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  be a metric space. A geodesic ray is a path  which minimizes distance everywhere along its length. i.e., for all , 
 
Equivalently, a ray is an isometry from the "canonical ray" (the set  equipped with the Euclidean metric) into the metric space X. Given a ray γ, the Busemann function  is defined by

Thus, when t is very large, the distance  is approximately equal to . Given a ray γ, its Busemann function is always well-defined: indeed the right hand side above , tends pointwise to the left hand side on compacta, since  is bounded above by  and non-increasing since, if ,

It is immediate from the triangle inequality that

so that  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  tend to  uniformly on compact sets as t tends to infinity. Example: Poincaré disk 

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

Then, for  and , the Busemann function is given by

where the term in brackets on the right hand side is the Poisson kernel for the unit disk and  corresponds to the radial geodesic  from the origin towards ,
. The computation of  can be reduced to that of , since the metric is invariant under Möbius transformations in ; the geodesics through  have the form  where  is the 1-parameter subgroup of ,

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  is convex, i.e. convex on geodesic segments . Explicitly this means that if  is the point which divides  in the ratio , then . For fixed  the function  is convex and hence so are its translates; in particular, if  is a geodesic ray in , then  is convex. Since the Busemann function  is the pointwise limit of ,

Busemann functions are convex on Hadamard spaces. On a Hadamard space, the functions  converge uniformly to  uniformly on any bounded subset of . Let . Since  is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function  is convex. Hence for 

Thus

so that

Letting t tend to ∞, it follows that

so convergence is uniform on bounded sets. Note that the inequality above for  (together with its proof) also holds for geodesic segments: if  is a geodesic segment starting at  and parametrised by arclength then

Next suppose that   are points in a Hadamard space, and let  be the geodesic through  with  and , where . This geodesic cuts the boundary of the closed ball  at the point . Thus if , there is a point  with  such that . 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  is a closed convex set in a Hadamard space , then every point  in  has a unique closest point  in  and ; moreover  is uniquely determined by the property that, for  in ,

so that the angle at  in the Euclidean comparison triangle for  is greater than or equal to . If  is a Busemann function on a Hadamard space, then, given  in  and , there is a unique point  with  such that . For fixed , the point  is the closest point of  to the closed convex  set of points  such that  and therefore depends continuously on . Let  be the closest point to  in . Then  and so  is minimised by  in  where  is the unique point where  is minimised. By the Lipschitz condition . To prove the assertion, it suffices to show that , i.e. . On the other hand,  is the uniform limit on any closed ball of functions . On , these are minimised by points  with . Hence the infimum of  on  is  and  tends to . Thus  with  and  tending towards . Let  be the closest point to  with . Let . Then , and, by the Lipschitz condition on , . In particular  tends to . Passing to a subsequence if necessary it can be assumed that  and  are both increasing (to ). The inequality for convex optimisation implies that for . so that  is a Cauchy sequence. If  is its limit, then  and . By uniqueness it follows that  and hence , as required. Uniform limits. The above argument proves more generally that if  tends to infinity and the functions  tend uniformly on bounded sets to , then  is convex, Lipschitz with Lipschitz constant 1 and, given  in  and , there is a unique point  with  such that . If on the other hand the sequence  is bounded, then the terms all lie in some closed ball and uniform convergence there implies that  is a Cauchy sequence so converges to some  in . So  tends uniformly to , 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  or even a semi-infinite interval . The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space. If  is a convex function, Lipschitz with constant 1 and  assumes its minimum on any closed ball centred on  with radius  at a unique point  on the boundary with , then for each  in  there is a unique geodesic ray  such that  and  cuts each closed convex set  with  at , so that . In particular this holds for each Busemann function. The third condition implies that  is the closest point to  in the closed convex set  of points  such that . Let  for  be the geodesic joining  to . Then  is a convex Lipschitz function on  with Lipschitz constant 1 satisfying  and  and . So  vanishes everywhere, since if  and . Hence . By uniqueness it follows that  is the closest point to  in  and that it is the unique point minimising  in . Uniqueness implies that these geodesics segments coincide for arbitrary  and therefore that  extends to a geodesic ray with the stated property. If , then the geodesic ray  starting at  satisfies . If  is another ray starting at  with  then . To prove the first assertion, it is enough to check this for  sufficiently large. In that case  and  are the projections of  and  onto the closed convex set . Therefore, . Hence  . The second assertion follows because  is convex and bounded on , so, if it vanishes at , must vanish everywhere. Suppose that  is a continuous convex function and for each  in  there is a unique geodesic ray  such that  and  cuts each closed convex set  at , so that ; then  is a Busemann function. is a constant function. Let  be the closed convex set of points  with . Since  is a Hadamard space for every point  in  there is a unique closest point  to  in . It depends continuously on  and if  lies outside , then  lies on the hypersurface —the boundary ∂ of —and  satisfies the inequality of convex optimisation. Let  be the geodesic ray starting at . Fix  in . Let  be the geodesic ray starting at . Let , the Busemann function for  with base point . In particular . It suffices to show that . Now take  with  and let  be the geodesic ray starting at  corresponding to . Then

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

Setting , , , , it follows that

so that

Hence . Similarly . Hence  on the level surface of  containing . Now for  and  in , let  the geodesic ray starting at . Then  and . Moreover, by boundedness, . The flow  can be used to transport this result to all the level surfaces of . For general , if , take  such that  and set . Then , where . But then , so that . Hence , as required. Similarly if , take
 such that . Let . Then , so . Hence , 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  and  differ by a constant if and only if . Suppose firstly that  and  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 , say. Let  be the closed convex set on which . Then  and similarly . Then for , the points  and   have closest points  and   in , so that . Hence . Now suppose that . Let  be the geodesic ray starting at  associated with . Then . Hence . Since  and  both start at , it follows that . By the previous result  and  differ by a constant; so  and  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  are equivalent:

 is a Busemann function. is a convex function, Lipschitz with constant  and  assumes its minimum on any closed ball centred on  with radius  at a unique point  on the boundary with . is a continuous convex function and for each  in  there is a unique geodesic ray  such that  and, for any , the ray  cuts each closed convex set  at . Bordification of a Hadamard space
In the previous section it was shown that if  is a Hadamard space and  is a fixed point in  then the union of the space of Busemann functions vanishing at  and the space of functions  is closed under taking uniform limits on bounded sets. This result can be formalised in the notion of bordification of . In this topology, the points  tend to a geodesic ray  starting at  if and only if  tends to  and for  arbitrarily large the sequence obtained by taking the point on each segment  at a distance  from  tends to . If  is a metric space, Gromov's bordification can be defined as follows. Fix a point  in  and let . Let  be the space of Lipschitz continuous functions on , i.e. those for which  for some constant . The space  can be topologised by the seminorms , 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  into the direct product of the Banach spaces  of continuous bounded functions on . It is give by the metric . The space  is embedded into  by sending  to the function . Let  be the closure of  in . Then  is metrisable, since  is, and contains  as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic. Let . Then  lies in . It is non-zero on  and vanishes only at . Hence it extends to a continuous function on  with zero set . It follows that  is closed in , as required. To check that  is independent of the basepoint, it suffices to show that  extends to a continuous function on . But , so, for  in , . Hence the correspondence between the compactifications for  and  is given by sending  in  to  in . When  is a Hadamard space, Gromov's ideal boundary  can be realised explicitly as "asymptotic limits" of geodesic rays using Busemann functions. If  is an unbounded sequence in  with  tending to  in , then  vanishes at , is convex, Lipschitz with Lipschitz constant  and has minimum  on any closed ball . Hence  is a Busemann function  corresponding to a unique geodesic ray  starting at . On the other hand,  tends to  uniformly on bounded sets if and only if  tends to  and for  arbitrarily large the sequence obtained by taking the point on each segment  at a distance  from  tends to . For , let  be the point in  with . Suppose first that  tends to  uniformly on . Then for ,
. This is a convex function. It vanishes as  and hence is increasing. So it is maximised at . So for each ,  tends towards 0. Let ,  and . Then  is close to  with  large. Hence in the Euclidean comparison triangle  is close to  with  large. So the angle at  is small. So the point  on  at the same distance as  lies close to . Hence, by the first comparison theorem for geodesic triangles,  is small. Conversely suppose that for fixed  and  sufficiently large  tends to 0. Then from the above  satisfies

so it suffices show that on any bounded set  is uniformly close to  for  sufficiently large. For a fixed ball , fix  so that . The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since

Hence, if  in  and  is sufficiently large that , then

Busemann functions on a Hadamard manifold 

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

If  is a Busemann function on a Hadamard manifold, then, given  in  and , there are unique points ,  with  such that  and  . For fixed , the points  and  depend continuously on . Taking a sequence  tending to  and , there are points  and  which satisfy these conditions for  for  sufficiently large. Passing to a subsequence if necessary, it can be assumed that  and  tend to  and . By continuity these points satisfy the conditions for . To prove uniqueness, note that by compactness  assumes its maximum and minimum on . The Lipschitz condition shows that the values of  there differ by at most . Hence  is minimized at  and maximized at . On the other hand,  and for  and  the points  and  are the unique points in  maximizing this distance. The Lipschitz condition on  then immediately implies  and  must be the unique points in  maximizing and minimizing . Now suppose that  tends to . Then the corresponding points  and  lie in a closed ball so admit convergent subsequences. But by uniqueness of  and  any such subsequences must tend to  and , so that  and  must tend to  and , establishing continuity. The above result holds more generally in a Hadamard space. If  is a Busemann function on a Hadamard manifold, then  is continuously differentiable with  for all . From the previous properties of , for each  there is a unique geodesic γ(t) parametrised by arclength with  such that  . It has the property that it cuts  at : in the previous notation  and  . The vector field  defined by the unit vector  at  is continuous, because  is a continuous function of  and the map  sending   to  is a diffeomorphism from  onto  by the Cartan-Hadamard theorem. Let  be another geodesic parametrised by arclength through  with . Then  . Indeed, let , so that . Then

Applying this with  and , it follows that for 

The outer terms tend to  as  tends to 0, so the middle term has the same limit, as claimed. A similar argument applies for . The assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows. Let 
 and , both unit vectors. Then for tangent vectors  and  at  in the unit ball

 
with  uniformly bounded. Let  and . Then

The right hand side here tends to  as  tends to 0 since

The same method works for the other terms. Hence it follows that  is a  function with  dual to the vector field , so that . The vector field  is thus the gradient vector field for . The geodesics through any point are the flow lines for the flow  for , so that  is the gradient flow for . THEOREM. On a Hadamard manifold  the following conditions on a continuous function  are equivalent:
  is a Busemann function. is a convex, Lipschitz function with constant 1, and for each  in  there are points  at a distance  from  such that . is a convex  function with . 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  in . Let  be the gradient flow for . It follows that  and that  is a geodesic through  parametrised by arclength with . Indeed, if , then

so that . Let , the Busemann function for  with base point . In particular . To prove (1), it suffices to show that . Let  be the closed convex set of points  with . Since  is a Hadamard space for every point  in  there is a unique closest point  to  in . It depends continuously on  and if  lies outside , then  lies on the hypersurface —the boundary  of —and the geodesic from  to  is orthogonal to . In this case the geodesic is just . Indeed, the fact that  is the gradient flow of  and the conditions  imply that the flow lines   are geodesics parametrised by arclength and cut the level curves of  orthogonally. Taking  with  and  ,

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

Setting , , , , it follows that

so that

Hence  on the level surface of  containing . The flow  can be used to transport this result to all the level surfaces of . For general  take  such that  and set . Then , where . But then , so that . Hence , as required. Note that this argument could be shortened using the fact that two Busemann functions  and  differ by a constant if and only if the corresponding geodesic rays satisfy . Indeed, all the geodesics defined by the flow  satisfy the latter condition, so differ by constants. Since along any of these geodesics  is linear with derivative 1,  must differ from these Busemann functions by constants. Compactification of a proper Hadamard space
 defined a compactification of a Hadamard manifold  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 , 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  is a proper metric space, Gromov's compactification can be defined as follows. Fix a point  in  and let . Let  be the space of Lipschitz continuous functions on , .e. those for which  for some constant . The space  can be topologised by the seminorms , 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 . It is give by the metric . The space  is embedded into  by sending  to the function . Let  be the closure of  in . Then  is compact (metrisable) and contains  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  is equicontinuous and uniformly bounded in norm by . Let  be a sequence in  tending to  in . Then all but finitely many terms must lie outside  since  is compact, so that any subsequence would converge to a point in ; so the sequence  must be unbounded in . Let . Then  lies in . It is non-zero on  and vanishes only at . Hence it extends to a continuous function on  with zero set  . It follows that  is closed in , as required. To check that the compactification   is independent of the basepoint, it suffices to show that  extends to a continuous function on . But , so, for  in , . Hence the correspondence between the compactifications for  and  is given by sending  in  to  in . When  is a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary  can be realised explicitly as "asymptotic limits" of geodesics by using Busemann functions. Fixing a base point , there is a unique geodesic  parametrised by arclength such that  and  is a given unit vector. If  is the corresponding Busemann function, then
 lies in  and induces a homeomorphism of the unit -sphere onto , sending  to . 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   with  is a map  into a metric space, not necessarily continuous, for which there are constants  and  such that for all  and :

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  depending on  and  such that any quasigeodesic segment  defined on a finite interval  is within a Hausdorff distance  of the geodesic segment . 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:

 can be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from  less than : break up the interval on which  is defined into equal subintervals of length  and take the geodesics between the images under  of the endpoints of the subintervals. Since  is piecewise geodesic,  is Lipschitz continuous with constant , , where  . 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  and ; the lower bound inequality holds by increasing ε by adding on twice this uniform bound. If  is a piecewise smooth curve segment lying outside an -neighbourhood of a geodesic line and  is the orthogonal projection onto the geodesic line then:

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  and . Hence the hypothesis implies , so that

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

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

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

since

Hence the assertion follows taking any  greater than . There is a constant  depending only on  and  such that the geodesic segment  lies within an -neighbourhood of . Every point of  lies within a distance  of . Thus orthogonal projection  carries each point of  onto a point in the closed convex set  at a distance less than . Since  is continuous and  connected, the map  must be onto since the image contains the endpoints of . But then every point of  is within a distance  of a point of . 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  from  for any  to the disk satisfying the inequalities is within a Hausdorff distance  of the geodesic segment . For then translating it may be assumed without loss of generality  is defined on  with  and then, taking  (the integer part of ), the result can be applied to  defined by . The Hausdorff distance between the images of  and  is evidently bounded by a constant  depending only on  and . Now the incircle of a geodesic triangle has diameter less than  where ; indeed it is strictly maximised by that of an ideal triangle where it equals . 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 , it follows that every side of a geodesic triangle is contained in a -neighbourhood of the other two sides. A simple induction argument shows that a geodesic polygon with  vertices for  has each side within a  neighbourhood of the other sides (such a polygon is made by combining two geodesic polygons with  sides along a common side). Hence if , the same estimate holds for a polygon with  sides. For  let , the largest radius for a closed ball centred on  which contains no  in its interior. This is a continuous function non-zero on  so attains its maximum  at some point  in this segment. Then  lies within an -neighbourhood of the image of  for any . It therefore suffices to find an upper bound for  independent of . Choose  and  in the segment  before and after  with  and  (or an endpoint if it within a distance of less than  from ). Then there are  with , . Hence , so that . By the triangle inequality all points on the segments  and  are at a distance  from . Thus there is a finite sequence of points starting at  and ending at , lying first on the segment , then proceeding through the points , before taking the segment . The successive points  are separated by a distance no greater than  and successive points on the geodesic segments can also be chosen to satisfy this condition. The minimum number  of points in such a sequence satisfies . These points form a geodesic polygon, with  as one of the sides. Take , so that the -neighbourhood of  does not contain all the other sides of the polygon. Hence, from the result above, it follows that . Hence

This inequality implies that  is uniformly bounded, independently of , as claimed. If all points  lie within  of the , the result follows. Otherwise the points which do not fall into maximal subsets  with . Thus points in  have a point   with  in the complement of  within a distance of . But the complement of , a disjoint union with  and . Connectivity of  implies there is a point  in the segment which is within a distance  of points  and  with  and . But then , so . Hence the points  for  in  lie within a distance from   of less than . Extension to quasigeodesic rays and lines
Recall that in a Hadamard space if  and  are two geodesic segments and the intermediate points  and  divide them in the ratio , then  is a convex function of . In particular if  and  are geodesic segments of unit speed defined on  starting at the same point then

In particular this implies the following:

In a CAT(–1) space , there is a constant  depending only on  and  such that any quasi-geodesic ray is within a bounded Hausdorff distance  of a geodesic ray. A similar result holds for quasigeodesic and geodesic lines. If  is a geodesic say with constant  and , let  be the unit speed geodesic for the segment . The estimate above shows that for fixed  and  sufficiently large,  is a Cauchy sequence in  with the uniform metric. Thus  tends to a geodesic ray  uniformly on compacta the bound on the Hausdorff distances between  and the segments  applies also to the limiting geodesic . The assertion for quasigeodesic lines follows by taking  corresponding to the geodesic segment . Efremovich–Tikhomirova theorem 
Before discussing CAT(-1) spaces, this section will describe the Efremovich–Tikhomirova theorem for the unit disk  with the Poincaré metric. It asserts that quasi-isometries of  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  and applied to bi-Lipschitz homeomorphisms of the unit disk for the Poincaré metric; earlier, in the posthumous paper , 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  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  extends uniquely to the boundary. By definition two self-mappings  of  are quasi-equivalent if , so that corresponding points are at a uniformly bounded distance of each other. A quasi-isometry  of  is a self-mapping of , not necessarily continuous, which has a quasi-inverse  such that  and  are quasi-equivalent to the appropriate identity maps and such that there are constants  and  such that for all  in  and both mappings

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  in , given a geodesic ray  starting at , the image  under a quasi-isometry  is a quasi-geodesic ray. By the Morse-Mostow lemma it is within a bounded distance of a unique geodesic ray  starting at . This defines a mapping  on the boundary  of , independent of the quasi-equivalence class of , such that . Thus there is a homomorphism of the group of quasi-isometries into the group of self-mappings of . To check that  is continuous, note that if  and  are geodesic rays that are uniformly close on , within a distance , then  and  lie within a distance   on , so that  and  lie within a distance ; hence on a smaller interval ,  and  lie within a distance  by convexity. On CAT(-1) spaces, a finer version of continuity asserts that  is a quasi-Möbius mapping with respect to a natural class of metric on , 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  is a quasi-isometry of the unit disk such that  is the identity. The assumption and the Morse lemma implies that if  is a geodesic line, then  lies in an -neighbourhood of . Now take a second geodesic line  such that  and  intersect orthogonally at a given point in . Then  lies in the intersection of -neighbourhoods of  and . Applying a Möbius transformation, it can be assumed that  is at the origin of the unit disk and the geodesics are the real and imaginary axes. By convexity, the -neighbourhoods of these axes intersect in a -neighbourhood of the origin: if  lies in both neighbourhoods, let  and  be the orthogonal projections of  onto the - and -axes; then  so taking projections onto the -axis, ; hence  . Hence , so that  is quasi-equivalent to the identity, as claimed. Cross ratio and distance between non-intersecting geodesic lines
Given two distinct points  on the unit circle or real axis there is a unique hyperbolic geodesic  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  in the extended complex plane their cross ratio is defined by

If  is a complex Möbius transformation then it leaves the cross ratio invariant: . Since the Möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number  in  such that  for a Möbius transformation . Since , ,  and  all appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of , ,  and , it suffices to consider permutations that fix , so only permute ,  and . The cross ratio transforms according to the anharmonic group of order 6 generated by the Möbius transformations sending  to  and . The other three transformations send  to , to  and to . Now let  be points on the unit circle or real axis in that order. Then the geodesics  and  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 , take . Then
 where . On the other hand, the geodesics  and  are the semicircles in the upper half plane of radius  and . The geodesic which cuts them orthogonally is the positive imaginary axis, so the distance between them is the hyperbolic distance between  and , . Let , then , so that there is a constant  such that, if , then

since  is bounded above and below in . Note that  are in order around the unit circle if and only if . 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  and  are the feet of the perpendiculars from  and  to the geodesic line , then . Since both sides are invariant under Möbius transformations, it suffices to check this in the case that , ,  and . In this case the geodesic line is the positive imaginary axis, right hand side equals ,  and . So the left hand side equals . Note that  and  are also the points where the incircles of the ideal triangles  and  touch . Proof of theorem
A homeomorphism  of the circle is quasisymmetric if there are constants  such that

It is quasi-Möbius is there are constants  such that

where

denotes the cross-ratio. It is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition. If  is quasisymmetric then it is also quasi-Möbius, with  and : this follows by multiplying the first inequality for  and . Conversely any quasi-Möbius homeomorphism  is quasisymmetric. To see this, it can be first be checked that  (and hence ) is Hölder continuous. Let  be the set of cube roots of unity, so that if  in , then . To prove a Hölder estimate, it can be assumed that  is uniformly small. Then both  and  are greater than a fixed distance away from  in  with , so the estimate follows by applying the quasi-Möbius inequality to . To verify that  is quasisymmetric, it suffices to find a uniform upper bound for  in the case of a triple with , uniformly small. In this case there is a point  at a distance greater than 1 from ,  and . Applying the quasi-Möbius inequality to , ,  and  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  then there are constants  such that for  distinct points on the unit circle

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

To prove this, it suffices to show that . From the previous section it suffices show . This follows from the fact that the images under  of  and  lie within -neighbourhoods of  and ; the minimal distance can be estimated using the quasi-isometry constants for  applied to the points on  and 
realising . Adjusting  and  if necessary, the inequality above applies also to . Replacing , ,  and  by their images under , it follows that

if , ,  and  are in order on the unit circle. Hence the same inequalities are valid for the three cyclic of the quadruple . If  and  are switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if  and  are switched. If both pairs are switched, the cross ratio remains unaltered. Hence the inequalities are also valid in this case. Finally if  and  are interchanged, the cross ratio changes from  to , 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 , ,  and , so that  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  and the special theory for the disk generalises completely to any proper CAT(-1) space . 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  such that has these points as the limits . The theory applies equally well to any CAT() space with  since these arise by scaling the metric on a CAT(-1) space by . On the hyperbolic unit disk  quasi-isometries of  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. Example: Poincaré disk

Applications in percolation theory

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

References

, Appendix. Geometry