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".
is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function g(t) = d(y,γ(t))2 − t2 is convex.
By the Lipschitz condition r = |h(y) − h(v)| ≤ R. To prove the assertion, it suffices to show that R = r, i.e. d(y,v) = r. On the other hand, h is the uniform limit on any closed ball of functions hn.
Then X is metrisable, since Y is, and contains X as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic.
When X is a Hadamard space, Gromov's ideal boundary ∂X = X \ X can be realised explicitly as "asymptotic limits" of geodesic rays using Busemann functions.
The Lipschitz condition on h then immediately implies u and v must be the unique points in B(y,r) maximizing and minimizing h. Now suppose that yn tends to y.
The assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows.
Note that this argument could be shortened using the fact that two Busemann functions hγ and hδ differ by a constant if and only if the corresponding geodesic rays satisfy supt ≥ 0 d(γ(t),δ(t)) < ∞.
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 X, 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.
This is the topology induced by the natural map of C(X) into the direct product of the Banach spaces C(XN).
Then X is compact (metrisable) and contains X as an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic.
To check that the compactification X = X(x0) is independent of the basepoint, it suffices to show that k(x) = d(x,x0) − d(x,x1) extends to a continuous function on X.
[18][19] 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.
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 P(z) = i | z | and | z | / Im z = cosh d(z,Pz).
The Hausdorff distance between the images of Γ and Δ is evidently bounded by a constant R2 depending only on λ and ε.
The estimate above shows that for fixed R > 0 and N sufficiently large, (ΓN) is a Cauchy sequence in C([0,R],X) with the uniform metric.
Before discussing CAT(-1) spaces, this section will describe the Efremovich–Tikhomirova theorem for the unit disk D with the Poincaré metric.
It asserts that quasi-isometries of D extend to quasi-Möbius homeomorphisms of the unit disk with the Euclidean metric.
Their original theorem was proved in a slightly less general and less precise form in Efremovich & Tikhomirova (1964) and applied to bi-Lipschitz homeomorphisms of the unit disk for the Poincaré metric;[27] earlier, in the posthumous paper Mori (1957), 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).
[28] If X 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 X extends uniquely to the boundary.
Applying a Möbius transformation, it can be assumed that a is at the origin of the unit disk and the geodesics are the real and imaginary axes.
Given two distinct points z, w on the unit circle or real axis there is a unique hyperbolic geodesic [z,w] joining them.
To compare the cross ratio and the distance between geodesics, Möbius invariance allows the calculation to be reduced to a symmetric configuration.
In this case the geodesic line is the positive imaginary axis, right hand side equals | log | x ||, p = | x | i and q = i.
Note that p and q are also the points where the incircles of the ideal triangles abc and abd touch ab.
This follows from the fact that the images under f of [a,b] and [c,d] lie within h-neighbourhoods of [F(a),F(b)] and [F(c),F(d)]; the minimal distance can be estimated using the quasi-isometry constants for f applied to the points on [a,b] and [c,d] realising d([a,b],[c,d]).
It is easy to check that using these transformations the inequalities are valid for all possible permutations of a, b, c and d, so that F and its inverse are quasi-Möbius homeomorphisms.
On the hyperbolic unit disk D quasi-isometries of D 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.