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-decreasing 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 ,