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.