{"history":[],"article":{"id":"c521887c-6d7c-46ed-a1bd-776f4bf672ed","title":"Busemann function","slug":"busemann-function","content":":::summary\n:::\n\n# \nIn [geometric topology](https://bliptext.com/articles/geometric-topology), **Busemann functions** are used to study the large-scale geometry of geodesics in [Hadamard spaces](https://bliptext.com/articles/hadamard-space) and in particular [Hadamard manifolds](https://bliptext.com/articles/hadamard-manifold) ([simply connected](https://bliptext.com/articles/simply-connected) complete [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) of nonpositive curvature).\nThey are named after [Herbert Busemann](https://bliptext.com/articles/herbert-busemann), who introduced them; he gave an extensive treatment of the topic in his 1955 book \"The geometry of geodesics\".\n\n# Definition and elementary properties\nLet (X,d) be a [metric space](https://bliptext.com/articles/metric-space).\nA [geodesic ray](https://bliptext.com/articles/ray-geometry) is a path which minimizes distance everywhere along its length.\ni.e., for all ,\nEquivalently, a ray is an isometry from the \"canonical ray\" (the set [0,\\infty) equipped with the Euclidean metric) into the metric space X.\nGiven a ray γ, the Busemann function is defined by\nThus, when t is very large, the distance is approximately equal to.\nGiven 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 s\\le t,\nIt is immediate from the triangle inequality that\nso that B_\\gamma is uniformly continuous.\nMore specifically, the above estimate above shows that\nBy [Dini's theorem](https://bliptext.com/articles/dini-s-theorem), the functions tend to B_\\gamma(x) uniformly on compact sets as t tends to infinity.\n\n# Example: Poincaré disk\nLet D be the unit disk in the complex plane with the [Poincaré metric](https://bliptext.com/articles/poincare-disk-model)\nThen, for |z| < 1 and |\\zeta| = 1, the Busemann function is given by\nwhere the term in brackets on the right hand side is the [Poisson kernel](https://bliptext.com/articles/poisson-kernel) for the unit disk and \\zeta corresponds to the radial geodesic \\gamma from the origin towards \\zeta,\n. The computation of d(x,y) can be reduced to that of, since the metric is invariant under [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) in SU(1,1); the geodesics through 0 have the form where g_t is the 1-parameter subgroup of SU(1,1),\nThe formula above also completely determines the Busemann function by Möbius invariance.\n\n# Busemann functions on a Hadamard space\nIn a [Hadamard space](https://bliptext.com/articles/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].\nExplicitly this means that if z(s) is the point which divides [x,y] in the ratio\ns : (1 − s)\n, then.\nFor 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.\nSince the Busemann function B_\\gamma is the pointwise limit of F_t,\nLet\nh(t) = d(y,γ(t)) − t = Ft(y)\n. Since \\gamma (t) is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function\ng(t) = d(y,γ(t))2 − t2\nis convex.\nHence for\n0< s < t\nThus\nso that\nLetting t tend to ∞, it follows that\nso convergence is uniform on bounded sets.\nNote that the inequality above for (together with its proof) also holds for geodesic segments: if\nγ(t)\nis a geodesic segment starting at\nx\nand parametrised by arclength then\nNext suppose that\nx, y\nare points in a Hadamard space, and let\nδ(s)\nbe the geodesic through\nx\nwith\nδ(0) = y\nand\nδ(t) = x\n, where\nt = d(x,y)\n. This geodesic cuts the boundary of the closed ball\nB(y,r)\nat the point\nδ(r)\n. Thus if\nd(x,y) > r\n, there is a point\nv\nwith\nd(y,v) = r\nsuch that\nd(x,v) = d(x,y) − r\n.\nThis condition persists for Busemann functions.\nThe 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](https://bliptext.com/articles/orthogonal-projection) in a [Hilbert space](https://bliptext.com/articles/hilbert-space): if\nC\nis a closed convex set in a Hadamard space\nX\n, then every point\nx\nin\nX\nhas a unique closest point\nP(x) ≡ PC(x)\nin\nC\nand\nd(P(x),P(y)) ≤ d(x,y)\na = P(x)\nis uniquely determined by the property that, for\ny\nin\nC\n,\nso that the angle at\na\nin the Euclidean [comparison triangle](https://bliptext.com/articles/comparison-triangle) for\na,x,y\nis greater than or equal to\nπ/2\n.\nh\nis a Busemann function on a Hadamard space, then, given\ny\nin\nX\nand\nr > 0\n, there is a unique point\nv\nwith\nd(y,v) = r\nsuch that\nh(v) = h(y) − r\n. For fixed\nr > 0\n, the point\nv\nis the closest point of\ny\nto the closed convex\nC\nset of points\nu\nsuch that\nh(u) ≤ h(y) − r\nand therefore depends continuously on\ny\n.'''\nLet\nv\nbe the closest point to\ny\nin ''\nC\n''. Then\nh(v) = h(y) − r\nand so ''\nh\n is minimised by \nv\n'' in\nB(y,R)\nwhere\nR = d(y,v) and v\nis the unique point where ''\nh\n'' is minimised.\nBy the Lipschitz condition\nh(y) − h(v)\n. To prove the assertion, it suffices to show that\nR = r\n, i.e.\nd(y,v) = r\n. On the other hand, ''\nh\n'' is the uniform limit on any closed ball of functions\nhn\n. On\nB(y,r)\n, these are minimised by points\nvn\nwith\nhn(vn) = hn(y) − r\n. Hence the infimum of ''\nh\n'' on\nB(y,r)\nis\nh(y) − r\nand\nh(vn)\ntends to\nh(y) − r\n. Thus\nh(vn) = h(y) − rn\nwith\nrn ≤ r\nand\nrn\ntending towards ''\nr\n''. Let\nun\nbe the closest point to\ny\nwith\nh(un) ≤ h(y) − rn\n. Let\nRn = d(y,un) ≤ r\n. Then\nh(un) = h(y) − rn\n, and, by the Lipschitz condition on\nh\n,\nRn ≥ rn\n. In particular\nRn\ntends to ''\nr\n''. Passing to a subsequence if necessary it can be assumed that\nrn\nand\nRn\nare both increasing (to ''\nr\n''). The inequality for convex optimisation implies that for\nn > m\n.\nso that\nun\nis a Cauchy sequence.\nIf ''\nu\n'' is its limit, then\nd(y,u) = r\nand\nh(u) = h(y) − r\n. By uniqueness it follows that\nu = v\nand hence\nd(y,v) = r\n, as required.\n**Uniform limits.** The above argument proves more generally that if\nd(xn,x0)\ntends to infinity and the functions\nhn(x) = d(x,xn) – d(xn,x0)\ntend uniformly on bounded sets to\nh(x)\n, then ''\nh\n is convex, Lipschitz with Lipschitz constant 1 and, given \ny\n in \nX\n'' and\nr > 0\n, there is a unique point ''\nv\n'' with\nd(y,v) = r\nsuch that\nh(v) = h(y) − r\n. If on the other hand the sequence\n(xn)\nis bounded, then the terms all lie in some closed ball and uniform convergence there implies that\n(xn)\nis a Cauchy sequence so converges to some\nx∞\nin ''\nX\n''. So\nhn\ntends uniformly to\nh∞(x) = d(x,x∞) – d(x∞,x0)\n, a function of the same form.\nThe 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.\n**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.\nIn particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of '''\nR\n''' or even a semi-infinite interval\n[0,∞)\n. The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space.\nh\nis a convex function, Lipschitz with constant 1 and\nh\nassumes its minimum on any closed ball centred on\ny\nwith radius\nr\nat a unique point\nv\non the boundary with\nh(v) = h(y) − r\n, then for each\ny\nin\nX\nthere is a unique geodesic ray\nδ\nsuch that\nδ(0) = y\nand\nδ\ncuts each closed convex set\nh ≤ h(y) – r\nwith\nr > 0\nat\nδ(r)\n, so that\nh(δ(t)) = h(y) – t\n. In particular this holds for each Busemann function.'''\nThe third condition implies that ''\nv\n is the closest point to \ny\n'' in the closed convex set\nCr\nof points u such that\nh(u) ≤ h(y) – r\n. Let\nδ(t)\nfor\n0 ≤ t ≤ r\nbe the geodesic joining ''\ny\n to \nv\n''. Then\nk(t) = h(δ(t)) - h(y)\nis a convex Lipschitz function on\n[0,r]\nwith Lipschitz constant 1 satisfying\nk(t) ≤ – t\nand\nk(0) = 0\nand\nk(r) = –r\n. So ''\nk\n'' vanishes everywhere, since if\n0 < s < r, k(s) ≤ –s\nand\n. Hence\nh(δ(t)) = h(y) – t\n. By uniqueness it follows that\nδ(t)\nis the closest point to\ny\nin\nCt\nand that it is the unique point minimising ''\nh\n'' in\nB(y,t)\n. Uniqueness implies that these geodesics segments coincide for arbitrary ''\nr\n'' and therefore that\nδ\nextends to a geodesic ray with the stated property.\nh = hγ\n, then the geodesic ray\nδ\nstarting at\ny\nsatisfies'''.\n'''If\nδ1\nis another ray starting at\ny\nwith** **then\nδ1 = δ\n.'''\nTo prove the first assertion, it is enough to check this for ''\nt\n'' sufficiently large.\nIn that case\nγ(t)\nand\nδ(t − h(y))\nare the projections of ''\nx\n and \ny\n'' onto the closed convex set\nh ≤ −t\n. Therefore,\nd(γ(t),δ(t − h(y))) ≤ d(x,y)\n. Hence\nd(γ(t),δ(t)) ≤ d(γ(t),δ(t − h(y))) + d(δ(t − h(y)),δ(t)) ≤ d(x,y) +\n. The second assertion follows because\nd(δ1(t),δ(t))\nis convex and bounded on\n[0,∞)\n, so, if it vanishes at\nt = 0\n, must vanish everywhere.\nh\nis a continuous convex function and for each\ny\nin\nX\nthere is a unique geodesic ray\nδ\nsuch that\nδ(0) = y\nand\nδ\ncuts each closed convex set\nh ≤ h(y) – r with r > 0\nat\nδ(r)\n, so that\nh(δ(t)) = h(y) – t\nh\nis a Busemann function.\nh − hδ\nis a constant function.'''\nLet\nCr\nbe the closed convex set of points ''\nz\n'' with\nh(z) ≤ −r\n. Since ''\nX\n is a Hadamard space for every point \ny\n in \nX\n'' there is a unique closest point\nPr(y)\nto\ny\nin\nCr\n. It depends continuously on ''\ny\n and if \ny\n'' lies outside\nCr\n, then\nPr(y)\nlies on the hypersurface\nh(z) = − r\n—the boundary ∂\nCr\nof\nCr\n—and\nPr(y)\nsatisfies the inequality of convex optimisation.\nLet\nδ(s)\nbe the geodesic ray starting at ''\ny\n''.\nFix ''\nx\n in \nX\n''. Let\nγ(s)\nbe the geodesic ray starting at ''\nx\n''. Let\ng(z) = hγ(z)\n, the Busemann function for\nγ\nwith base point ''\nx\n''. In particular\ng(x) = 0\n. It suffices to show that\ng = h – h(y)1\n. Now take ''\ny\n'' with\nh(x) = h(y)\nand let\nδ(t)\nbe the geodesic ray starting at ''\ny\n corresponding to \nh\n''. Then\nOn the other hand, for any four points ''\na\n, \nb\n, \nc\n, \nd\n'' in a Hadamard space, the following quadrilateral inequality of [Reshetnyak](https://bliptext.com/articles/yurii-reshetnyak) holds:\nSetting\na = x\n,\nb = y\n,\nc = γ(t)\n,\nd = δ(t)\n, it follows that\nso that\nHence\nhγ(y) = 0\n. Similarly\nhδ(x) = 0\n. Hence\nhγ(y) = 0\non the level surface of ''\nh\n containing \nx\n''. Now for\nt ≥ 0\nand ''\nz\n in \nX\n'', let\nαt(z) = γ1(t)\nthe geodesic ray starting at ''\nz\n''. Then\nαs + t = αs ∘ αt\nand\nh ∘ αt = h − t\n. Moreover, by boundedness,\nd(αt(u),αt(v)) ≤ d(u,v)\n. The flow\nαs\ncan be used to transport this result to all the level surfaces of ''\nh\n''. For general\ny1\n, if\nh(y1) < h(x)\n, take\ns > 0\nsuch that\nh(αs(x)) = h(y1)\nand set\nx1 = αs(x)\n. Then\nhγ 1 (y1) = 0\n, where\nγ1(t) = αt(x1) = γ(s + t)\n. But then\nhγ 1 = hγ – s\n, so that\nhγ(y1) = s\n. Hence\ng(y1) = s = h((αs(x)) – h(x) = h(y1) – h(x)\n, as required.\nSimilarly if\nh(y1) > h(x)\n, take\ns > 0\nsuch that\nh(αs(y1)) = h(x)\n. Let\ny = αs(y1)\n. Then\nhγ(y) = 0\n, so\nhγ(y1) = –s\n. Hence\ng(y1) = –s = h(y1) – h(x)\n, as required.\nFinally there are necessary and sufficient conditions for two geodesics to define the same Busemann function up to constant:\nSuppose firstly that\nγ\nand\nδ\nare two geodesic rays with Busemann functions differing by a constant.\nShifting the argument of one of the geodesics by a constant, it may be assumed that\nBγ = Bδ = B\n, say.\nLet ''\nC\n'' be the closed convex set on which\nB(x) ≤ −r\n. Then\nB(γ(t)) = Bγ(γ(t)) = −t\nand similarly\nB(δ(t)) = − t\n. Then for\ns ≤ r\n, the points\nγ(s)\nand\nδ(s)\nhave closest points\nγ(r)\nand\nδ(r)\nin ''\nC\n'', so that\nd(γ(r), δ(r)) ≤ d(γ(s), δ(s))\n. Hence\nsupt ≥ 0 d(γ(t), δ(t)) < ∞\n.\nNow suppose that\nsupt ≥ 0 d(γ1(t), γ2(t)) < ∞\n. Let\nδi(t)\nbe the geodesic ray starting at ''\ny\n'' associated with\nhγ i\n. Then\nsupt ≥ 0 d(γi(t), δi(t)) < ∞\n. Hence\nsupt ≥ 0 d(δ1(t), δ2(t)) < ∞\n. Since\nδ1\nand\nδ2\nboth start at ''\ny\n'', it follows that\nδ1(t) ≡ δ2(t)\n. By the previous result\nhγ i\nand\nhδ i\ndiffer by a constant; so\nhγ 1\nand\nhγ 2\ndiffer by a constant.\nTo summarise, the above results give the following characterisation of Busemann functions on a Hadamard space:\n**THEOREM.** '''On a Hadamard space, the following conditions on a function\nf\nare equivalent:'''\nh\nis a Busemann function.'''\nh\nis a convex function, Lipschitz with constant\n1\nand\nh\nassumes its minimum on any closed ball centred on\ny\nwith radius\nr\nat a unique point\nv\non the boundary with\nh(v) = h(y) − r\n.'''\nh\nis a continuous convex function and for each\ny\nin\nX\nthere is a unique geodesic ray\nδ\nsuch that\nδ(0) = y\nand, for any\nr > 0\n, the ray\nδ\ncuts each closed convex set\nh ≤ h(y) – r\nat\nδ(r)\n.'''\n\n# Bordification of a Hadamard space\nIn the previous section it was shown that if\nX\nis a Hadamard space and\nx0\nis a fixed point in\nX\nthen the union of the space of Busemann functions vanishing at\nx0\nand the space of functions\nhy(x) = d(x,y) − d(x0,y)\nis closed under taking uniform limits on bounded sets.\nThis result can be formalised in the notion of **bordification** of\nX\n. In this topology, the points\nxn\ntend to a geodesic ray\nγ\nstarting at\nx0\nif and only if\nd(x0,xn)\ntends to\n∞\nand for\nt > 0\narbitrarily large the sequence obtained by taking the point on each segment\n[x0,xn]\nat a distance\nt\nfrom\nx0\ntends to\nγ(t)\n.\nIf\nX\nis a metric space, Gromov's bordification can be defined as follows.\nFix a point\nx0\nin\nX\nand let\nXN = B(x0,N)\n. Let\nY = C(X)\nbe the space of Lipschitz continuous functions on\nX\n, i.e. those for which\nfor some constant\nA > 0\n. The space\nY\ncan be topologised by the seminorms\nf\n, the topology of uniform convergence on bounded sets.\nThe seminorms are finite by the Lipschitz conditions.\nThis is the topology induced by the natural map of\nC(X)\ninto the direct product of the Banach spaces\nCb(XN)\nof continuous bounded functions on\nXN\n. It is give by the metric\nD(f,g) = Σ 2−N f − gN(1 +f − gN)−1\n.\nThe space\nX\nis embedded into\nY\nby sending\nx\nto the function\nfx(y) = d(y,x) – d(x0,x)\n. Let\nX\nbe the closure of\nX\nin\nY\n. Then\nX\nis metrisable, since\nY\nis, and contains\nX\nas an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic.\nLet\nh(x) = (d(x,x0) + 1)−1\n. Then\nh\nlies in\nC0(X)\n. It is non-zero on\nX\nand vanishes only at\n∞\n. Hence it extends to a continuous function on\nX\nwith zero set\nX \\ X\n. It follows that\nX \\ X\nis closed in\nX\n, as required.\nTo check that\nX = X(x0)\nis independent of the basepoint, it suffices to show that\nk(x) = d(x,x0) − d(x,x1)\nextends to a continuous function on\nX\n. But\nk(x) = fx(x1)\n, so, for\ng\nin\nX\n,\nk(g) = g(x1)\n. Hence the correspondence between the compactifications for\nx0\nand\nx1\nis given by sending\ng\nin\nX(x0)\nto\ng + g(x1)1\nin\nX(x1)\n.\nWhen\nX\nis a Hadamard space, Gromov's ideal boundary\n∂X = X \\ X\ncan be realised explicitly as \"asymptotic limits\" of geodesic rays using Busemann functions.\nIf\nxn\nis an unbounded sequence in\nX\nwith\nhn(x) = d(x,xn) − d(xn,x0)\ntending to\nh\nin\nY\n, then\nh\nvanishes at\nx0\n, is convex, Lipschitz with Lipschitz constant\n1\nand has minimum\nh(y) − r\non any closed ball\nB(y,r)\n. Hence\nh\nis a Busemann function\nBγ\ncorresponding to a unique geodesic ray\nγ\nstarting at\nx0\n.\nOn the other hand,\nhn\ntends to\nBγ\nuniformly on bounded sets if and only if\nd(x0,xn)\ntends to\n∞\nand for\nt > 0\narbitrarily large the sequence obtained by taking the point on each segment\n[x0,xn]\nat a distance\nt\nfrom\nx0\ntends to\nγ(t)\n. For\nd(x0,xn) ≥ t\n, let\nxn(t)\nbe the point in\n[x0,xn]\nwith\nd(x0,xn(t)) = t\n. Suppose first that\nhn\ntends to\nBγ\nuniformly on\nB(x0,R)\n. Then for\nt ≤ R\n,\nhn(γ(t)) – Bγ(γ(t))\n. This is a convex function.\nIt vanishes as\nt = 0\nand hence is increasing.\nSo it is maximised at\nt = R\n. So for each\nt\n,\ntends towards 0.\nLet\na = X0\n,\nb = γ(t)\nand\nc = xn\n. Then\nd(c,a) – d(c,b)\nis close to\nd(a,b)\nwith\nd(c,a)\nlarge.\nHence in the Euclidean comparison triangle\nCA - CB\nis close to\nAB\nwith\nCA\nlarge.\nSo the angle at\nA\nis small.\nSo the point\nD\non\nAC\nat the same distance as\nAB\nlies close to\nB\n. Hence, by the first comparison theorem for geodesic triangles,\nd(xn(t),γ(t))\nis small.\nConversely suppose that for fixed\nt\nand\nn\nsufficiently large\nd(xn(t),γ(t))\ntends to 0.\nThen from the above\nFs(y) = d(y,γ(s)) – s\nsatisfies\nso it suffices show that on any bounded set\nhn(y) = d(y,xn) – d(x0,xn)\nis uniformly close to\nFs(y)\nfor\nn\nsufficiently large.\nFor a fixed ball\nB(x0,R)\n, fix\ns\nso that\nR2/s ≤ ε\n. The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since\nHence, if\ny\nin\nB(x0,R)\nand\nn\nis sufficiently large that\nd(xn(s),γ(s)) ≤ ε\n, then\n\n# Busemann functions on a Hadamard manifold\nSuppose that\nx, y\nare points in a Hadamard manifold and let\nγ(s)\nbe the geodesic through\nx\nwith\nγ(0) = y\n. This geodesic cuts the boundary of the closed ball\nB(y,r)\nat the two points\nγ(±r)\n. Thus if\nd(x,y) > r\n, there are points\nu, v\nwith\nd(y,u) = d(y,v) = r\nsuch that\nd(x,u) − d(x,v)\n. By continuity this condition persists for Busemann functions:\nh\nis a Busemann function on a Hadamard manifold, then, given\ny\nin\nX\nand\nr > 0\n, there are unique points\nu\n,\nv\nwith\nd(y,u) = d(y,v) = r\nsuch that\nh(u) = h(y) + r\nand\nh(v) = h(y) − r\n. For fixed\nr > 0\n, the points\nu\nand\nv\ndepend continuously on\ny\n.'''\nTaking a sequence\ntn\ntending to\n∞\nand\nhn = Ft n\n, there are points\nun\nand\nvn\nwhich satisfy these conditions for\nhn\nfor\nn\nsufficiently large.\nPassing to a subsequence if necessary, it can be assumed that\nun\nand\nvn\ntend to\nu\nand\nv\n. By continuity these points satisfy the conditions for\nh\n. To prove uniqueness, note that by compactness\nh\nassumes its maximum and minimum on\nB(y,r)\n. The Lipschitz condition shows that the values of\nh\nthere differ by at most\n2r\n. Hence\nh\nis minimized at\nv\nand maximized at\nu\n. On the other hand,\nd(u,v) = 2r\nand for\nu\nand\nv\nthe points\nv\nand\nu\nare the unique points in\nB(y,r)\nmaximizing this distance.\nThe Lipschitz condition on\nh\nthen immediately implies\nu\nand\nv\nmust be the unique points in\nB(y,r)\nmaximizing and minimizing\nh\n. Now suppose that\nyn\ntends to\ny\n. Then the corresponding points\nun\nand\nvn\nlie in a closed ball so admit convergent subsequences.\nBut by uniqueness of\nu\nand\nv\nany such subsequences must tend to\nu\nand\nv\n, so that\nun\nand\nvn\nmust tend to\nu\nand\nv\n, establishing continuity.\nThe above result holds more generally in a Hadamard space.\nh\nis a Busemann function on a Hadamard manifold, then\nh\nis continuously differentiable with\ndh(y) = 1\nfor all\ny\n.'''\nFrom the previous properties of\nh\n, for each\ny\nthere is a unique geodesic γ(t) parametrised by arclength with\nγ(0) = y\nsuch that\nh ∘ γ(t) = h(y) + t\n. It has the property that it cuts\n∂B(y,r)\nat\nt = ±r\nγ(r) = u\nand\nγ(–r) = v\n. The vector field\nVh\ndefined by the unit vector at\ny\nis continuous, because\nu\nis a continuous function of\ny\nand the map sending\n(x,v)\nto\n(x,expx v)\nis a diffeomorphism from\nTX\nonto\nX × X\nby the [Cartan-Hadamard theorem](https://bliptext.com/articles/cartan-hadamard-theorem).\nLet\nδ(s)\nbe another geodesic parametrised by arclength through\ny\nwith\nδ(0) = y\n. Then\ndh ∘ δ (0)/ ds =\n. Indeed, let\nH(x) = h(x) − h(y)\n, so that\nH(y) = 0\n. Then\nApplying this with\nx = u\nand\nv\n, it follows that for\ns > 0\nThe outer terms tend to as\ns\ntends to 0, so the middle term has the same limit, as claimed.\nA similar argument applies for\ns < 0\n.\nThe assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows.\nLet\nand, both unit vectors.\nThen for tangent vectors\np\nand\nq\nat\ny\nin the unit ball\nwith\nε\nuniformly bounded.\nLet\ns = t3\nand\nr = t2\n. Then\nThe right hand side here tends to\n(a,b)\nas\nt\ntends to 0 since\nThe same method works for the other terms.\nHence it follows that\nh\nis a\nC1\nfunction with\ndh\ndual to the vector field\nVh\n, so that\ndh(y) = 1\n. The vector field\nVh\nis thus the [gradient vector field](https://bliptext.com/articles/gradient-vector-field) for\nh\n. The geodesics through any point are the flow lines for the flow\nαt\nfor\nVh\n, so that\nαt\nis the [gradient flow](https://bliptext.com/articles/gradient-flow) for\nh\n.\n**THEOREM.** '''On a Hadamard manifold\nX\nthe following conditions on a continuous function\nh\nare equivalent:'''\nh\nis a Busemann function.'''\nh\nis a convex, Lipschitz function with constant 1, and for each\ny\nin\nX\nthere are points\nu±\nat a distance\nr\nfrom\ny\nsuch that\nh(u±) = h(y) ± r\n.'''\nh\nis a convex\nC1\nfunction with\ndh(x) ≡ 1\n.'''\nIt has already been proved that (1) implies (2).\nThe arguments above show [mutatis mutandi](https://bliptext.com/articles/mutatis-mutandi) that (2) implies (3).\nIt therefore remains to show that (3) implies (1).\nFix\nx\nin\nX\n. Let\nαt\nbe the gradient flow for\nh\n. It follows that\nh ∘ αt (x) = h(x) + t\nand that\nγ(t) = αt(x)\nis a geodesic through\nx\nparametrised by arclength with\nγ(0) = x\n. Indeed, if\ns < t\n, then\nso that\ns − t\n. Let\ng(y) = hγ(y)\n, the Busemann function for\nγ\nwith base point\nx\n. In particular\ng(x) = 0\n. To prove (1), it suffices to show that\ng = h – h(x)1\n.\nLet\nC(−r)\nbe the closed convex set of points\nz\nwith\nh(z) ≤ −r\n. Since\nX\nis a Hadamard space for every point\ny\nin\nX\nthere is a unique closest point\nPr(y)\nto\ny\nin\nC(-r)\n. It depends continuously on\ny\nand if\ny\nlies outside\nC(-r)\n, then\nPr(y)\nlies on the hypersurface\nh(z) = − r\n—the boundary\n∂C(–r)\nof\nC(–r)\n—and the geodesic from\ny\nto\nPr(y)\nis orthogonal to\n∂C(–r)\n. In this case the geodesic is just\nαt(y)\n. Indeed, the fact that\nαt\nis the gradient flow of\nh\nand the conditions\ndh(y) ≡ 1\nimply that the flow lines\nαt(y)\nare geodesics parametrised by arclength and cut the level curves of\nh\northogonally.\nTaking\ny\nwith\nh(y) = h(x)\nand\nt > 0\n,\nOn the other hand, for any four points\na\n,\nb\n,\nc\n,\nd\nin a Hadamard space, the following quadrilateral inequality of [Reshetnyak](https://bliptext.com/articles/yurii-reshetnyak) holds:\nSetting\na = x\n,\nb = y\n,\nc = αt(x)\n,\nd = αt(y)\n, it follows that\nso that\nHence\nhγ(y) = 0\non the level surface of\nh\ncontaining\nx\n. The flow\nαs\ncan be used to transport this result to all the level surfaces of\nh\n. For general\ny1\ntake\ns\nsuch that\nh(αs(x)) = h(y1)\nand set\nx1 = αs(x)\n. Then\nhγ 1 (y1) = 0\n, where\nγ1(t) = αt(x1) = γ(s + t)\n. But then\nhγ 1 = hγ – s\n, so that\nhγ(y1) = s\n. Hence\ng(y1) = s = h((αs(x)) – h(x) = h(y1) – h(x)\n, as required.\nNote that this argument could be shortened using the fact that two Busemann functions\nhγ\nand\nhδ\ndiffer by a constant if and only if the corresponding geodesic rays satisfy\nsupt ≥ 0 d(γ(t),δ(t)) < ∞\n. Indeed, all the geodesics defined by the flow\nαt\nsatisfy the latter condition, so differ by constants.\nSince along any of these geodesics\nh\nis linear with derivative 1,\nh\nmust differ from these Busemann functions by constants.\n\n# Compactification of a proper Hadamard space\ndefined a compactification of a [Hadamard manifold](https://bliptext.com/articles/hadamard-manifold)\nX\nwhich uses Busemann functions.\nTheir construction, which can be extended more generally to proper (i.e. locally compact) [Hadamard spaces](https://bliptext.com/articles/hadamard-space), 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](https://bliptext.com/articles/proper-metric-space)\nX\n, those for which every closed ball is compact.\nNote that, since any Cauchy sequence is contained in a closed ball, any proper metric space is automatically complete.\nThe ideal boundary is a special case of the ideal boundary for a metric space.\nIn 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.\nIf\nX\nis a proper metric space, Gromov's compactification can be defined as follows.\nFix a point\nx0\nin\nX\nand let\nXN = B(x0,N)\n. Let\nY = C(X)\nbe the space of Lipschitz continuous functions on\nX\n, .e. those for which\nfor some constant\nA > 0\n. The space\nY\ncan be topologised by the seminorms\nf\n, the topology of uniform convergence on compacta.\nThis is the topology induced by the natural map of C(X) into the direct product of the Banach spaces\nC(XN)\n. It is give by the metric\nD(f,g) = Σ 2−N f − g N(1 + f − g N)−1\n.\nThe space\nX\nis embedded into\nY\nby sending\nx\nto the function\nfx(y) = d(y,x) – d(x0,x)\n. Let\nX\nbe the closure of\nX\nin\nY\n. Then\nX\nis compact (metrisable) and contains\nX\nas an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic.\nCompactness follows from the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) since the image in\nC(XN)\nis [equicontinuous](https://bliptext.com/articles/equicontinuous) and uniformly bounded in norm by\nN\n. Let\nxn\nbe a sequence in\nX ⊂ X\ntending to\ny\nin\nX \\ X\n. Then all but finitely many terms must lie outside\nXN\nsince\nXN\nis compact, so that any subsequence would converge to a point in\nXN\nxn\nmust be unbounded in\nX\n. Let\nh(x) = (d(x,x0) + 1)−1\n. Then\nh\nlies in\nC0(X)\n. It is non-zero on\nX\nand vanishes only at\n∞\n. Hence it extends to a continuous function on\nX\nwith zero set\nX \\ X\n. It follows that\nX \\ X\nis closed in\nX\n, as required.\nTo check that the compactification\nX = X(x0)\nis independent of the basepoint, it suffices to show that\nk(x) = d(x,x0) − d(x,x1)\nextends to a continuous function on\nX\n. But\nk(x) = fx(x1)\n, so, for\ng\nin\nX\n,\nk(g) = g(x1)\n. Hence the correspondence between the compactifications for\nx0\nand\nx1\nis given by sending\ng\nin\nX(x0)\nto\ng + g(x1)1\nin\nX(x1)\n.\nWhen\nX\nis a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary\n∂X = X \\ X\ncan be realised explicitly as \"asymptotic limits\" of geodesics by using Busemann functions.\nFixing a base point\nx0\n, there is a unique geodesic\nγ(t)\nparametrised by arclength such that\nγ(0) = x0\nand is a given unit vector.\nIf\nBγ\nis the corresponding Busemann function, then\nBγ\nlies in\n∂X(x0)\nand induces a homeomorphism of the unit\n(n − 1)\n-sphere onto\n∂X(x0)\n, sending to\nBγ\n.\n\n# Quasigeodesics in the Poincaré disk, CAT(-1) and hyperbolic spaces\n\n## Morse–Mostow lemma\nIn 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.\nThis structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays.\nQuasigeodesics were first studied for negatively curved surfaces—in particular the hyperbolic upper halfplane and unit disk—by [Morse](https://bliptext.com/articles/marston-morse) and generalised to negatively curved [symmetric spaces](https://bliptext.com/articles/symmetric-space) by [Mostow](https://bliptext.com/articles/george-mostow), for his work on the [rigidity of discrete groups](https://bliptext.com/articles/mostow-rigidity-theorem).\nThe basic result is the **Morse–Mostow lemma** on the stability of geodesics.\nBy definition a **[quasigeodesic](https://bliptext.com/articles/quasigeodesic)** Γ defined on an interval\n[a,b]\nwith\n−∞ ≤ a < b ≤ ∞\nis a map\nΓ(t)\ninto a metric space, not necessarily continuous, for which there are constants\nλ ≥ 1\nand\nε > 0\nsuch that for all\ns\nand\nt\nThe following result is essentially due to [Marston Morse](https://bliptext.com/articles/marston-morse) (1924).\n**Morse's lemma on stability of geodesics.** In the hyperbolic disk there is a constant\nR\ndepending on\nλ\nand\nε\nsuch that any quasigeodesic segment\nΓ\ndefined on a finite interval\n[a,b]\nis within a [Hausdorff distance](https://bliptext.com/articles/hausdorff-distance)\nR\nof the geodesic segment\n[Γ(a),Γ(b)]\n.\n\n## Classical proof for Poincaré disk\nThe 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.\nΓ\ncan be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from\nΓ\nless than\nc = (2λ2 + 1)ε\nΓ\nis defined into equal subintervals of length\n2λε\nand take the geodesics between the images under\nΓ\nof the endpoints of the subintervals.\nSince\nΔ\nis piecewise geodesic,\nΔ\nis Lipschitz continuous with constant\nλ1\n,\nd(Δ(s),Δ(t)) ≤ λ1\n, where\nλ1 ≤ λ + ε\n. The lower bound is automatic at the endpoints of intervals.\nBy construction the other values differ from these by a uniformly bounded depending only on\nλ\nand\nε\nγ\nis a piecewise smooth curve segment lying outside an\ns\n-neighbourhood of a geodesic line and\nP\nis the orthogonal projection onto the geodesic line then:\nApplying 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\nz\nand\nz\n. Hence the hypothesis implies\n, so that\nh > 0\ndepending only on\nλ\nand\nε\nsuch that\nΓ[a,b]\nlies within an\nh\n-neighbourhood of the geodesic segment\n[Γ(a),Γ(b)]\n.\nLet\nγ(t)\nbe the geodesic line containing the geodesic segment\n[Γ(a),Γ(b)]\n. Then there is a constant\nh > 0\ndepending only on\nλ\nand\nε\nsuch that\nh\n-neighbourhood\nΓ[a,b]\nlies within an\nh\n-neighbourhood of\nγ(R)\n. Indeed for any\ns > 0\n, the subset of\n[a,b]\nfor which\nΓ(t)\nlies outside the closure of the\ns\n-neighbourhood of\nγ(R)\nis open, so a countable union of open intervals\n(c,d)\n. Then\nλ\nand\ns + s1\nof\nγ(R)\n. To deduce the assertion, note that the subset of\n[a,b]\nfor which\nΓ(t)\nlies outside the closure of the\ns\n-neighbourhood of\n[Γ(a),Γ(b)] ⊂ γ(R)\nis open, so a union of intervals\n(c,d)\nwith\nΓ(c)\nand\nΓ(d)\nboth at a distance\ns + s1\nfrom either\nΓ(a)\nor\nΓ(b)\n. Then\nh\ngreater than\ns +s1 + s2\n.\nh > 0\ndepending only on\nλ\nand\nε\nsuch that the geodesic segment\n[Γ(a),Γ(b)]\nlies within an\nh\n-neighbourhood of\nΓ[a,b]\n.\nEvery point of\nΓ[a,b]\nlies within a distance\nh\nof\n[Γ(a),Γ(b)]\n. Thus orthogonal projection\nP\ncarries each point of\nΓ[a,b]\nonto a point in the closed convex set\n[Γ(a),Γ(b)]\nat a distance less than\nh\n. Since\nP\nis continuous and\nΓ[a,b]\nconnected, the map\nP\nmust be onto since the image contains the endpoints of\n[Γ(a),Γ(b)]\n. But then every point of\n[Γ(a),Γ(b)]\nis within a distance\nh\nof a point of\nΓ[a,b]\n.\n\n## Gromov's proof for Poincaré disk\nThe 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.\nThere is also a generalisation for the more general class of [hyperbolic metric spaces](https://bliptext.com/articles/hyperbolic-metric-space) due to Gromov.\nGromov'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](https://bliptext.com/articles/mutatis-mutandi) to CAT(-1) or hyperbolic metric spaces.\nSince this is a large-scale phenomenon, it is enough to check that any maps\nΔ\nfrom\nfor any\nN > 0\nto the disk satisfying the inequalities is within a Hausdorff distance\nR1\nof the geodesic segment\n[Δ(0),Δ(N)]\n. For then translating it may be assumed without loss of generality\nΓ\nis defined on\n[0,r]\nwith\nr > 1\nand then, taking\nN = [r]\n(the integer part of\nr\n), the result can be applied to\nΔ\ndefined by\nΔ(i) = Γ(i)\n. The Hausdorff distance between the images of\nΓ\nand\nΔ\nis evidently bounded by a constant\nR2\ndepending only on\nλ\nand\nε\n.\nδ\nwhere\nδ = 2 log 3\n2 log 3\n. 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\nδ\n, it follows that every side of a geodesic triangle is contained in a\nδ\n-neighbourhood of the other two sides.\nA simple induction argument shows that a geodesic polygon with\n2k + 2\nvertices for\nk ≥ 0\nhas each side within a\n(k + 1)δ\nneighbourhood of the other sides (such a polygon is made by combining two geodesic polygons with\n2k−1 + 1\nsides along a common side). Hence if\nM ≤ 2k + 2\n, the same estimate holds for a polygon with\nM\nsides.\nyi = Δ(i)\nlet\nf(x) = min d(x,yi)\n, the largest radius for a closed ball centred on\nx\nwhich contains no\nyi\nin its interior.\nThis is a continuous function non-zero on\n[Δ(0),Δ(N)]\nso attains its maximum\nh\nat some point\nx\nin this segment.\nThen\n[Δ(0),Δ(N)]\nlies within an\nh1\n-neighbourhood of the image of\nΔ\nfor any\nh1 > h\n. It therefore suffices to find an upper bound for\nh\nindependent of\nN\n.\ny\nand\nz\nin the segment\n[Δ(0),Δ(N)]\nbefore and after\nx\nwith\nd(x,y) = 2h\nand\nd(x,z) = 2h\n(or an endpoint if it within a distance of less than\n2h\nfrom\nx\n). Then there are\ni, j\nwith\nd(y,Δ(i))\n,\nd(z,Δ(j)) ≤ h\n. Hence\nd(Δ(i),Δ(j)) ≤ 6h\n, so that\n. By the triangle inequality all points on the segments\n[y,Δ(i)]\nand\n[z,Δ(j)]\nare at a distance\n≥ h\nfrom\nx\n. Thus there is a finite sequence of points starting at\ny\nand ending at\nz\n, lying first on the segment\n[y,Δ(i)]\n, then proceeding through the points\nΔ(i), Δ(i+1), ..., Δ(j)\n, before taking the segment\n[Δ(j),z]\n. The successive points\nΔ(i), Δ(i+1), ..., Δ(j)\nare separated by a distance no greater than\nλ + ε\nand successive points on the geodesic segments can also be chosen to satisfy this condition.\nThe minimum number\nK\nof points in such a sequence satisfies\nK ≤\n. These points form a geodesic polygon, with\n[y,z]\nas one of the sides.\nTake\nL = [h/δ]\n, so that the\n(L − 1)δ\n-neighbourhood of\n[y,z]\ndoes not contain all the other sides of the polygon.\nHence, from the result above, it follows that\nK > 2L − 1 + 2\n. Hence\nh\nis uniformly bounded, independently of\nN\n, as claimed.\nΔ(i)\nlie within\nh1\nof the\n[Δ(0),Δ(N)]\n, the result follows.\nOtherwise the points which do not fall into maximal subsets\nS =\nwith\nr < s\n. Thus points in\n[Δ(0),Δ(N)]\nhave a point\nΔ(i)\nwith\ni\nin the complement of\nS\nwithin a distance of\nh1\n. But the complement of\nS = S1 ∐ S2\n, a disjoint union with\nS1 =\nand\nS2 =\n. [Connectivity](https://bliptext.com/articles/connected-space) of\n[Δ(0),Δ(N)]\nimplies there is a point\nx\nin the segment which is within a distance\nh1\nof points\nΔ(i)\nand\nΔ(j)\nwith\ni < r\nand\nj > s\n. But then\nd(Δ(i),Δ(j)) < 2 h1\n, so\n. Hence the points\nΔ(k)\nfor\nk\nin\nS\nlie within a distance from\n[Δ(0),Δ(N)]\nof less than\ni − j\n.\n\n## Extension to quasigeodesic rays and lines\nRecall that in a Hadamard space if\n[a1,b1]\nand\n[a2,b2]\nare two geodesic segments and the intermediate points\nc1(t)\nand\nc2(t)\ndivide them in the ratio\nt:(1 – t)\n, then\nd(c1(t),c2(t))\nis a convex function of\nt\n. In particular if\nΓ1(t)\nand\nΓ2(t)\nare geodesic segments of unit speed defined on\n[0,R]\nstarting at the same point then\nIn particular this implies the following:\nX\n, there is a constant\nh > 0\ndepending only on\nλ\nand\nε\nsuch that any quasi-geodesic ray is within a bounded Hausdorff distance\nh\nof a geodesic ray.\nA similar result holds for quasigeodesic and geodesic lines.'''\nIf\nΓ(t)\nis a geodesic say with constant\nλ\nand\nε\n, let\nΓN(t)\nbe the unit speed geodesic for the segment\n[Γ(0),Γ(N)]\n. The estimate above shows that for fixed\nR > 0\nand\nN\nsufficiently large,\n(ΓN)\nis a Cauchy sequence in\nC([0,R],X)\nwith the uniform metric.\nThus\nΓN\ntends to a geodesic ray\nγ\nuniformly on compacta the bound on the Hausdorff distances between\nΓ\nand the segments\nΓN\napplies also to the limiting geodesic\nγ\n. The assertion for quasigeodesic lines follows by taking\nΓN\ncorresponding to the geodesic segment\n[Γ(–N),Γ(N)]\n.\n\n# Efremovich–Tikhomirova theorem\nBefore discussing CAT(-1) spaces, this section will describe the **Efremovich–Tikhomirova theorem** for the unit disk\nD\nwith the Poincaré metric.\nIt asserts that quasi-isometries of\nD\nextend to quasi-Möbius homeomorphisms of the unit disk with the Euclidean metric.\nThe theorem forms the prototype for the more general theory of CAT(-1) spaces.\nTheir 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](https://bliptext.com/articles/teichm-ller-theory) assuring that every [quasiconformal homeomorphism](https://bliptext.com/articles/quasiconformal-mapping) of the disk is [Hölder continuous](https://bliptext.com/articles/h-lder-continuous) and therefore extends continuously to a homeomorphism of the unit circle (it is known that this extension is quasi-Möbius).\n\n## Extension of quasi-isometries to boundary\nIf\nX\nis the Poincaré unit disk, or more generally a CAT(-1) space, the Morse lemma on stability of quasigeodesics implies that every quasi-isometry of\nX\nextends uniquely to the boundary.\nBy definition two self-mappings\nf, g\nof\nX\nare quasi-equivalent if\nsupX d(f(x),g(x)) < ∞\n, so that corresponding points are at a uniformly bounded distance of each other.\nA quasi-isometry\nf1\nof\nX\nis a self-mapping of\nX\n, not necessarily continuous, which has a quasi-inverse\nf2\nsuch that\nf1 ∘ f2\nand\nf2 ∘ f1\nare quasi-equivalent to the appropriate identity maps and such that there are constants\nλ ≥ 1\nand\nε > 0\nsuch that for all\nx, y\nin\nX\nand both mappings\nNote 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.\nFixing a point\nx\nin\nX\n, given a geodesic ray\nγ\nstarting at\nx\n, the image\nf ∘ γ\nunder a quasi-isometry\nf\nis a quasi-geodesic ray.\nBy the Morse-Mostow lemma it is within a bounded distance of a unique geodesic ray\nδ\nstarting at\nx\n. This defines a mapping\n∂f\non the boundary\n∂X\nof\nX\n, independent of the quasi-equivalence class of\nf\n, such that\n∂(f ∘ g) = ∂f ∘ ∂g\n. Thus there is a homomorphism of the group of quasi-isometries into the group of self-mappings of\n∂X\n.\nTo check that\n∂f\nis continuous, note that if\nγ1\nand\nγ2\nare geodesic rays that are uniformly close on\n[0,R]\n, within a distance\nη\n, then\nf ∘ γ1\nand\nf ∘ γ2\nlie within a distance\nλη + ε\non\n[0,R]\n, so that\nδ1\nand\nδ2\nlie within a distance\nλη + ε + 2h(λ,ε)\n[0,r]\n,\nδ1\nand\nδ2\nlie within a distance\n(r/R)⋅[λη + ε + 2h(λ,ε)]\nby convexity.\nOn CAT(-1) spaces, a finer version of continuity asserts that\n∂f\nis a quasi-Möbius mapping with respect to a natural class of metric on\n∂X\n, the \"visual metrics\" generalising the Euclidean metric on the unit circle and its transforms under the Möbius group.\nThese visual metrics can be defined in terms of Busemann functions.\nIn 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](https://bliptext.com/articles/douady-earle-extension)): it follows that the homomorphism from the quasi-isometry group into the quasi-Möbius group is surjective.\nIn the other direction, it is straightforward to prove that the homomorphism is injective.\nSuppose that\nf\nis a quasi-isometry of the unit disk such that\n∂f\nis the identity.\nThe assumption and the Morse lemma implies that if\nγ(R)\nis a geodesic line, then\nf(γ(R))\nlies in an\nh\n-neighbourhood of\nγ(R)\n. Now take a second geodesic line\nδ\nsuch that\nδ\nand\nγ\nintersect orthogonally at a given point in\na\n. Then\nf(a)\nlies in the intersection of\nh\n-neighbourhoods of\nδ\nand\nγ\n. Applying a Möbius transformation, it can be assumed that\na\nis at the origin of the unit disk and the geodesics are the real and imaginary axes.\nBy convexity, the\nh\n-neighbourhoods of these axes intersect in a\n3h\n-neighbourhood of the origin: if\nz\nlies in both neighbourhoods, let\nx\nand\ny\nbe the orthogonal projections of\nz\nonto the\nx\n- and\ny\n-axes; then\nd(z,x) ≤ h\nso taking projections onto the\ny\n-axis,\nd(0,y) ≤ h\nd(z,0) ≤ d(z,y) + d(y,0) ≤ 2h\n. Hence\nd(a,f(a)) ≤ 2h\n, so that\nf\nis quasi-equivalent to the identity, as claimed.\n\n## Cross ratio and distance between non-intersecting geodesic lines\nGiven two distinct points\nz, w\non the unit circle or real axis there is a unique hyperbolic geodesic\n[z,w]\njoining them.\nIt is given by the circle (or straight line) which cuts the unit circle unit circle or real axis orthogonally at those two points.\nGiven four distinct points\na, b, c, d\nin the extended complex plane their [cross ratio](https://bliptext.com/articles/cross-ratio) is defined by\nIf\ng\nis a complex [Möbius transformation](https://bliptext.com/articles/m-bius-transformation) then it leaves the cross ratio invariant:\n(g(a),g(b);g(c),g(d)) = (a,b:c,d)\n. Since the Möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number\nz\nin\nC\\\nsuch that\ng(a) = 0, g(b) = 1, g(c) = λ, g(d) = ∞\nfor a Möbius transformation\ng\n.\nSince\na\n,\nb\n,\nc\nand\nd\nall appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of\na\n,\nb\n,\nc\nand\nd\n, it suffices to consider permutations that fix\nd\n, so only permute\na\n,\nb\nand\nc\n. The cross ratio transforms according to the [anharmonic group](https://bliptext.com/articles/anharmonic-group) of order 6 generated by the Möbius transformations sending\nλ\nto\n1 – λ\nand\nλ−1\n. The other three transformations send\nλ\nto\n1 – λ−1\n, to\nλ(λ – 1)−1\nand to\n(1 – λ)−1\n.\nNow let\na, b, c, d\nbe points on the unit circle or real axis in that order.\nThen the geodesics\n[a,b]\nand\n[c,d]\ndo 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.\nIt is evidently invariant under real Möbius transformations.\nTo compare the cross ratio and the distance between geodesics, Möbius invariance allows the calculation to be reduced to a symmetric configuration.\nFor\n0 < r < R\n, take\na = –R, b = −r, c = r, d = R\n. Then\nλ = (a,b;c,d) = (R + r)2/4rR = (t + 1)2/4t\nwhere\nt = R/r > 1\n. On the other hand, the geodesics\n[a,d]\nand\n[b,c]\nare the semicircles in the upper half plane of radius\nr\nand\nR\n. The geodesic which cuts them orthogonally is the positive imaginary axis, so the distance between them is the hyperbolic distance between\nir\nand\niR\n,\nd(ir,iR) = log R/r = log t\n. Let\ns = log t\n, then\nλ = cosh2(s/2)\n, so that there is a constant\nC > 0\nsuch that, if\n(a,b;c,d) > 1\n, then\nsince\nlog[cosh(x)/expx)] = log (1 + exp(–2x))/2\nis bounded above and below in\nx ≥ 0\n. Note that\na, b, c, d\nare in order around the unit circle if and only if\n(a,b;c,d) > 1\n.\nA 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.\np\nand\nq\nare the feet of the perpendiculars from\nc\nand\nd\nto the geodesic line\nab\n, then\nlog\n.'''\nSince both sides are invariant under Möbius transformations, it suffices to check this in the case that\na = 0\n,\nb = ∞\n,\nc = x\nand\nd = 1\n. In this case the geodesic line is the positive imaginary axis, right hand side equals\n,\nx\nand\nq = i\n. So the left hand side equals\n. Note that\np\nand\nq\nare also the points where the incircles of the ideal triangles\nabc\nand\nabd\ntouch\nab\n.\n\n## Proof of theorem\nA homeomorphism\nF\nof the circle is quasisymmetric if there are constants\na, b > 0\nsuch that\nIt is quasi-Möbius is there are constants\nc, d > 0\nsuch that\nwhere\ndenotes the [cross-ratio](https://bliptext.com/articles/cross-ratio).\nIt is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition.\nIf\nF\nis quasisymmetric then it is also quasi-Möbius, with\nc = a2\nand\nd = b\n(z1,z3,z4)\nand\n(z2,z4,z3)\n. Conversely any quasi-Möbius homeomorphism\nF\nis quasisymmetric.\nTo see this, it can be first be checked that\nF\n(and hence\nF−1\n) is [Hölder continuous](https://bliptext.com/articles/h-lder-continuous). Let\nS\nbe the set of cube roots of unity, so that if\na ≠ b\nin\nS\n, then\na − b\n. To prove a Hölder estimate, it can be assumed that\nx – y\nis uniformly small.\nThen both\nx\nand\ny\nare greater than a fixed distance away from\na, b\nin\nS\nwith\na ≠ b\n, so the estimate follows by applying the quasi-Möbius inequality to\nx, a, y, b\n. To verify that\nF\nis quasisymmetric, it suffices to find a uniform upper bound for\nin the case of a triple with\nx − z\n, uniformly small.\nIn this case there is a point\nw\nat a distance greater than 1 from\nx\n,\ny\nand\nz\n. Applying the quasi-Möbius inequality to\nx\n,\nw\n,\ny\nand\nz\nyields the required upper bound.\nTo summarise:\nTo prove the theorem it suffices to prove that if\nF = ∂f\nthen there are constants\nA, B > 0\nsuch that for\na, b, c, d\ndistinct points on the unit circle\nIt has already been checked that\nF\n(and is inverse) are continuous.\nComposing\nf\n, and hence\nF\n, with complex conjugation if necessary, it can further be assumed that\nF\npreserves the orientation of the circle.\nIn this case, if\na,b, c,d\nare in order on the circle, so too are there images under\nF\n(a,b;c,d)\nand\n(F(a),F(b);F(c),F(d))\nare real and greater than one.\nIn this case\nTo prove this, it suffices to show that\nlog (F(a),F(b);F(c),F(d)) ≤ B log (a,b;c,d) + C\n. From the previous section it suffices show\nd([F(a),F(b)],[F(c),F(d)]) ≤ P d([a,b],[c,d]) + Q\n. This follows from the fact that the images under\nf\nof\n[a,b]\nand\n[c,d]\nlie within\nh\n-neighbourhoods of\n[F(a),F(b)]\nand\n[F(c),F(d)]\nf\napplied to the points on\n[a,b]\nand\n[c,d]\nrealising\nd([a,b],[c,d])\n.\nAdjusting\nA\nand\nB\nif necessary, the inequality above applies also to\nF−1\n. Replacing\na\n,\nb\n,\nc\nand\nd\nby their images under\nF\n, it follows that\nif\na\n,\nb\n,\nc\nand\nd\nare in order on the unit circle.\nHence the same inequalities are valid for the three cyclic of the quadruple\na, b, c, d\n. If\na\nand\nb\nare switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if\nc\nand\nd\nare switched.\nIf both pairs are switched, the cross ratio remains unaltered.\nHence the inequalities are also valid in this case.\nFinally if\nb\nand\nc\nare interchanged, the cross ratio changes from\nλ\nto\nλ–1\n(λ – 1) = 1 – λ–1\n, which lies between 0 and 1.\nHence again the same inequalities are valid.\nIt is easy to check that using these transformations the inequalities are valid for all possible permutations of\na\n,\nb\n,\nc\nand\nd\n, so that\nF\nand its inverse are quasi-Möbius homeomorphisms.\n\n# Busemann functions and visual metrics for CAT(-1) spaces\nBusemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces.\nThese 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.\nIn the case of the unit disk the chordal metric can be recovered directly using Busemann functions\nBγ\nand the special theory for the disk generalises completely to any proper CAT(-1) space\nX\n. 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.\nFrom 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\nγ\nsuch that has these points as the limits\nγ(±∞)\n. The theory applies equally well to any CAT(\n−κ\n) space with\nκ > 0\nsince these arise by scaling the metric on a CAT(-1) space by\nκ−1/2\n. On the hyperbolic unit disk\nD\nquasi-isometries of\nD\ninduce quasi-Möbius homeomorphisms of the boundary in a functorial way.\nThere is a more general theory of Gromov hyperbolic spaces, a similar statement holds, but with less precise control on the homeomorphisms of the boundary.\n\n## Example: Poincaré disk\n\n# Applications in percolation theory\nMore recently Busemann functions have been used by [probabilists](https://bliptext.com/articles/probability-theory) to study asymptotic properties in models of [first-passage percolation](https://bliptext.com/articles/first-passage-percolation) and directed last-passage percolation.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:31:54.928Z","updated_at":"2025-01-25T22:31:54.928Z","search_vector":"'busemann':1 'function':2","search_content":" \n \n\n# \nIn [geometric topology](https://bliptext.com/articles/geometric-topology), **Busemann functions** are used to study the large-scale geometry of geodesics in [Hadamard spaces](https://bliptext.com/articles/hadamard-space) and in particular [Hadamard manifolds](https://bliptext.com/articles/hadamard-manifold) ([simply connected](https://bliptext.com/articles/simply-connected) complete [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) of nonpositive curvature).\nThey are named after [Herbert Busemann](https://bliptext.com/articles/herbert-busemann), who introduced them; he gave an extensive treatment of the topic in his 1955 book \"The geometry of geodesics\".\n\n# Definition and elementary properties\nLet (X,d) be a [metric space](https://bliptext.com/articles/metric-space).\nA [geodesic ray](https://bliptext.com/articles/ray-geometry) is a path which minimizes distance everywhere along its length.\ni.e., for all ,\nEquivalently, a ray is an isometry from the \"canonical ray\" (the set [0,\\infty) equipped with the Euclidean metric) into the metric space X.\nGiven a ray γ, the Busemann function is defined by\nThus, when t is very large, the distance is approximately equal to.\nGiven 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 s\\le t,\nIt is immediate from the triangle inequality that\nso that B_\\gamma is uniformly continuous.\nMore specifically, the above estimate above shows that\nBy [Dini's theorem](https://bliptext.com/articles/dini-s-theorem), the functions tend to B_\\gamma(x) uniformly on compact sets as t tends to infinity.\n\n# Example: Poincaré disk\nLet D be the unit disk in the complex plane with the [Poincaré metric](https://bliptext.com/articles/poincare-disk-model)\nThen, for |z| r\n, there is a point\nv\nwith\nd(y,v) = r\nsuch that\nd(x,v) = d(x,y) − r\n.\nThis condition persists for Busemann functions.\nThe 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](https://bliptext.com/articles/orthogonal-projection) in a [Hilbert space](https://bliptext.com/articles/hilbert-space): if\nC\nis a closed convex set in a Hadamard space\nX\n, then every point\nx\nin\nX\nhas a unique closest point\nP(x) ≡ PC(x)\nin\nC\nand\nd(P(x),P(y)) ≤ d(x,y)\na = P(x)\nis uniquely determined by the property that, for\ny\nin\nC\n,\nso that the angle at\na\nin the Euclidean [comparison triangle](https://bliptext.com/articles/comparison-triangle) for\na,x,y\nis greater than or equal to\nπ/2\n.\nh\nis a Busemann function on a Hadamard space, then, given\ny\nin\nX\nand\nr > 0\n, there is a unique point\nv\nwith\nd(y,v) = r\nsuch that\nh(v) = h(y) − r\n. For fixed\nr > 0\n, the point\nv\nis the closest point of\ny\nto the closed convex\nC\nset of points\nu\nsuch that\nh(u) ≤ h(y) − r\nand therefore depends continuously on\ny\n.'''\nLet\nv\nbe the closest point to\ny\nin ''\nC\n''. Then\nh(v) = h(y) − r\nand so ''\nh\n is minimised by \nv\n'' in\nB(y,R)\nwhere\nR = d(y,v) and v\nis the unique point where ''\nh\n'' is minimised.\nBy the Lipschitz condition\nh(y) − h(v)\n. To prove the assertion, it suffices to show that\nR = r\n, i.e.\nd(y,v) = r\n. On the other hand, ''\nh\n'' is the uniform limit on any closed ball of functions\nhn\n. On\nB(y,r)\n, these are minimised by points\nvn\nwith\nhn(vn) = hn(y) − r\n. Hence the infimum of ''\nh\n'' on\nB(y,r)\nis\nh(y) − r\nand\nh(vn)\ntends to\nh(y) − r\n. Thus\nh(vn) = h(y) − rn\nwith\nrn ≤ r\nand\nrn\ntending towards ''\nr\n''. Let\nun\nbe the closest point to\ny\nwith\nh(un) ≤ h(y) − rn\n. Let\nRn = d(y,un) ≤ r\n. Then\nh(un) = h(y) − rn\n, and, by the Lipschitz condition on\nh\n,\nRn ≥ rn\n. In particular\nRn\ntends to ''\nr\n''. Passing to a subsequence if necessary it can be assumed that\nrn\nand\nRn\nare both increasing (to ''\nr\n''). The inequality for convex optimisation implies that for\nn > m\n.\nso that\nun\nis a Cauchy sequence.\nIf ''\nu\n'' is its limit, then\nd(y,u) = r\nand\nh(u) = h(y) − r\n. By uniqueness it follows that\nu = v\nand hence\nd(y,v) = r\n, as required.\n**Uniform limits.** The above argument proves more generally that if\nd(xn,x0)\ntends to infinity and the functions\nhn(x) = d(x,xn) – d(xn,x0)\ntend uniformly on bounded sets to\nh(x)\n, then ''\nh\n is convex, Lipschitz with Lipschitz constant 1 and, given \ny\n in \nX\n'' and\nr > 0\n, there is a unique point ''\nv\n'' with\nd(y,v) = r\nsuch that\nh(v) = h(y) − r\n. If on the other hand the sequence\n(xn)\nis bounded, then the terms all lie in some closed ball and uniform convergence there implies that\n(xn)\nis a Cauchy sequence so converges to some\nx∞\nin ''\nX\n''. So\nhn\ntends uniformly to\nh∞(x) = d(x,x∞) – d(x∞,x0)\n, a function of the same form.\nThe 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.\n**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.\nIn particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of '''\nR\n''' or even a semi-infinite interval\n[0,∞)\n. The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space.\nh\nis a convex function, Lipschitz with constant 1 and\nh\nassumes its minimum on any closed ball centred on\ny\nwith radius\nr\nat a unique point\nv\non the boundary with\nh(v) = h(y) − r\n, then for each\ny\nin\nX\nthere is a unique geodesic ray\nδ\nsuch that\nδ(0) = y\nand\nδ\ncuts each closed convex set\nh ≤ h(y) – r\nwith\nr > 0\nat\nδ(r)\n, so that\nh(δ(t)) = h(y) – t\n. In particular this holds for each Busemann function.'''\nThe third condition implies that ''\nv\n is the closest point to \ny\n'' in the closed convex set\nCr\nof points u such that\nh(u) ≤ h(y) – r\n. Let\nδ(t)\nfor\n0 ≤ t ≤ r\nbe the geodesic joining ''\ny\n to \nv\n''. Then\nk(t) = h(δ(t)) - h(y)\nis a convex Lipschitz function on\n[0,r]\nwith Lipschitz constant 1 satisfying\nk(t) ≤ – t\nand\nk(0) = 0\nand\nk(r) = –r\n. So ''\nk\n'' vanishes everywhere, since if\n0 0\nat\nδ(r)\n, so that\nh(δ(t)) = h(y) – t\nh\nis a Busemann function.\nh − hδ\nis a constant function.'''\nLet\nCr\nbe the closed convex set of points ''\nz\n'' with\nh(z) ≤ −r\n. Since ''\nX\n is a Hadamard space for every point \ny\n in \nX\n'' there is a unique closest point\nPr(y)\nto\ny\nin\nCr\n. It depends continuously on ''\ny\n and if \ny\n'' lies outside\nCr\n, then\nPr(y)\nlies on the hypersurface\nh(z) = − r\n—the boundary ∂\nCr\nof\nCr\n—and\nPr(y)\nsatisfies the inequality of convex optimisation.\nLet\nδ(s)\nbe the geodesic ray starting at ''\ny\n''.\nFix ''\nx\n in \nX\n''. Let\nγ(s)\nbe the geodesic ray starting at ''\nx\n''. Let\ng(z) = hγ(z)\n, the Busemann function for\nγ\nwith base point ''\nx\n''. In particular\ng(x) = 0\n. It suffices to show that\ng = h – h(y)1\n. Now take ''\ny\n'' with\nh(x) = h(y)\nand let\nδ(t)\nbe the geodesic ray starting at ''\ny\n corresponding to \nh\n''. Then\nOn the other hand, for any four points ''\na\n, \nb\n, \nc\n, \nd\n'' in a Hadamard space, the following quadrilateral inequality of [Reshetnyak](https://bliptext.com/articles/yurii-reshetnyak) holds:\nSetting\na = x\n,\nb = y\n,\nc = γ(t)\n,\nd = δ(t)\n, it follows that\nso that\nHence\nhγ(y) = 0\n. Similarly\nhδ(x) = 0\n. Hence\nhγ(y) = 0\non the level surface of ''\nh\n containing \nx\n''. Now for\nt ≥ 0\nand ''\nz\n in \nX\n'', let\nαt(z) = γ1(t)\nthe geodesic ray starting at ''\nz\n''. Then\nαs + t = αs ∘ αt\nand\nh ∘ αt = h − t\n. Moreover, by boundedness,\nd(αt(u),αt(v)) ≤ d(u,v)\n. The flow\nαs\ncan be used to transport this result to all the level surfaces of ''\nh\n''. For general\ny1\n, if\nh(y1) 0\nsuch that\nh(αs(x)) = h(y1)\nand set\nx1 = αs(x)\n. Then\nhγ 1 (y1) = 0\n, where\nγ1(t) = αt(x1) = γ(s + t)\n. But then\nhγ 1 = hγ – s\n, so that\nhγ(y1) = s\n. Hence\ng(y1) = s = h((αs(x)) – h(x) = h(y1) – h(x)\n, as required.\nSimilarly if\nh(y1) > h(x)\n, take\ns > 0\nsuch that\nh(αs(y1)) = h(x)\n. Let\ny = αs(y1)\n. Then\nhγ(y) = 0\n, so\nhγ(y1) = –s\n. Hence\ng(y1) = –s = h(y1) – h(x)\n, as required.\nFinally there are necessary and sufficient conditions for two geodesics to define the same Busemann function up to constant:\nSuppose firstly that\nγ\nand\nδ\nare two geodesic rays with Busemann functions differing by a constant.\nShifting the argument of one of the geodesics by a constant, it may be assumed that\nBγ = Bδ = B\n, say.\nLet ''\nC\n'' be the closed convex set on which\nB(x) ≤ −r\n. Then\nB(γ(t)) = Bγ(γ(t)) = −t\nand similarly\nB(δ(t)) = − t\n. Then for\ns ≤ r\n, the points\nγ(s)\nand\nδ(s)\nhave closest points\nγ(r)\nand\nδ(r)\nin ''\nC\n'', so that\nd(γ(r), δ(r)) ≤ d(γ(s), δ(s))\n. Hence\nsupt ≥ 0 d(γ(t), δ(t)) 0\n, the ray\nδ\ncuts each closed convex set\nh ≤ h(y) – r\nat\nδ(r)\n.'''\n\n# Bordification of a Hadamard space\nIn the previous section it was shown that if\nX\nis a Hadamard space and\nx0\nis a fixed point in\nX\nthen the union of the space of Busemann functions vanishing at\nx0\nand the space of functions\nhy(x) = d(x,y) − d(x0,y)\nis closed under taking uniform limits on bounded sets.\nThis result can be formalised in the notion of **bordification** of\nX\n. In this topology, the points\nxn\ntend to a geodesic ray\nγ\nstarting at\nx0\nif and only if\nd(x0,xn)\ntends to\n∞\nand for\nt > 0\narbitrarily large the sequence obtained by taking the point on each segment\n[x0,xn]\nat a distance\nt\nfrom\nx0\ntends to\nγ(t)\n.\nIf\nX\nis a metric space, Gromov's bordification can be defined as follows.\nFix a point\nx0\nin\nX\nand let\nXN = B(x0,N)\n. Let\nY = C(X)\nbe the space of Lipschitz continuous functions on\nX\n, i.e. those for which\nfor some constant\nA > 0\n. The space\nY\ncan be topologised by the seminorms\nf\n, the topology of uniform convergence on bounded sets.\nThe seminorms are finite by the Lipschitz conditions.\nThis is the topology induced by the natural map of\nC(X)\ninto the direct product of the Banach spaces\nCb(XN)\nof continuous bounded functions on\nXN\n. It is give by the metric\nD(f,g) = Σ 2−N f − gN(1 +f − gN)−1\n.\nThe space\nX\nis embedded into\nY\nby sending\nx\nto the function\nfx(y) = d(y,x) – d(x0,x)\n. Let\nX\nbe the closure of\nX\nin\nY\n. Then\nX\nis metrisable, since\nY\nis, and contains\nX\nas an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic.\nLet\nh(x) = (d(x,x0) + 1)−1\n. Then\nh\nlies in\nC0(X)\n. It is non-zero on\nX\nand vanishes only at\n∞\n. Hence it extends to a continuous function on\nX\nwith zero set\nX \\ X\n. It follows that\nX \\ X\nis closed in\nX\n, as required.\nTo check that\nX = X(x0)\nis independent of the basepoint, it suffices to show that\nk(x) = d(x,x0) − d(x,x1)\nextends to a continuous function on\nX\n. But\nk(x) = fx(x1)\n, so, for\ng\nin\nX\n,\nk(g) = g(x1)\n. Hence the correspondence between the compactifications for\nx0\nand\nx1\nis given by sending\ng\nin\nX(x0)\nto\ng + g(x1)1\nin\nX(x1)\n.\nWhen\nX\nis a Hadamard space, Gromov's ideal boundary\n∂X = X \\ X\ncan be realised explicitly as \"asymptotic limits\" of geodesic rays using Busemann functions.\nIf\nxn\nis an unbounded sequence in\nX\nwith\nhn(x) = d(x,xn) − d(xn,x0)\ntending to\nh\nin\nY\n, then\nh\nvanishes at\nx0\n, is convex, Lipschitz with Lipschitz constant\n1\nand has minimum\nh(y) − r\non any closed ball\nB(y,r)\n. Hence\nh\nis a Busemann function\nBγ\ncorresponding to a unique geodesic ray\nγ\nstarting at\nx0\n.\nOn the other hand,\nhn\ntends to\nBγ\nuniformly on bounded sets if and only if\nd(x0,xn)\ntends to\n∞\nand for\nt > 0\narbitrarily large the sequence obtained by taking the point on each segment\n[x0,xn]\nat a distance\nt\nfrom\nx0\ntends to\nγ(t)\n. For\nd(x0,xn) ≥ t\n, let\nxn(t)\nbe the point in\n[x0,xn]\nwith\nd(x0,xn(t)) = t\n. Suppose first that\nhn\ntends to\nBγ\nuniformly on\nB(x0,R)\n. Then for\nt ≤ R\n,\nhn(γ(t)) – Bγ(γ(t))\n. This is a convex function.\nIt vanishes as\nt = 0\nand hence is increasing.\nSo it is maximised at\nt = R\n. So for each\nt\n,\ntends towards 0.\nLet\na = X0\n,\nb = γ(t)\nand\nc = xn\n. Then\nd(c,a) – d(c,b)\nis close to\nd(a,b)\nwith\nd(c,a)\nlarge.\nHence in the Euclidean comparison triangle\nCA - CB\nis close to\nAB\nwith\nCA\nlarge.\nSo the angle at\nA\nis small.\nSo the point\nD\non\nAC\nat the same distance as\nAB\nlies close to\nB\n. Hence, by the first comparison theorem for geodesic triangles,\nd(xn(t),γ(t))\nis small.\nConversely suppose that for fixed\nt\nand\nn\nsufficiently large\nd(xn(t),γ(t))\ntends to 0.\nThen from the above\nFs(y) = d(y,γ(s)) – s\nsatisfies\nso it suffices show that on any bounded set\nhn(y) = d(y,xn) – d(x0,xn)\nis uniformly close to\nFs(y)\nfor\nn\nsufficiently large.\nFor a fixed ball\nB(x0,R)\n, fix\ns\nso that\nR2/s ≤ ε\n. The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since\nHence, if\ny\nin\nB(x0,R)\nand\nn\nis sufficiently large that\nd(xn(s),γ(s)) ≤ ε\n, then\n\n# Busemann functions on a Hadamard manifold\nSuppose that\nx, y\nare points in a Hadamard manifold and let\nγ(s)\nbe the geodesic through\nx\nwith\nγ(0) = y\n. This geodesic cuts the boundary of the closed ball\nB(y,r)\nat the two points\nγ(±r)\n. Thus if\nd(x,y) > r\n, there are points\nu, v\nwith\nd(y,u) = d(y,v) = r\nsuch that\nd(x,u) − d(x,v)\n. By continuity this condition persists for Busemann functions:\nh\nis a Busemann function on a Hadamard manifold, then, given\ny\nin\nX\nand\nr > 0\n, there are unique points\nu\n,\nv\nwith\nd(y,u) = d(y,v) = r\nsuch that\nh(u) = h(y) + r\nand\nh(v) = h(y) − r\n. For fixed\nr > 0\n, the points\nu\nand\nv\ndepend continuously on\ny\n.'''\nTaking a sequence\ntn\ntending to\n∞\nand\nhn = Ft n\n, there are points\nun\nand\nvn\nwhich satisfy these conditions for\nhn\nfor\nn\nsufficiently large.\nPassing to a subsequence if necessary, it can be assumed that\nun\nand\nvn\ntend to\nu\nand\nv\n. By continuity these points satisfy the conditions for\nh\n. To prove uniqueness, note that by compactness\nh\nassumes its maximum and minimum on\nB(y,r)\n. The Lipschitz condition shows that the values of\nh\nthere differ by at most\n2r\n. Hence\nh\nis minimized at\nv\nand maximized at\nu\n. On the other hand,\nd(u,v) = 2r\nand for\nu\nand\nv\nthe points\nv\nand\nu\nare the unique points in\nB(y,r)\nmaximizing this distance.\nThe Lipschitz condition on\nh\nthen immediately implies\nu\nand\nv\nmust be the unique points in\nB(y,r)\nmaximizing and minimizing\nh\n. Now suppose that\nyn\ntends to\ny\n. Then the corresponding points\nun\nand\nvn\nlie in a closed ball so admit convergent subsequences.\nBut by uniqueness of\nu\nand\nv\nany such subsequences must tend to\nu\nand\nv\n, so that\nun\nand\nvn\nmust tend to\nu\nand\nv\n, establishing continuity.\nThe above result holds more generally in a Hadamard space.\nh\nis a Busemann function on a Hadamard manifold, then\nh\nis continuously differentiable with\ndh(y) = 1\nfor all\ny\n.'''\nFrom the previous properties of\nh\n, for each\ny\nthere is a unique geodesic γ(t) parametrised by arclength with\nγ(0) = y\nsuch that\nh ∘ γ(t) = h(y) + t\n. It has the property that it cuts\n∂B(y,r)\nat\nt = ±r\nγ(r) = u\nand\nγ(–r) = v\n. The vector field\nVh\ndefined by the unit vector at\ny\nis continuous, because\nu\nis a continuous function of\ny\nand the map sending\n(x,v)\nto\n(x,expx v)\nis a diffeomorphism from\nTX\nonto\nX × X\nby the [Cartan-Hadamard theorem](https://bliptext.com/articles/cartan-hadamard-theorem).\nLet\nδ(s)\nbe another geodesic parametrised by arclength through\ny\nwith\nδ(0) = y\n. Then\ndh ∘ δ (0)/ ds =\n. Indeed, let\nH(x) = h(x) − h(y)\n, so that\nH(y) = 0\n. Then\nApplying this with\nx = u\nand\nv\n, it follows that for\ns > 0\nThe outer terms tend to as\ns\ntends to 0, so the middle term has the same limit, as claimed.\nA similar argument applies for\ns 0\n,\nOn the other hand, for any four points\na\n,\nb\n,\nc\n,\nd\nin a Hadamard space, the following quadrilateral inequality of [Reshetnyak](https://bliptext.com/articles/yurii-reshetnyak) holds:\nSetting\na = x\n,\nb = y\n,\nc = αt(x)\n,\nd = αt(y)\n, it follows that\nso that\nHence\nhγ(y) = 0\non the level surface of\nh\ncontaining\nx\n. The flow\nαs\ncan be used to transport this result to all the level surfaces of\nh\n. For general\ny1\ntake\ns\nsuch that\nh(αs(x)) = h(y1)\nand set\nx1 = αs(x)\n. Then\nhγ 1 (y1) = 0\n, where\nγ1(t) = αt(x1) = γ(s + t)\n. But then\nhγ 1 = hγ – s\n, so that\nhγ(y1) = s\n. Hence\ng(y1) = s = h((αs(x)) – h(x) = h(y1) – h(x)\n, as required.\nNote that this argument could be shortened using the fact that two Busemann functions\nhγ\nand\nhδ\ndiffer by a constant if and only if the corresponding geodesic rays satisfy\nsupt ≥ 0 d(γ(t),δ(t)) 0\n. The space\nY\ncan be topologised by the seminorms\nf\n, the topology of uniform convergence on compacta.\nThis is the topology induced by the natural map of C(X) into the direct product of the Banach spaces\nC(XN)\n. It is give by the metric\nD(f,g) = Σ 2−N f − g N(1 + f − g N)−1\n.\nThe space\nX\nis embedded into\nY\nby sending\nx\nto the function\nfx(y) = d(y,x) – d(x0,x)\n. Let\nX\nbe the closure of\nX\nin\nY\n. Then\nX\nis compact (metrisable) and contains\nX\nas an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic.\nCompactness follows from the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) since the image in\nC(XN)\nis [equicontinuous](https://bliptext.com/articles/equicontinuous) and uniformly bounded in norm by\nN\n. Let\nxn\nbe a sequence in\nX ⊂ X\ntending to\ny\nin\nX \\ X\n. Then all but finitely many terms must lie outside\nXN\nsince\nXN\nis compact, so that any subsequence would converge to a point in\nXN\nxn\nmust be unbounded in\nX\n. Let\nh(x) = (d(x,x0) + 1)−1\n. Then\nh\nlies in\nC0(X)\n. It is non-zero on\nX\nand vanishes only at\n∞\n. Hence it extends to a continuous function on\nX\nwith zero set\nX \\ X\n. It follows that\nX \\ X\nis closed in\nX\n, as required.\nTo check that the compactification\nX = X(x0)\nis independent of the basepoint, it suffices to show that\nk(x) = d(x,x0) − d(x,x1)\nextends to a continuous function on\nX\n. But\nk(x) = fx(x1)\n, so, for\ng\nin\nX\n,\nk(g) = g(x1)\n. Hence the correspondence between the compactifications for\nx0\nand\nx1\nis given by sending\ng\nin\nX(x0)\nto\ng + g(x1)1\nin\nX(x1)\n.\nWhen\nX\nis a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary\n∂X = X \\ X\ncan be realised explicitly as \"asymptotic limits\" of geodesics by using Busemann functions.\nFixing a base point\nx0\n, there is a unique geodesic\nγ(t)\nparametrised by arclength such that\nγ(0) = x0\nand is a given unit vector.\nIf\nBγ\nis the corresponding Busemann function, then\nBγ\nlies in\n∂X(x0)\nand induces a homeomorphism of the unit\n(n − 1)\n-sphere onto\n∂X(x0)\n, sending to\nBγ\n.\n\n# Quasigeodesics in the Poincaré disk, CAT(-1) and hyperbolic spaces\n\n## Morse–Mostow lemma\nIn 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.\nThis structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays.\nQuasigeodesics were first studied for negatively curved surfaces—in particular the hyperbolic upper halfplane and unit disk—by [Morse](https://bliptext.com/articles/marston-morse) and generalised to negatively curved [symmetric spaces](https://bliptext.com/articles/symmetric-space) by [Mostow](https://bliptext.com/articles/george-mostow), for his work on the [rigidity of discrete groups](https://bliptext.com/articles/mostow-rigidity-theorem).\nThe basic result is the **Morse–Mostow lemma** on the stability of geodesics.\nBy definition a **[quasigeodesic](https://bliptext.com/articles/quasigeodesic)** Γ defined on an interval\n[a,b]\nwith\n−∞ ≤ a 0\nsuch that for all\ns\nand\nt\nThe following result is essentially due to [Marston Morse](https://bliptext.com/articles/marston-morse) (1924).\n**Morse's lemma on stability of geodesics.** In the hyperbolic disk there is a constant\nR\ndepending on\nλ\nand\nε\nsuch that any quasigeodesic segment\nΓ\ndefined on a finite interval\n[a,b]\nis within a [Hausdorff distance](https://bliptext.com/articles/hausdorff-distance)\nR\nof the geodesic segment\n[Γ(a),Γ(b)]\n.\n\n## Classical proof for Poincaré disk\nThe 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.\nΓ\ncan be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from\nΓ\nless than\nc = (2λ2 + 1)ε\nΓ\nis defined into equal subintervals of length\n2λε\nand take the geodesics between the images under\nΓ\nof the endpoints of the subintervals.\nSince\nΔ\nis piecewise geodesic,\nΔ\nis Lipschitz continuous with constant\nλ1\n,\nd(Δ(s),Δ(t)) ≤ λ1\n, where\nλ1 ≤ λ + ε\n. The lower bound is automatic at the endpoints of intervals.\nBy construction the other values differ from these by a uniformly bounded depending only on\nλ\nand\nε\nγ\nis a piecewise smooth curve segment lying outside an\ns\n-neighbourhood of a geodesic line and\nP\nis the orthogonal projection onto the geodesic line then:\nApplying 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\nz\nand\nz\n. Hence the hypothesis implies\n, so that\nh > 0\ndepending only on\nλ\nand\nε\nsuch that\nΓ[a,b]\nlies within an\nh\n-neighbourhood of the geodesic segment\n[Γ(a),Γ(b)]\n.\nLet\nγ(t)\nbe the geodesic line containing the geodesic segment\n[Γ(a),Γ(b)]\n. Then there is a constant\nh > 0\ndepending only on\nλ\nand\nε\nsuch that\nh\n-neighbourhood\nΓ[a,b]\nlies within an\nh\n-neighbourhood of\nγ(R)\n. Indeed for any\ns > 0\n, the subset of\n[a,b]\nfor which\nΓ(t)\nlies outside the closure of the\ns\n-neighbourhood of\nγ(R)\nis open, so a countable union of open intervals\n(c,d)\n. Then\nλ\nand\ns + s1\nof\nγ(R)\n. To deduce the assertion, note that the subset of\n[a,b]\nfor which\nΓ(t)\nlies outside the closure of the\ns\n-neighbourhood of\n[Γ(a),Γ(b)] ⊂ γ(R)\nis open, so a union of intervals\n(c,d)\nwith\nΓ(c)\nand\nΓ(d)\nboth at a distance\ns + s1\nfrom either\nΓ(a)\nor\nΓ(b)\n. Then\nh\ngreater than\ns +s1 + s2\n.\nh > 0\ndepending only on\nλ\nand\nε\nsuch that the geodesic segment\n[Γ(a),Γ(b)]\nlies within an\nh\n-neighbourhood of\nΓ[a,b]\n.\nEvery point of\nΓ[a,b]\nlies within a distance\nh\nof\n[Γ(a),Γ(b)]\n. Thus orthogonal projection\nP\ncarries each point of\nΓ[a,b]\nonto a point in the closed convex set\n[Γ(a),Γ(b)]\nat a distance less than\nh\n. Since\nP\nis continuous and\nΓ[a,b]\nconnected, the map\nP\nmust be onto since the image contains the endpoints of\n[Γ(a),Γ(b)]\n. But then every point of\n[Γ(a),Γ(b)]\nis within a distance\nh\nof a point of\nΓ[a,b]\n.\n\n## Gromov's proof for Poincaré disk\nThe 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.\nThere is also a generalisation for the more general class of [hyperbolic metric spaces](https://bliptext.com/articles/hyperbolic-metric-space) due to Gromov.\nGromov'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](https://bliptext.com/articles/mutatis-mutandi) to CAT(-1) or hyperbolic metric spaces.\nSince this is a large-scale phenomenon, it is enough to check that any maps\nΔ\nfrom\nfor any\nN > 0\nto the disk satisfying the inequalities is within a Hausdorff distance\nR1\nof the geodesic segment\n[Δ(0),Δ(N)]\n. For then translating it may be assumed without loss of generality\nΓ\nis defined on\n[0,r]\nwith\nr > 1\nand then, taking\nN = [r]\n(the integer part of\nr\n), the result can be applied to\nΔ\ndefined by\nΔ(i) = Γ(i)\n. The Hausdorff distance between the images of\nΓ\nand\nΔ\nis evidently bounded by a constant\nR2\ndepending only on\nλ\nand\nε\n.\nδ\nwhere\nδ = 2 log 3\n2 log 3\n. 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\nδ\n, it follows that every side of a geodesic triangle is contained in a\nδ\n-neighbourhood of the other two sides.\nA simple induction argument shows that a geodesic polygon with\n2k + 2\nvertices for\nk ≥ 0\nhas each side within a\n(k + 1)δ\nneighbourhood of the other sides (such a polygon is made by combining two geodesic polygons with\n2k−1 + 1\nsides along a common side). Hence if\nM ≤ 2k + 2\n, the same estimate holds for a polygon with\nM\nsides.\nyi = Δ(i)\nlet\nf(x) = min d(x,yi)\n, the largest radius for a closed ball centred on\nx\nwhich contains no\nyi\nin its interior.\nThis is a continuous function non-zero on\n[Δ(0),Δ(N)]\nso attains its maximum\nh\nat some point\nx\nin this segment.\nThen\n[Δ(0),Δ(N)]\nlies within an\nh1\n-neighbourhood of the image of\nΔ\nfor any\nh1 > h\n. It therefore suffices to find an upper bound for\nh\nindependent of\nN\n.\ny\nand\nz\nin the segment\n[Δ(0),Δ(N)]\nbefore and after\nx\nwith\nd(x,y) = 2h\nand\nd(x,z) = 2h\n(or an endpoint if it within a distance of less than\n2h\nfrom\nx\n). Then there are\ni, j\nwith\nd(y,Δ(i))\n,\nd(z,Δ(j)) ≤ h\n. Hence\nd(Δ(i),Δ(j)) ≤ 6h\n, so that\n. By the triangle inequality all points on the segments\n[y,Δ(i)]\nand\n[z,Δ(j)]\nare at a distance\n≥ h\nfrom\nx\n. Thus there is a finite sequence of points starting at\ny\nand ending at\nz\n, lying first on the segment\n[y,Δ(i)]\n, then proceeding through the points\nΔ(i), Δ(i+1), ..., Δ(j)\n, before taking the segment\n[Δ(j),z]\n. The successive points\nΔ(i), Δ(i+1), ..., Δ(j)\nare separated by a distance no greater than\nλ + ε\nand successive points on the geodesic segments can also be chosen to satisfy this condition.\nThe minimum number\nK\nof points in such a sequence satisfies\nK ≤\n. These points form a geodesic polygon, with\n[y,z]\nas one of the sides.\nTake\nL = [h/δ]\n, so that the\n(L − 1)δ\n-neighbourhood of\n[y,z]\ndoes not contain all the other sides of the polygon.\nHence, from the result above, it follows that\nK > 2L − 1 + 2\n. Hence\nh\nis uniformly bounded, independently of\nN\n, as claimed.\nΔ(i)\nlie within\nh1\nof the\n[Δ(0),Δ(N)]\n, the result follows.\nOtherwise the points which do not fall into maximal subsets\nS =\nwith\nr s\n. But then\nd(Δ(i),Δ(j)) 0\ndepending only on\nλ\nand\nε\nsuch that any quasi-geodesic ray is within a bounded Hausdorff distance\nh\nof a geodesic ray.\nA similar result holds for quasigeodesic and geodesic lines.'''\nIf\nΓ(t)\nis a geodesic say with constant\nλ\nand\nε\n, let\nΓN(t)\nbe the unit speed geodesic for the segment\n[Γ(0),Γ(N)]\n. The estimate above shows that for fixed\nR > 0\nand\nN\nsufficiently large,\n(ΓN)\nis a Cauchy sequence in\nC([0,R],X)\nwith the uniform metric.\nThus\nΓN\ntends to a geodesic ray\nγ\nuniformly on compacta the bound on the Hausdorff distances between\nΓ\nand the segments\nΓN\napplies also to the limiting geodesic\nγ\n. The assertion for quasigeodesic lines follows by taking\nΓN\ncorresponding to the geodesic segment\n[Γ(–N),Γ(N)]\n.\n\n# Efremovich–Tikhomirova theorem\nBefore discussing CAT(-1) spaces, this section will describe the **Efremovich–Tikhomirova theorem** for the unit disk\nD\nwith the Poincaré metric.\nIt asserts that quasi-isometries of\nD\nextend to quasi-Möbius homeomorphisms of the unit disk with the Euclidean metric.\nThe theorem forms the prototype for the more general theory of CAT(-1) spaces.\nTheir 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](https://bliptext.com/articles/teichm-ller-theory) assuring that every [quasiconformal homeomorphism](https://bliptext.com/articles/quasiconformal-mapping) of the disk is [Hölder continuous](https://bliptext.com/articles/h-lder-continuous) and therefore extends continuously to a homeomorphism of the unit circle (it is known that this extension is quasi-Möbius).\n\n## Extension of quasi-isometries to boundary\nIf\nX\nis the Poincaré unit disk, or more generally a CAT(-1) space, the Morse lemma on stability of quasigeodesics implies that every quasi-isometry of\nX\nextends uniquely to the boundary.\nBy definition two self-mappings\nf, g\nof\nX\nare quasi-equivalent if\nsupX d(f(x),g(x)) 0\nsuch that for all\nx, y\nin\nX\nand both mappings\nNote 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.\nFixing a point\nx\nin\nX\n, given a geodesic ray\nγ\nstarting at\nx\n, the image\nf ∘ γ\nunder a quasi-isometry\nf\nis a quasi-geodesic ray.\nBy the Morse-Mostow lemma it is within a bounded distance of a unique geodesic ray\nδ\nstarting at\nx\n. This defines a mapping\n∂f\non the boundary\n∂X\nof\nX\n, independent of the quasi-equivalence class of\nf\n, such that\n∂(f ∘ g) = ∂f ∘ ∂g\n. Thus there is a homomorphism of the group of quasi-isometries into the group of self-mappings of\n∂X\n.\nTo check that\n∂f\nis continuous, note that if\nγ1\nand\nγ2\nare geodesic rays that are uniformly close on\n[0,R]\n, within a distance\nη\n, then\nf ∘ γ1\nand\nf ∘ γ2\nlie within a distance\nλη + ε\non\n[0,R]\n, so that\nδ1\nand\nδ2\nlie within a distance\nλη + ε + 2h(λ,ε)\n[0,r]\n,\nδ1\nand\nδ2\nlie within a distance\n(r/R)⋅[λη + ε + 2h(λ,ε)]\nby convexity.\nOn CAT(-1) spaces, a finer version of continuity asserts that\n∂f\nis a quasi-Möbius mapping with respect to a natural class of metric on\n∂X\n, the \"visual metrics\" generalising the Euclidean metric on the unit circle and its transforms under the Möbius group.\nThese visual metrics can be defined in terms of Busemann functions.\nIn 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](https://bliptext.com/articles/douady-earle-extension)): it follows that the homomorphism from the quasi-isometry group into the quasi-Möbius group is surjective.\nIn the other direction, it is straightforward to prove that the homomorphism is injective.\nSuppose that\nf\nis a quasi-isometry of the unit disk such that\n∂f\nis the identity.\nThe assumption and the Morse lemma implies that if\nγ(R)\nis a geodesic line, then\nf(γ(R))\nlies in an\nh\n-neighbourhood of\nγ(R)\n. Now take a second geodesic line\nδ\nsuch that\nδ\nand\nγ\nintersect orthogonally at a given point in\na\n. Then\nf(a)\nlies in the intersection of\nh\n-neighbourhoods of\nδ\nand\nγ\n. Applying a Möbius transformation, it can be assumed that\na\nis at the origin of the unit disk and the geodesics are the real and imaginary axes.\nBy convexity, the\nh\n-neighbourhoods of these axes intersect in a\n3h\n-neighbourhood of the origin: if\nz\nlies in both neighbourhoods, let\nx\nand\ny\nbe the orthogonal projections of\nz\nonto the\nx\n- and\ny\n-axes; then\nd(z,x) ≤ h\nso taking projections onto the\ny\n-axis,\nd(0,y) ≤ h\nd(z,0) ≤ d(z,y) + d(y,0) ≤ 2h\n. Hence\nd(a,f(a)) ≤ 2h\n, so that\nf\nis quasi-equivalent to the identity, as claimed.\n\n## Cross ratio and distance between non-intersecting geodesic lines\nGiven two distinct points\nz, w\non the unit circle or real axis there is a unique hyperbolic geodesic\n[z,w]\njoining them.\nIt is given by the circle (or straight line) which cuts the unit circle unit circle or real axis orthogonally at those two points.\nGiven four distinct points\na, b, c, d\nin the extended complex plane their [cross ratio](https://bliptext.com/articles/cross-ratio) is defined by\nIf\ng\nis a complex [Möbius transformation](https://bliptext.com/articles/m-bius-transformation) then it leaves the cross ratio invariant:\n(g(a),g(b);g(c),g(d)) = (a,b:c,d)\n. Since the Möbius group acts simply transitively on triples of points, the cross ratio can alternatively be described as the complex number\nz\nin\nC\\\nsuch that\ng(a) = 0, g(b) = 1, g(c) = λ, g(d) = ∞\nfor a Möbius transformation\ng\n.\nSince\na\n,\nb\n,\nc\nand\nd\nall appear in the numerator defining the cross ratio, to understand the behaviour of the cross ratio under permutations of\na\n,\nb\n,\nc\nand\nd\n, it suffices to consider permutations that fix\nd\n, so only permute\na\n,\nb\nand\nc\n. The cross ratio transforms according to the [anharmonic group](https://bliptext.com/articles/anharmonic-group) of order 6 generated by the Möbius transformations sending\nλ\nto\n1 – λ\nand\nλ−1\n. The other three transformations send\nλ\nto\n1 – λ−1\n, to\nλ(λ – 1)−1\nand to\n(1 – λ)−1\n.\nNow let\na, b, c, d\nbe points on the unit circle or real axis in that order.\nThen the geodesics\n[a,b]\nand\n[c,d]\ndo 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.\nIt is evidently invariant under real Möbius transformations.\nTo compare the cross ratio and the distance between geodesics, Möbius invariance allows the calculation to be reduced to a symmetric configuration.\nFor\n0 1\n. On the other hand, the geodesics\n[a,d]\nand\n[b,c]\nare the semicircles in the upper half plane of radius\nr\nand\nR\n. The geodesic which cuts them orthogonally is the positive imaginary axis, so the distance between them is the hyperbolic distance between\nir\nand\niR\n,\nd(ir,iR) = log R/r = log t\n. Let\ns = log t\n, then\nλ = cosh2(s/2)\n, so that there is a constant\nC > 0\nsuch that, if\n(a,b;c,d) > 1\n, then\nsince\nlog[cosh(x)/expx)] = log (1 + exp(–2x))/2\nis bounded above and below in\nx ≥ 0\n. Note that\na, b, c, d\nare in order around the unit circle if and only if\n(a,b;c,d) > 1\n.\nA 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.\np\nand\nq\nare the feet of the perpendiculars from\nc\nand\nd\nto the geodesic line\nab\n, then\nlog\n.'''\nSince both sides are invariant under Möbius transformations, it suffices to check this in the case that\na = 0\n,\nb = ∞\n,\nc = x\nand\nd = 1\n. In this case the geodesic line is the positive imaginary axis, right hand side equals\n,\nx\nand\nq = i\n. So the left hand side equals\n. Note that\np\nand\nq\nare also the points where the incircles of the ideal triangles\nabc\nand\nabd\ntouch\nab\n.\n\n## Proof of theorem\nA homeomorphism\nF\nof the circle is quasisymmetric if there are constants\na, b > 0\nsuch that\nIt is quasi-Möbius is there are constants\nc, d > 0\nsuch that\nwhere\ndenotes the [cross-ratio](https://bliptext.com/articles/cross-ratio).\nIt is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition.\nIf\nF\nis quasisymmetric then it is also quasi-Möbius, with\nc = a2\nand\nd = b\n(z1,z3,z4)\nand\n(z2,z4,z3)\n. Conversely any quasi-Möbius homeomorphism\nF\nis quasisymmetric.\nTo see this, it can be first be checked that\nF\n(and hence\nF−1\n) is [Hölder continuous](https://bliptext.com/articles/h-lder-continuous). Let\nS\nbe the set of cube roots of unity, so that if\na ≠ b\nin\nS\n, then\na − b\n. To prove a Hölder estimate, it can be assumed that\nx – y\nis uniformly small.\nThen both\nx\nand\ny\nare greater than a fixed distance away from\na, b\nin\nS\nwith\na ≠ b\n, so the estimate follows by applying the quasi-Möbius inequality to\nx, a, y, b\n. To verify that\nF\nis quasisymmetric, it suffices to find a uniform upper bound for\nin the case of a triple with\nx − z\n, uniformly small.\nIn this case there is a point\nw\nat a distance greater than 1 from\nx\n,\ny\nand\nz\n. Applying the quasi-Möbius inequality to\nx\n,\nw\n,\ny\nand\nz\nyields the required upper bound.\nTo summarise:\nTo prove the theorem it suffices to prove that if\nF = ∂f\nthen there are constants\nA, B > 0\nsuch that for\na, b, c, d\ndistinct points on the unit circle\nIt has already been checked that\nF\n(and is inverse) are continuous.\nComposing\nf\n, and hence\nF\n, with complex conjugation if necessary, it can further be assumed that\nF\npreserves the orientation of the circle.\nIn this case, if\na,b, c,d\nare in order on the circle, so too are there images under\nF\n(a,b;c,d)\nand\n(F(a),F(b);F(c),F(d))\nare real and greater than one.\nIn this case\nTo prove this, it suffices to show that\nlog (F(a),F(b);F(c),F(d)) ≤ B log (a,b;c,d) + C\n. From the previous section it suffices show\nd([F(a),F(b)],[F(c),F(d)]) ≤ P d([a,b],[c,d]) + Q\n. This follows from the fact that the images under\nf\nof\n[a,b]\nand\n[c,d]\nlie within\nh\n-neighbourhoods of\n[F(a),F(b)]\nand\n[F(c),F(d)]\nf\napplied to the points on\n[a,b]\nand\n[c,d]\nrealising\nd([a,b],[c,d])\n.\nAdjusting\nA\nand\nB\nif necessary, the inequality above applies also to\nF−1\n. Replacing\na\n,\nb\n,\nc\nand\nd\nby their images under\nF\n, it follows that\nif\na\n,\nb\n,\nc\nand\nd\nare in order on the unit circle.\nHence the same inequalities are valid for the three cyclic of the quadruple\na, b, c, d\n. If\na\nand\nb\nare switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if\nc\nand\nd\nare switched.\nIf both pairs are switched, the cross ratio remains unaltered.\nHence the inequalities are also valid in this case.\nFinally if\nb\nand\nc\nare interchanged, the cross ratio changes from\nλ\nto\nλ–1\n(λ – 1) = 1 – λ–1\n, which lies between 0 and 1.\nHence again the same inequalities are valid.\nIt is easy to check that using these transformations the inequalities are valid for all possible permutations of\na\n,\nb\n,\nc\nand\nd\n, so that\nF\nand its inverse are quasi-Möbius homeomorphisms.\n\n# Busemann functions and visual metrics for CAT(-1) spaces\nBusemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces.\nThese 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.\nIn the case of the unit disk the chordal metric can be recovered directly using Busemann functions\nBγ\nand the special theory for the disk generalises completely to any proper CAT(-1) space\nX\n. 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.\nFrom 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\nγ\nsuch that has these points as the limits\nγ(±∞)\n. The theory applies equally well to any CAT(\n−κ\n) space with\nκ > 0\nsince these arise by scaling the metric on a CAT(-1) space by\nκ−1/2\n. On the hyperbolic unit disk\nD\nquasi-isometries of\nD\ninduce quasi-Möbius homeomorphisms of the boundary in a functorial way.\nThere is a more general theory of Gromov hyperbolic spaces, a similar statement holds, but with less precise control on the homeomorphisms of the boundary.\n\n## Example: Poincaré disk\n\n# Applications in percolation theory\nMore recently Busemann functions have been used by [probabilists](https://bliptext.com/articles/probability-theory) to study asymptotic properties in models of [first-passage percolation](https://bliptext.com/articles/first-passage-percolation) and directed last-passage percolation.","revisionCount":0},"hasMore":false,"nextCursor":null}

History