:Math in this segment. Then Template:Math lies within an Template:Math-neighbourhood of the image of Template:Math for any Template:Math. It therefore suffices to find an upper bound for Template:Math independent of Template:Math.

Choose Template:Math and Template:Math in the segment Template:Math before and after Template:Math with Template:Math and Template:Math (or an endpoint if it within a distance of less than Template:Math from Template:Math). Then there are Template:Math with Template:Math, Template:Math. Hence Template:Math, so that Template:Math. By the triangle inequality all points on the segments Template:Math and Template:Math are at a distance Template:Math from Template:Math. Thus there is a finite sequence of points starting at Template:Math and ending at Template:Math, lying first on the segment Template:Math, then proceeding through the points Template:Math, before taking the segment Template:Math. The successive points Template:Math are separated by a distance no greater than Template:Math and successive points on the geodesic segments can also be chosen to satisfy this condition. The minimum number Template:Math of points in such a sequence satisfies Template:Math. These points form a geodesic polygon, with Template:Math as one of the sides. Take Template:Math, so that the Template:Math-neighbourhood of Template:Math does not contain all the other sides of the polygon. Hence, from the result above, it follows that Template:Math. Hence

$$3+2(\lambda +\epsilon {)}^{-1}h+6\lambda h+\epsilon >2^{h/\delta }+2.$$

This inequality implies that Template:Math is uniformly bounded, independently of Template:Math, as claimed.

If all points Template:Math lie within Template:Math of the Template:Math, the result follows. Otherwise the points which do not fall into maximal subsets Template:Math with Template:Math. Thus points in Template:Math have a point Template:Math with Template:Math in the complement of Template:Math within a distance of Template:Math. But the complement of Template:Math, a disjoint union with Template:Math and Template:Math. Connectivity of Template:Math implies there is a point Template:Math in the segment which is within a distance Template:Math of points Template:Math and Template:Math with Template:Math and Template:Math. But then Template:Math, so Template:Math. Hence the points Template:Math for Template:Math in Template:Math lie within a distance from Template:Math of less than Template:Math.

Extension to quasigeodesic rays and lines

Recall that in a Hadamard space if Template:Math and Template:Math are two geodesic segments and the intermediate points Template:Math and Template:Math divide them in the ratio Template:Math, then Template:Math is a convex function of Template:Math. In particular if Template:Math and Template:Math are geodesic segments of unit speed defined on Template:Math starting at the same point then

$$d({\mathrm{\Gamma }}_1(t),{\mathrm{\Gamma }}_2(t))\le \frac{t}{R}d({\mathrm{\Gamma }}_1(R),{\mathrm{\Gamma }}_2(R)).$$

In particular this implies the following:

• In a CAT(–1) space Template:Math, there is a constant Template:Math depending only on Template:Math and Template:Math such that any quasi-geodesic ray is within a bounded Hausdorff distance Template:Math of a geodesic ray. A similar result holds for quasigeodesic and geodesic lines.

If Template:Math is a geodesic say with constant Template:Math and Template:Math, let Template:Math be the unit speed geodesic for the segment Template:Math. The estimate above shows that for fixed Template:Math and Template:Math sufficiently large, Template:Math is a Cauchy sequence in Template:Math with the uniform metric. Thus Template:Math tends to a geodesic ray Template:Math uniformly on compacta the bound on the Hausdorff distances between Template:Math and the segments Template:Math applies also to the limiting geodesic Template:Math. The assertion for quasigeodesic lines follows by taking Template:Math corresponding to the geodesic segment Template:Math.

Efremovich–Tikhomirova theorem

Before discussing CAT(-1) spaces, this section will describe the Efremovich–Tikhomirova theorem for the unit disk Template:Math with the Poincaré metric. It asserts that quasi-isometries of Template:Math extend to quasi-Möbius homeomorphisms of the unit disk with the Euclidean metric. The theorem forms the prototype for the more general theory of CAT(-1) spaces. Their original theorem was proved in a slightly less general and less precise form in Template:Harvtxt and applied to bi-Lipschitz homeomorphisms of the unit disk for the Poincaré metric; earlier, in the posthumous paper Template:Harvtxt, the Japanese mathematician Akira Mori had proved a related result within Teichmüller theory assuring that every quasiconformal homeomorphism of the disk is Hölder continuous and therefore extends continuously to a homeomorphism of the unit circle (it is known that this extension is quasi-Möbius).

Extension of quasi-isometries to boundary

If Template:Math is the Poincaré unit disk, or more generally a CAT(-1) space, the Morse lemma on stability of quasigeodesics implies that every quasi-isometry of Template:Math extends uniquely to the boundary. By definition two self-mappings Template:Math of Template:Math are quasi-equivalent if Template:Math, so that corresponding points are at a uniformly bounded distance of each other. A quasi-isometry Template:Math of Template:Math is a self-mapping of Template:Math, not necessarily continuous, which has a quasi-inverse Template:Math such that Template:Math and Template:Math are quasi-equivalent to the appropriate identity maps and such that there are constants Template:Math and Template:Math such that for all Template:Math in Template:Math and both mappings

$${\lambda }^{-1}d(x,y)-\epsilon \le d(f_k(x),f_k(y))\le \lambda d(x,y)+\epsilon .$$

Note that quasi-inverses are unique up to quasi-equivalence; that equivalent definition could be given using possibly different right and left-quasi inverses, but they would necessarily be quasi-equivalent; that quasi-isometries are closed under composition which up to quasi-equivalence depends only the quasi-equivalence classes; and that, modulo quasi-equivalence, the quasi-isometries form a group.

Fixing a point Template:Math in Template:Math, given a geodesic ray Template:Math starting at Template:Math, the image Template:Math under a quasi-isometry Template:Math is a quasi-geodesic ray. By the Morse-Mostow lemma it is within a bounded distance of a unique geodesic ray Template:Math starting at Template:Math. This defines a mapping Template:Math on the boundary Template:Math of Template:Math, independent of the quasi-equivalence class of Template:Math, such that Template:Math. Thus there is a homomorphism of the group of quasi-isometries into the group of self-mappings of Template:Math.

To check that Template:Math is continuous, note that if Template:Math and Template:Math are geodesic rays that are uniformly close on Template:Math, within a distance Template:Math, then Template:Math and Template:Math lie within a distance Template:Math on Template:Math, so that Template:Math and Template:Math lie within a distance Template:Math; hence on a smaller interval Template:Math, Template:Math and Template:Math lie within a distance Template:Math by convexity.

On CAT(-1) spaces, a finer version of continuity asserts that Template:Math is a quasi-Möbius mapping with respect to a natural class of metric on Template:Math, the "visual metrics" generalising the Euclidean metric on the unit circle and its transforms under the Möbius group. These visual metrics can be defined in terms of Busemann functions.

In the case of the unit disk, Teichmüller theory implies that the homomorphism carries quasiconformal homeomorphisms of the disk onto the group of quasi-Möbius homeomorphisms of the circle (using for example the Ahlfors–Beurling or Douady–Earle extension): it follows that the homomorphism from the quasi-isometry group into the quasi-Möbius group is surjective.

In the other direction, it is straightforward to prove that the homomorphism is injective. Suppose that Template:Math is a quasi-isometry of the unit disk such that Template:Math is the identity. The assumption and the Morse lemma implies that if Template:Math is a geodesic line, then Template:Math lies in an Template:Math-neighbourhood of Template:Math. Now take a second geodesic line Template:Math such that Template:Math and Template:Math intersect orthogonally at a given point in Template:Math. Then Template:Math lies in the intersection of Template:Math-neighbourhoods of Template:Math and Template:Math. Applying a Möbius transformation, it can be assumed that Template:Math is at the origin of the unit disk and the geodesics are the real and imaginary axes. By convexity, the Template:Math-neighbourhoods of these axes intersect in a Template:Math-neighbourhood of the origin: if Template:Math lies in both neighbourhoods, let Template:Math and Template:Math be the orthogonal projections of Template:Math onto the Template:Math- and Template:Math-axes; then Template:Math so taking projections onto the Template:Math-axis, Template:Math; hence Template:Math. Hence Template:Math, so that Template:Math is quasi-equivalent to the identity, as claimed.

Cross ratio and distance between non-intersecting geodesic lines

Given two distinct points Template:Math on the unit circle or real axis there is a unique hyperbolic geodesic Template:Math joining them. It is given by the circle (or straight line) which cuts the unit circle unit circle or real axis orthogonally at those two points. Given four distinct points Template:Math in the extended complex plane their cross ratio is defined by

$$(a,b;c,d)=\frac{(a-c)(b-d)}{(a-d)(b-c)}.$$

If Template:Math is a complex Möbius transformation then it leaves the cross ratio invariant: Template:Math. Since the Möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number Template:Math in Template:Math such that Template:Math for a Möbius transformation Template:Math.

Since Template:Math, Template:Math, Template:Math and Template:Math all appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of Template:Math, Template:Math, Template:Math and Template:Math, it suffices to consider permutations that fix Template:Math, so only permute Template:Math, Template:Math and Template:Math. The cross ratio transforms according to the anharmonic group of order 6 generated by the Möbius transformations sending Template:Math to Template:Math and Template:Math. The other three transformations send Template:Math to Template:Math, to Template:Math and to Template:Math.

Now let Template:Math be points on the unit circle or real axis in that order. Then the geodesics Template:Math and Template:Math do not intersect and the distance between these geodesics is well defined: there is a unique geodesic line cutting these two geodesics orthogonally and the distance is given by the length of the geodesic segment between them. It is evidently invariant under real Möbius transformations. To compare the cross ratio and the distance between geodesics, Möbius invariance allows the calculation to be reduced to a symmetric configuration. For Template:Math, take Template:Math. Then Template:Math where Template:Math. On the other hand, the geodesics Template:Math and Template:Math are the semicircles in the upper half plane of radius Template:Math and Template:Math. The geodesic which cuts them orthogonally is the positive imaginary axis, so the distance between them is the hyperbolic distance between Template:Math and Template:Math, Template:Math. Let Template:Math, then Template:Math, so that there is a constant Template:Math such that, if Template:Math, then

$$d([a,d];[b,c])-C\le \log (a,b;c,d)\le d([a,d];[b,c])+C,$$

since Template:Math is bounded above and below in Template:Math. Note that Template:Math are in order around the unit circle if and only if Template:Math.

A more general and precise geometric interpretation of the cross ratio can be given using projections of ideal points on to a geodesic line; it does not depend on the order of the points on the circle and therefore whether or not geodesic lines intersect.

• If Template:Math and Template:Math are the feet of the perpendiculars from Template:Math and Template:Math to the geodesic line Template:Math, then Template:Math.

Since both sides are invariant under Möbius transformations, it suffices to check this in the case that Template:Math, Template:Math, Template:Math and Template:Math. In this case the geodesic line is the positive imaginary axis, right hand side equals Template:Math, Template:Math and Template:Math. So the left hand side equals Template:Math. Note that Template:Math and Template:Math are also the points where the incircles of the ideal triangles Template:Math and Template:Math touch Template:Math.

Proof of theorem

A homeomorphism Template:Math of the circle is quasisymmetric if there are constants Template:Math such that

$$\frac{|F(z_1)-F(z_2)|}{|F(z_1)-F(z_3)|}\le a\frac{|z_1-z_2{|}^b}{|z_1-z_3{|}^b}.$$

It is quasi-Möbius is there are constants Template:Math such that

$$|(F(z_1),F(z_2);F(z_3),F(z_4))|\le c|(z_1,z_2;z_3,z_4){|}^d,$$

where

$$(z_1,z_2;z_3,z_4)=\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}$$

denotes the cross-ratio.

It is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition.

If Template:Math is quasisymmetric then it is also quasi-Möbius, with Template:Math and Template:Math: this follows by multiplying the first inequality for Template:Math and Template:Math. Conversely any quasi-Möbius homeomorphism Template:Math is quasisymmetric. To see this, it can be first be checked that Template:Math (and hence Template:Math) is Hölder continuous. Let Template:Math be the set of cube roots of unity, so that if Template:Math in Template:Math, then Template:Math. To prove a Hölder estimate, it can be assumed that Template:Math is uniformly small. Then both Template:Math and Template:Math are greater than a fixed distance away from Template:Math in Template:Math with Template:Math, so the estimate follows by applying the quasi-Möbius inequality to Template:Math. To verify that Template:Math is quasisymmetric, it suffices to find a uniform upper bound for Template:Math in the case of a triple with Template:Math, uniformly small. In this case there is a point Template:Math at a distance greater than 1 from Template:Math, Template:Math and Template:Math. Applying the quasi-Möbius inequality to Template:Math, Template:Math, Template:Math and Template:Math yields the required upper bound. To summarise:

• A homeomorphism of the circle is quasi-Möbius if and only if it is quasisymmetric. In this case it and its inverse are Hölder continuous. The quasi-Möbius homeomorphisms form a group under composition.

To prove the theorem it suffices to prove that if Template:Math then there are constants Template:Math such that for Template:Math distinct points on the unit circle

$$|(F(a),F(b);F(c),F(d))|\le A|(a,b;c,d){|}^B.$$

It has already been checked that Template:Math (and is inverse) are continuous. Composing Template:Math, and hence Template:Math, with complex conjugation if necessary, it can further be assumed that Template:Math preserves the orientation of the circle. In this case, if Template:Math are in order on the circle, so too are there images under Template:Math; hence both Template:Math and Template:Math are real and greater than one. In this case

$$(F(a),F(b);F(c),F(d))\le A(a,b;c,d{)}^B.$$

To prove this, it suffices to show that Template:Math. From the previous section it suffices show Template:Math. This follows from the fact that the images under Template:Math of Template:Math and Template:Math lie within Template:Math-neighbourhoods of Template:Math and Template:Math; the minimal distance can be estimated using the quasi-isometry constants for Template:Math applied to the points on Template:Math and Template:Math realising Template:Math.

Adjusting Template:Math and Template:Math if necessary, the inequality above applies also to Template:Math. Replacing Template:Math, Template:Math, Template:Math and Template:Math by their images under Template:Math, it follows that

$$A^{-1}|(a,b;c,d){|}^{-B}\le |(F(a),F(b);F(c),F(d))|\le A|(a,b;c,d){|}^B$$

if Template:Math, Template:Math, Template:Math and Template:Math are in order on the unit circle. Hence the same inequalities are valid for the three cyclic of the quadruple Template:Math. If Template:Math and Template:Math are switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if Template:Math and Template:Math are switched. If both pairs are switched, the cross ratio remains unaltered. Hence the inequalities are also valid in this case. Finally if Template:Math and Template:Math are interchanged, the cross ratio changes from Template:Math to Template:Math, which lies between 0 and 1. Hence again the same inequalities are valid. It is easy to check that using these transformations the inequalities are valid for all possible permutations of Template:Math, Template:Math, Template:Math and Template:Math, so that Template:Math and its inverse are quasi-Möbius homeomorphisms.

Busemann functions and visual metrics for CAT(-1) spaces

Busemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces. These are complete geodesic metric spaces in which the distances between points on the boundary of a geodesic triangle are less than or equal to the comparison triangle in the hyperbolic upper half plane or equivalently the unit disk with the Poincaré metric. In the case of the unit disk the chordal metric can be recovered directly using Busemann functions Template:Math and the special theory for the disk generalises completely to any proper CAT(-1) space Template:Math. The hyperbolic upper half plane is a CAT(0) space, as lengths in a hyperbolic geodesic triangle are less than lengths in the Euclidean comparison triangle: in particular a CAT(-1) space is a CAT(0) space, so the theory of Busemann functions and the Gromov boundary applies. From the theory of the hyperbolic disk, it follows in particular that every geodesic ray in a CAT(-1) space extends to a geodesic line and given two points of the boundary there is a unique geodesic Template:Math such that has these points as the limits Template:Math. The theory applies equally well to any CAT(Template:Math) space with Template:Math since these arise by scaling the metric on a CAT(-1) space by Template:Math. On the hyperbolic unit disk Template:Math quasi-isometries of Template:Math induce quasi-Möbius homeomorphisms of the boundary in a functorial way. There is a more general theory of Gromov hyperbolic spaces, a similar statement holds, but with less precise control on the homeomorphisms of the boundary.

Applications in percolation theory

Template:See More recently Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation and directed last-passage percolation.

References

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