Two-dimensional metric spaces with curvature bounded above I
aa r X i v : . [ m a t h . M G ] F e b TWO-DIMENSIONAL METRIC SPACES WITHCURVATURE BOUNDED ABOVE I
KOICHI NAGANO, TAKASHI SHIOYA, AND TAKAO YAMAGUCHI
Abstract.
We determine the local geometric structure of two-dimensional metric spaces with curvature bounded above as theunion of finitely many properly embedded/branched immersed Lip-schitz disks. As a result, we obtain a graph structure of the topo-logical singular point set of such a singular surface.
Contents
1. Introduction 12. Basic properties of CAT( κ )-spaces 63. Basic properties of ruled surfaces 104. Thin ruled surfaces 164.1. Behavior of ruling geodesics 184.2. Canonical balls 264.3. Spaces of directions 304.4. Proof of Theorem 4.1 325. Filling via CAT( κ )-disks 366. Graph structure of singular set 41Appendix A. Alexandrov’s result on ruled surfaces 47A.1. Finite sequences of ruling geodesics 48A.2. Curvature bounds on ruled surfaces 51References 541. Introduction
Let X be a locally compact, geodesically complete Alexandrov spacewith curvature bounded above. In this paper, we are concerned withthe local structure of X . In general X may have very complicatedlocal geometry. For instance, X may have no polyhedral structureeven in local. There is such a two-dimensional space constructed by Date : February 2, 2021.2010
Mathematics Subject Classification.
Primary 53C20, 53C23.
Key words and phrases.
Upper curvature bound; ruled surface; singular set.This work was supported by JSPS KAKENHI Grant Numbers 18H01118,15H05739, 19K03459,15K13436, 26610012, 21740036,17204003,18740023 .
Kleiner (cf.[18]). In the present paper, we completely describe the localgeometry of such spaces in dimension two.The study of metric spaces with cuvature bounded above beganwith the work of Alexandrov [3]. For the dimensions of such spaces X , Kleiner [14] proved that the topological dimension coincides withthe maximal dimension of topological manifolds embedded in X . For geodesically complete metric spaces X with curvature bounded above,Otsu-Tanoue [21] implicitly showed that the topological dimension co-incides with the Hausdorff dimension, which has been verified via adifferent method by a recent work due to Lytchak-Nagano [16]. [16]has also clarified that the local geometric properties of geodesicallycomplete metric spaces X with cuvature bounded above have a lot ofanalogues to those of Alexandrov spaces with curvature bounded be-low (see also Remarks 1.4 and 1.6 below). Lytchak-Stadler [17] haverecently proved that for every convex open ball in a CAT( κ )-spacethere exists a complete CAT( − κ )-metric; in particular, in local con-siderations on topological properties of CAT( κ )-spaces, we may assume κ to be − boundedcurvature . They constructed the curvature measure on such surfacesand established the Gauss-Bonnet theorem. See also Reshetnyak [24]for the work from an analytic point of view. Generalizing [6] andsucceeding the works of Balmann-Buyalo [9] and Arsinova-Buyalo [2],Burago-Buyalo [12] established the theory of two-dimensional polyhe-dra with curvature bounded above.Here it should be emphasized that there were no general resultsdetermining local structure even in dimension two. The purpose ofthis paper is to determine the general local geometric structure oftwo-dimensional geodesically complete metric spaces with curvaturebounded above.Let X be a two-dimensional locally compact, geodesically completemetric space with curvature ≤ κ for a constant κ . For every p ∈ X , thespace of directions Σ p = Σ p ( X ) is the disjoint union of finitely manypoints and connected finite graphs. Since we are interested in the localstructure, we assume the most essential case when Σ p is a connectedgraph, called a CAT(1)-graph (see Section 2). We shall determine thegeometry of the closed r -ball B ( p, r ) around p for small enough r > WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 3
Let S ( X ) denote the set of all topological singular points in X . For ℓ ≥ π and r >
0, we denote by D ( ℓ ; r ) the closed disk of radius r around the vertex O in the Euclidean cone over the circle of length ℓ .For x ∈ ∂D ( ℓ ; r ) and ǫ >
0, let Ω( x, ǫ ) denote the sector spanned bythe vertex and the subarc of ∂D ( ℓ ; r ) around x of length ǫ . A map f : D ( ℓ ; r ) → B ( p, r ) is called proper if f − ( ∂B ( p, r )) = ∂D ( ℓ ; r ). Let τ p ( r ) denote a function depending on p and r satisfying lim r → τ p ( r ) =0. Let S ( p, r ) denote the metric sphere ∂B ( p, r ).The main result in this paper is stated as follows. Theorem 1.1.
For every p ∈ X such that Σ p is a connected graph,there exists a positive number r such that for every < r ≤ r , B ( p, r ) is a union of images Im f i of finitely many proper Lipschitz immersions f i : D ( ℓ i ; r ) → B ( p, r ) , ℓ i ≥ π , possibly with branch point f − i ( p ) = { } satisfying the following :(1) With respect to the length metric induced from X , Im f i are CAT( κ ) -spaces ;(2) Either f i is an embedding, or else there are x + , x − ∈ ∂D ( ℓ i ; r ) such that for any multiple point q ∈ Im f i , f − i ( q ) consists ofexactly two points q + ∈ Ω( x + , τ p ( r ) r ) and q − ∈ Ω( x − , τ p ( r ) r ) .In the latter case, ℓ i ≥ π and ∠ x + Ox − > π − τ p ( r ) .Moreover, S ( X ) ∩ B ( p, r ) consists of finitely many simple Lipschitz arcsstarting from p and reaching S ( p, r ) . Remark 1.2. (1) The bi-Lipschitz constant of f i is less than 1 + τ p ( r )when f i is an embedding. If f i is a branched immersion, the localbi-Lipschitz constant of f i except { p } is less than 1 + τ p ( r ).(2) One might ask if it is possible to fill the ball B ( p, r ) with thoseIm f i that are convex in X or properly embedded disks. However, bothare impossible in general. For example, take the domain D on the xy -plane bounded by the curves C : y = x ( x ≥
0) and C : y = − x ( x ≥
0) containing the positive x -axis. Let R and R be two copiesof R , and let f i : C i → R i ( i = 1 ,
2) be isometric maps preservinglength such that the images f i ( C i ) coincide with the positive x -axisesof R i . Then the glued space X = R ∪ f D ∪ f R is a CAT(0)-space.Note that any metric ball around the origin of X can not be written asa union of properly embedded disks as described in Theorem 1.1 (seealso Examples 4.5, 5.4 and 5.5).Using Theorem 1.1, we can define a metric graph structure on S ( X )in a generalized sense (see Definition 6.8), and we have Corollary 1.3.
Suppose that Σ p is a connected graph for every p ∈ X .Then with respect to the induced length structure, S ( X ) is isometric toa metric graph having (possibly locally uncountably many vertices, but)the vertices of locally finite order. K.NAGANO, T.SHIOYA, AND T.YAMAGUCHI
Remark 1.4.
In the general dimension, [16] has characterized the sin-gular set in the k -dimensional part as a countably ( k − µ )-graph ( µ >
0) ifevery non-contractible loop in Σ has length ≥ π/ √ µ . Corollary 1.5.
For a given p ∈ X such that Σ p is a connected graph,there exists a positive number r p such that for every < r ≤ r p , S ( p, r ) with the interior metric is a CAT( µ κ ( r )) -graph having the same homo-topy type as Σ p , where µ κ ( r ) is given by the sharp constant µ κ ( r ) = (cid:16) sin √ κr √ κ (cid:17) − if κ > , r − if κ = 0 , (cid:16) sinh √− κr √− κ (cid:17) − if κ < . Remark 1.6.
A result in [16] shows that for every small r , S ( p, r ) hasthe same homotopy type as Σ p in the general dimension. Corollary 1.5gives a refinement of this result in dimension two.The idea of the proof of the main result is as follows. We know thestructure of the space Σ p of directions at p , which is completely charac-terized as a CAT(1)-graph. If we rescale the metric of X by the factor1 /r , then ( r X, p ) converges to the tangent cone ( K p , o p ) at p as r → sing p be asmall neighborhood of the vertices of the graph Σ p , and Σ reg p the com-plement of Σ sing p . Now the convergence theorem ([19]) applied to theunit cone K (Σ reg p ) over Σ reg p yields the existence of a Lipschitz domain B reg ( p, r ) of B ( p, r ) consisting of finitely many sectors correspondingto sectors of K (Σ reg p ). One can consider B reg ( p, r ) as a regular part of B ( p, r ). The main problem is to determine the structure of the singularpart B sing ( p, r ), the complement of B reg ( p, r ) in B ( p, r ). To carry outthis, we consider finitely many thin ruled surfaces S ij and fill B sing ( p, r )using them. A key is to show that those ruled surfaces are CAT( κ )-spaces with respect to the interior metrics and are homeomorphic toa disk. According to Alexandrov’s result in [4], every ruled surfacein a CAT( κ )-space is also a CAT( κ )-space with respect to the pull-back metric . Obviously, the interior metric and the pullback metric arecompletely different from each other in general. Therefore we have toshow that in our thin ruled surfaces pullback metrics coincide with theinterior metrics. After achieving this, it turns out that the topologicalsingular point set S ( X ) locally arises from the intersections of thosethin ruled surfaces S ij . We investigate how those ruled surfaces meeteach other to get the structure of S ( X ) ∩ B ( p, r ) as the union of finitely WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 5 many Lipschitz curves. Combining the structures of both B reg ( p, r )and B sing ( p, r ) and considering the graph structure of Σ p , we definethe embeddings or the branched immersions f i : D ( ℓ i ; r ) → B ( p, r ) asdescribed in Theorem 1.1.As related studies on ruled surfaces, Petrunin-Stadler [22] have provedthat for metric minimizing disks in CAT(0)-spaces, the pullback met-rics on the disks are CAT(0), which are generalization of Alexandrov’sresult [4] on ruled surfaces in the CAT(0)-setting. According to Stadler[25, Theorem 2], for any Jordan triangle in a CAT(0)-space, every min-imal disk filling of the triangle is an embedd disk that is CAT(0) withrespect to the interior metric.The organization of the paper is as follows.In Section 2, we recall and verify basic results for locally compact,geodesically complete Alexandrov spaces with curvature bounded above.In Section 3, we give basic properties of a ruled surface S in aCAT( κ )-space. We discuss the pullback metric, the induced metric,the interior metric of S and their relations. In the original argumentin Alexandrov [4], there are several unclear points for the authors. Forinstance, there is no description in [4] about continuous monotone rep-resentations. We make clear all these points.In Section 4, which is a key section, we investigate a thin ruledsurface S in a two dimensional space, and prove that S actually admitsthe induced metric and therefore becomes a CAT( κ )-space with respectto the interior metric. Then we obtain the crucial property that S ishomeomorphic to a disk.In Section 5, we fill B ( p, r ) via those imbedded/branched immerseddisks using thin ruled surfaces essentially.In Section 6, we describe S ( X ) ∩ B ( p, r ) as a union of finitely manyLipschitz curves starting from p and reaching points of S ( p, r ). Thestructure of generalized metric graph of S ( X ) is also discussed there.In Appendix A, following the basic idea of [4], we give the proofof Alexandov’s result on ruled sufaces in CAT( κ )-spaces based on theresults proved in Section 3.In the second part [15] of our works, we show that(a) each singular curve in the statement of the second-half of Theorem1.1 has turn of bounded variation in the sense of [6].We should point out that Burago-Buyalo [12] gave a complete charac-terization of two-dimensional polyhedra of curvature bounded above.In [15], we show that the conclusions in Theorem 1.1 together with theabove (a) completely characterize two-dimensional metric spaces withcurvature bounded above.In [15], we also show that(b) any pointed two-dimensional geodesically complete locally CAT( κ )-space ( X, p ) can be approximated by a sequence of two-dimensional
K.NAGANO, T.SHIOYA, AND T.YAMAGUCHI pointed geodesically complete, polyhedral locally CAT( κ )-spaces ( X n , p n )having the same homotopy types as X with respect to the pointedGromov-Hausdorff topology. This solves a problem raised in Burago-Buyalo [12].(c) we establish a Gauss-Bonnet type theorem for two-dimensionalgeodesically complete locally CAT( κ )-space.Most results in the present paper were announced in [27]. Acknowledgements
First of all, the authors would like to thankBruce Kleiner. The outline of the results in this paper came fromthe discussions with him on the basic idea many years ago. We wouldalso like to thank Alexander Lytchak for informing recent related re-sults on ruled surfaces and minimal filling disks in CAT(0)-spaces. Wewould also like to thank Werner Ballmann, Yuri Burago, Sergei Buyalo,Misha Gromov for their interest to this work. This work was partiallysupported by IHES, while the last named author was in residence there,during the summer of 2005.2.
Basic properties of
CAT( κ ) -spaces For some basic results in this section, we refer to [10], [11].The distance between two points x, y in a metric space X is denotedby | x, y | or | x, y | X , and d ( x, y ) or d X ( x, y ) sometimes. Let X be a locallycompact, complete geodesic space with curvature ≤ κ . By definition,for each point p ∈ X , there exists a positive number r > r ≤ π/ √ κ when κ > r -ball B ( p, r ) around p isconvex and having the following properties: Let M κ be the simplyconnected complete surface of constant curvature κ , called the κ -plane in short. For any geodesic triangle △ xyz in B ( p, r ) with vertices x, y and z , we denote by ˜ △ xyz a comparison triangle in M κ having thesame side lengths as △ xyz . Then the natural mapping ˜ △ xyz → △ xyz is non-expanding. A convex domain with this property is called aCAT( κ ) -domain . Such a space X with curvature ≤ κ is called a locallyCAT( κ )-space, and X is called a CAT( κ )-space if X itself is a CAT( κ )-domain. For arbitrary x and y in B ( p, r ), let γ x,y : [0 , | x, y | ] → X denote a unique minimal geodesic joining x to y . The angle betweenthe geodesics γ y,x and γ y,z is denoted by ∠ xyz , and the correspondingangle of ˜ △ xyz by ˜ ∠ xyz . The space of directions and the tangent coneof X at p are denoted by Σ p = Σ p ( X ) and K p = K p ( X ) respectively.We shall occasionally use the identification Σ p = Σ p × { } ⊂ K p . Wedenote by ˙ γ x,y (0), γ ′ x,y (0) or ↑ yx , the direction at x defined by γ x,y . Forevery ξ ∈ Σ p ( X ), γ ξ denotes a geodesic with ˙ γ ξ (0) = ξ . For a path-connected subset S ⊂ X and x, y ∈ S , we denote by γ Sx,y a shortestcurve in S joining x to y if it exists. Occasionally, we identify a geodesic WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 7 with its image, and write as x ∈ γ for instance. The length metric of S induced from X is denoted by d S or | , | S .For a closed subset A of X and for an accumulation point p of A ,the set of all directions ξ ∈ Σ p ( X ) such that there is a sequence a i in A \ { p } satisfying a i → p and lim γ ′ p,a i (0) = ξ is denoted by Σ p ( A ) andcalled the space of directions of A at p .The upper semi-continuity of angle is fundamental in the geometryof spaces with curvature bounded above. Lemma 2.1.
Suppose that sequences p i , q i and r i converge to p , q and r respectively in a CAT( κ ) -domain. Then we have lim sup ∠ p i q i r i ≤ ∠ pqr . Let d p denote the distance function from p . For every x = p , letus denote by ( ∇ d p )( x ) the set of all directions ξ ∈ Σ x ( X ) such that ∠ ( ξ, ↑ px ) = π . For simplicity, we set − ( ∇ d p )( x ) = ↑ px . The followinglemma, which describes local geometry around a given point, is basic inour study of local structure of surfaces with curvature bounded above.We denote by τ p ( ǫ , . . . , ǫ k ) a function depending on p and ǫ , . . . , ǫ k satisfying lim ǫ ,...,ǫ k → τ p ( ǫ , . . . , ǫ k ) = 0. Lemma 2.2.
For every p ∈ X , there exists a positive number r suchthat for every r with < r ≤ r , B ( p, r ) satisfies the following :(1) diam (( ∇ d p )( x )) < τ p ( r ) for every x ∈ B ( p, r ) \ { p } ;(2) For any two geodesics γ and γ starting at p with angle θ , andfor every s ∈ [0 , r ] , the geodesic σ s ( t ) joining γ ( s ) to γ ( s ) satisfies that | ∠ ( − ( ∇ d p )( σ s ( t )) , σ ′ s ( t )) − π/ | < τ p ( θ, s ) . Proof. (1) is due to [21] (see also [16, Prop.7.3]). (2) easily follows from(1), and hence the proof is omitted. (cid:3)
The following lemma is fundamental, and plays an important role asin the case of Alexandrov space with curvature bounded below ([13]).For the proof, see [19, Lemma 3.6].
Lemma 2.3 (Jack Lemma) . For every p ∈ X , there exists a positivenumber r such that if x = y ∈ B ( p, r ) and q satisfy that ˜ ∠ pxq > π − ǫ and | x, y | < ǫ min {| p, x | , | q, x |} , then we have | ∠ pxy − ˜ ∠ pxy | < τ p ( | p, x | , ǫ ) . In the study of spaces of curvature bounded below, the theory ofthe Gromov-Hausdorff convergence has been useful. We apply it in ourcase of curvature bounded above.We denote by H n the n -dimensional Hausdorff measure, and set ω n := H n ( S n (1)), where S n (1) is the unit n -sphere. K.NAGANO, T.SHIOYA, AND T.YAMAGUCHI
Theorem 2.4 ([19], Compare [26]) . For each positive integer n , thereis a positive number ǫ n satisfying the following: Let X i , i = 1 , , . . . ,and X be n -dimensional locally compact, geodesically complete, pointedAlexandrov spaces with curvature ≤ κ , and suppose that a compact CAT( κ ) -domain U i of X i converges to a compact CAT( κ ) -domain U of X with respect to the Gromov-Hausdorff distance. Then for everycompact domain V in int U satisfying H n (Σ x ( X )) < ω n + ǫ with ǫ ≤ ǫ n for all x ∈ V , there is a compact domain V i in int U i and a τ ( ǫ, /i ) -almost isometry ϕ i : V i → V in the sense that (cid:12)(cid:12)(cid:12)(cid:12) | ϕ i ( x ) , ϕ i ( y ) || x, y | − (cid:12)(cid:12)(cid:12)(cid:12) < τ ( ǫ, /i ) , for all x, y ∈ V i . A point p in X is called a topological singular point of X if anyneighborhood of p is not homeomorphic to a disk, and the set of alltopological singular point of X is denoted by S ( X ). It is proved in [16]that if dim X = n , then dim H S ( X ) ≤ n −
1. In particular X \ S ( X )has full measure with respect to H n ([21]).Next we shortly discuss the connectivity of a small neighborhoodof a given point in X . For each point p ∈ X , the set of componentsof Σ p are in one to one correspondence with the set of components of B ( p, r ) \ { p } if B ( p, r ) is a CAT( κ )-domain. We call the number ofcomponents of Σ p ( X ) the order of p . Lemma 2.5.
For a point p ∈ X , suppose that Σ p ( X ) has no isolatedpoints. Then there exists a positive number r such that every point x in B ( p, r ) \ { p } has order one.Proof. Suppose that Σ x ( X ) is disconnected for some x ∈ B ( p, r ) \ { p } for sufficiently small r . Then Lemma 2.2 (1) implies that ( −∇ d p )( x )and ( ∇ d p )( x ) lie in different components in Σ x ( X ) respectively. Extend γ p,x in the direction of one of ( ∇ d p )( x ) and let y := γ p,x (2 s ), where s = | p, x | . Take ξ ∈ Σ p ( X ) \ { γ ′ p,x (0) } with ∠ ( ξ, γ ′ p,x (0)) being small,and set z := γ ξ (2 s ). By Lemma 2.2(2), γ z,y does not meet x . Thereforejoining x to each point of the broken geodesic γ p,z ∪ γ z,y by a geodesic, wehave a one-parameter family of directions in Σ x ( X ) joining ( −∇ d p )( x )and ( ∇ d p )( x ), which is a contradiction. (cid:3) Now we state the gluing theorem proved by [23], which is convenientto construct spaces with curvature bounded above. The proof is alsofound in [10, p.347].
Theorem 2.6.
Let D i , i = 1 , , be a closed convex subset in an Alexan-drov space X i with curvature ≤ κ . If there is an isometry f : D → D ,then the identification space X ∪ f X is an Alexandrov space with cur-vature ≤ κ with respect to the natural length metric. WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 9
Two-dimensional case.
By a result of Otsu-Tanoue [21], the Haus-dorff dimension of every relatively compact open domain of X is aninteger. See [16] for a different proof. It is also known that Σ p ( X ) isa compact geodesically complete CAT(1)-space for every p ∈ X . Nowwe assume X has dimension 2. Then any component Σ of Σ p has di-mension ≤
1. If dim Σ = 1, then Σ has the structure of a finite graph,where a point of Σ is a vertex of the graph if and only if it has order ≥
3. Furthermore Σ is a so called CAT(1) -graph in the sense that it hasno endpoints (that is, points of order 1) and each simple closed curvein Σ has length at least 2 π . If dim Σ = 0, then Σ is a point and thecomponent of B ( p, r ) \ { p } corresponding to Σ is an arc for any smallenough r . Therefore the study of local structure around p reduces tothe case when Σ p is a connected CAT(1)-graph. Lemma 2.7.
A neighborhood of p ∈ X is homeomorphic to a two-dimensional disk if and only if Σ p ( X ) is a circle.Proof. If Σ p ( X ) is a circle, then Theorem 2.4 applied to B ( p, r ) \ { p } implies that B ( p, r ) must be bi-Lipschitz homeomorphic to a two-dimensional disk for a sufficiently small r (see [20, Prop.3.1] for thedetail).Conversely, if a neighborhood of p ∈ X is homeomorphic to a two-dimensional disk, then it follows from of [6, Theorem 11 in Chapter II]for instance that Σ p ( X ) is a circle. Note that a two-dimensional man-ifold with curvature bounded above is a surface of bounded curvaturein the sense of [6]. (cid:3) Lemma 2.8.
Let p ∈ S ( X ) . Then Σ p ( S ( X )) is contained in the set V (Σ p ) of all vertices of the graph Σ p ( X ) .Proof. For every v ∈ Σ p ( S ( X )), take a sequence x i in S ( X ) convergingto p such that lim ∠ ( γ ′ p,x i (0) , v ) = 0. If v is not a vertex of Σ p ( X ),choose ǫ > ǫ -neighborhood of v contains no verticesof Σ p ( X ). Let δ i := | x i , p | . Theorem 2.4 applied to the convergence( δ i X, x i ) → ( K p ( X ) , v ) yields that a small neighborhood of x i is almostisometric to a neighborhood in R . This is a contradiction. (cid:3) Remark 2.9.
The converse to Lemma 2.8 is also true. This will beproved in Lemma 5.1
Lemma 2.10.
Let p ∈ S ( X ) . For any x ∈ S ( X ) ∩ ( B ( p, r ) \ { p } ) , V (Σ x ( X )) is contained in the τ p ( r ) -neighborhood of { ( −∇ d p )( x ) , ( ∇ d p )( x ) } .Therefore there is a positive integer m ≥ such that the Gromov-Hausdorff distance between Σ x ( X ) and the spherical suspension over m points is less than τ p ( r ) .Proof. If the lemma does not hold, we have a sequence x i → p in S ( X ) such that for some vertex v i of Σ x i ( X ), we have 0 < c ≤ ∠ (( −∇ d p )( x i ) , v i ) ≤ π − c , where c is some positive number inde-pendent of i . Let C i be a circle in Σ x i ( X ) of length less than 2 π + ǫ i containing ( −∇ d p )( x i ), an element of ∇ d p ( x i ) and v i . We may assumelim ǫ i = 0. By the CAT(1)-property of Σ x i ( X ), one can take an arc I i of Σ x i ( X ) meeting C i only at v i such that ∠ (( −∇ d p )( x i ) , w i ) = π for some w i ∈ I i . Since ∠ (( ∇ d p )( x i ) , w i ) ≥ c , this contradicts Lemma2.2(1). (cid:3) As an immediate consequence of Lemmas 2.8 and 2.10, we have
Corollary 2.11.
Let p ∈ S ( X ) . For every x ∈ S ( X ) ∩ ( B ( p, r ) \ { p } ) , Σ x ( S ( X )) is contained in a τ p ( r ) -neighborhood of { ( −∇ d p )( x ) , ( ∇ d p )( x ) } . Finally in this subsection, we shortly discuss the cardinality of sin-gular points in a two-dimensional manifold X with curvature ≤ κ . Let ǫ >
0. We say that x ∈ X is an ǫ -singular point if L (Σ x ( X )) ≥ π + ǫ .We also say that x is a singular point if it is ǫ -singular for some ǫ > Lemma 2.12. ( cf. [6] , [12, Prop.4.5]) For a bounded domain D of atwo-dimensional manifold X with curvature ≤ κ , the set of all singularpoints contained in D is at most countable.Proof. By Lemma 2.2(1), the set of all ǫ -singular points contained in D is finite for every ǫ >
0, which immediately yields the conclusion ofthe lemma. (cid:3) Basic properties of ruled surfaces
We recall the notion of ruled surfaces in metric spaces introduced byAlexandrov [4]. The metric on a ruled surface discussed in [4] is thepullback metric defined below, although an explicit definition was notgiven in [4]. For instance, the property corresponding to Sublemma 3.2was assumed as one of the conditions of the metric on the ruled surfaceunder consideration. In this section, we provide some fundamentalproperties of the pullback metric, most of which are not contained in[4]. These are used in the proof of Alexandrov’s result (Theorem 3.6),which is presented in Appendix A. There are related results in [22,Section 2].For our purpose, it is sufficient to consider ruled surfaces in spaceswith curvature bounded above. We fix a rectangle R := [0 , ℓ ] × [0 ,
1] inthis section.
Ruled surfaces.
Let X be a locally compact, complete geodesic spacewith curvature ≤ κ with metric d X . A continuous map σ : R → X iscalled a ruled surface in X if(1) for every s ∈ [0 , ℓ ] the t -curve λ s : [0 , → X of σ definedas λ s ( t ) := σ ( s, t ) is a minimal geodesic in X from σ ( s,
0) to σ ( s, WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 11 (2) for some t ∈ [0 ,
1] the s -curve σ t : [0 , ℓ ] → X of σ defined as σ t ( s ) := σ ( s, t ) is rectifiable with respect to d X .As usual, the subset S of X defined as S := σ ( R ) is also called a ruledsurface in X . For each s ∈ [0 , ℓ ], the minimal geodesic λ s : [0 , → X is called a generator of σ , or a ruling geodesic of σ . For each t ∈ [0 , σ t : [0 , ℓ ] → X is called a directrix of σ at t . Pullback metrics and induced metrics on ruled surfaces
Let σ : R → X be a ruled surface in X defined as above. We denoteby e σ the pull-back metric on R induced from σ defined as e σ ( u, u ′ ) := inf c L ( σ ◦ c ) , where c runs over all continuous curves in R from u to u ′ , and L denotes the length of curves with respect to d X . Note that the metric e σ is certainly finite since our ruled surface σ has a rectifiable directrix.We denote by R ∗ the quotient metric space( R ∗ , e σ ) := ( R, e σ ) / { e σ = 0 } . Let π : R → R ∗ be the projection. Obviously, e σ ( u, u ′ ) = 0 implies σ ( u ) = σ ( u ′ ). Therefore, we can define a continuous map σ ∗ : R ∗ → X such that σ = σ ∗ ◦ π .We say that a curve c : [ a, b ] → R is monotone if p ◦ c is monotonenon-decreasing or monotone non-increasing where p : R → [0 , ℓ ] is theprojection. Similarly, c is said to be strictly monotone if p ◦ c is strictlymonotone.For s ∈ [0 , ℓ ], we set I s := { s } × [0 , ⊂ R . Lemma 3.1.
For arbitrary u, u ′ ∈ R , there is a continuous monotonecurve c in R from u to u ′ such that e σ ( u, u ′ ) = L ( σ ◦ c ) .Proof. We may assume u = ( s , t ) , u ′ = ( s ′ , t ′ ) with s < s ′ . Take asequence of continuous curves c n : [0 , → R from u to u ′ such that e σ ( u, u ′ ) = lim n →∞ L ( σ ◦ c n ). Note that the length of c n with respectto e σ is uniformly bounded. Therefore passing to a subsequence, wemay assume that π ◦ c n converges to a continuous curve c ∗ : [0 , → R ∗ from π ( u ) to π ( u ′ ) satisfying L ( c ∗ ) ≤ lim inf n →∞ L ( π ◦ c n ) = e σ ( π ( u ) , π ( u ′ )) . Thus c ∗ is a e σ -shortest curve from π ( u ) to π ( u ′ ).First we consider the special case when e σ ( u, u ′ ) = 0. Sublemma 3.2. If e σ ( u, u ′ ) = 0 , then there is a continuous strictlymonotone curve c : [0 , → R joining u to u ′ such that π ( c ) = π ( u ) = π ( u ′ ) .Proof. In this case, c ∗ is a point and σ ( c ∗ ) = x , where x := σ ( u ) = σ ( u ′ ). We show that π − ( c ∗ ) defines a continuous curve in R containing u, u ′ . For each s ≤ s ≤ s ′ , take v n ( s ) ∈ c n ∩ I s . Passing to a subsequence, we may assume that v n ( s ) converges to a point v ( s ) ∈ I s .Note that π ( v ( s )) = x and v ( s ) is unique since π is injective on I s .Therefore v ( s ) ( s ≤ s ≤ s ′ ) defines a continuous strictly monotonecurve in R joining u to u ′ such that v ( s ) ∈ π − ( c ∗ ). (cid:3) Let us go back to the general case, and set I ∗ s := π ( I s ). We showthat π − ( c ∗ ) defines a continuous monotone curve joining u to u ′ . Wedivide the argument into several steps.1). For every s ∈ [ s , s ′ ], π − ( c ∗ ) ∩ I s is nonempty. This follows froma limit argument via π ( c n ) → c ∗ .2). If v, v ′ ∈ π − ( c ∗ ) ∩ I s , then the segment [ v, v ′ ] in I s is also containedin π − ( c ∗ ) ∩ I s . This follows from the fact that the subarc of λ s between σ ( v ) and σ ( v ′ ) is a unique shortest arc in X joining the endpoints.3). We set J := { s ∈ [ s , s ′ ] | ♯ ( π − ( c ∗ ) ∩ I s ) ≥ } . For each s ∈ J , wedenote by K s the closed interval π − ( c ∗ ) ∩ I s in I s . For each 1 > ǫ > J ǫ := { s ∈ J | L ( σ ( K s )) ≥ ǫ } . Since J ǫ is finite, J is at mostcountable. Then we have X s ∈ J L ( σ ( K s )) ≤ L ( c ∗ ) < ∞ . J := [ s , s ′ ] \ J . For each s ∈ J , we set v ( s ) := π − ( c ∗ ) ∩ I s .Clearly, v ( s ) is continuous on J . Set A := { v ( s ) | s ∈ J } . Note that¯ J \ J ⊂ J ,5). We show that if s k ∈ J converges to s ∈ J while keeping s k < s (resp. s k > s ), then v ( s k ) converges to an endpoint of K s (resp. theother endpoint of K s ). Suppose that a subsequence of v ( s k ) convergesto an interior point v of the interval K s . Passing to a subsequence,we may assume that two points, say a n , b n , of c n converges to v andan endpoint of K s respectively. It turns out that the length of π ( c n )diverges to ∞ . This is a contradiction. A similar argument also impliesthat v ( s k ) converges to an endpoint of K s (resp. the other endpoint of K s ) if s k < s (resp. if s k > s ).6). We denote by M ǫ the union of K s for all s ∈ J ǫ , and set M := S ǫ> M ǫ . Let E := [ s , s ′ + L ( σ ( M ))]. Consider a certain parameterproportional to arc-length on each K s for s ∈ J ǫ defined by σ ( K s ), andtranslate each separated part of J by J ǫ properly. Thus, in view ofStep 5), we can define a natural monotone parameter on C ǫ := A ∪ M ǫ as C ǫ = C ǫ ( t ) for t ∈ E ǫ , where E ǫ is a subset of the closed interval E with ∂E ⊂ E ǫ , such that C ǫ ( t ) is continuous on E ǫ . Note thateach component H of E \ E ǫ corresponds to an K s in M \ M ǫ with L ( H ) = L ( σ ( K s )). In particular E ǫ is τ ( ǫ )-dense in E . Note that E ǫ ⊂ E ǫ ′ for ǫ ′ < ǫ .7). Let C be the union S ǫ> C ǫ . Since C ǫ converges to C and L ( C \ C ǫ ) → ǫ →
0, in view of Steps 4), 5) and 6), the parametrization C ǫ ( t ) of C ǫ on E ǫ naturally extends to a parametrization C ( t ) of C WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 13 on E . Note that c ( t ) := C ( t ) is monotone and continuous on E , andthat π ( c ) = c ∗ , e σ ( u, u ′ ) = L ( c ∗ ) = L ( σ ◦ c ). This completes the proofof Lemma 3.1 . (cid:3) In Lemma 3.1, essentially we have proved the following:
Corollary 3.3.
Let c ∗ : [ a, b ] → ( R ∗ , e σ ) be a continuous curve from π ( u ) to π ( u ′ ) satisfying (1) for every s ∈ [ p ( u ) , p ( u ′ )] , π − ( c ∗ ) ∩ I s is nonempty ;(2) if v, v ′ ∈ π − ( c ∗ ) ∩ I s , then the segment [ v, v ′ ] in I s is alsocontained in π − ( c ∗ ) ∩ I s ;(3) L ( c ∗ ) < ∞ .Then there is a continuous monotone curve c in R from u to u ′ suchthat c ∗ = π ◦ c up to monotone parametrization. Proposition 3.4.
Let c ∗ : [ a, b ] → ( R ∗ , e σ ) be a shortest curve from π ( u ) to π ( u ′ ) . Then there is a continuous monotone curve c in R from u to u ′ such that c ∗ = π ◦ c up to monotone parametrization.Proof. In view of Corollary 3.3, it suffices to show that c ∗ ∩ I ∗ s isnonempty for every s ∈ [ p ( u ) , p ( u ′ )]. Suppose that c ∗ ∩ I ∗ s is empty forsome s ∈ [ p ( u ) , p ( u ′ )]. Then there are some s − < s + in [ p ( u ) , p ( u ′ )]satisfying c ∗ ⊂ π ([0 , s − ] × [0 , ∪ π ([ s + , ℓ ] × [0 , . Set R − := π ([0 , s − ] × [0 , R + := π ([ s + , ℓ ] × [0 , t − := sup { t | c ∗ ([0 , t ]) ⊂ R − } , t + := inf { t | c ∗ ([ t, ⊂ R + } . If t − = t + , then c ∗ ( t ± ) ∈ R − ∩ R + . It follows from Sublemma 3.2that p ( π − ( c ∗ ( t ± )) is a closed interval containing [ s − , s + ]. This is acontradiction. Thus we have t − < t + . Then for some sequence t i > t − with lim i →∞ t i = t − , we have c ∗ ( t i ) ∈ S + . It turns out that c ∗ ( t i ) ∈ I ∗ s i for some s i ≥ s + . This yields that c ∗ ( t − ) ∈ I ∗ s ∞ for some s ∞ ≥ s + .It follows from Sublemma 3.2 that p ( π − ( c ∗ ( t − ))) is a closed intervalcontaining [ s − , s + ]. This is a contradiction. (cid:3) Next, we give an explicite formulation of the pullback metric e σ . For u = ( s , t ) and u ′ = ( s ′ , t ′ ) with s < s ′ in R , let ∆ : s ≤ s ≤ · · · ≤ s n = s ′ be a decomposition of [ s , s ′ ], and set | ∆ | = max {| s i − s i − | | ≤ i ≤ n } . We consider e ∆ σ ( π ( u ) ,π ( u ′ )):= inf ( n X i =1 | x i − , x i | | x = σ ( u ) , x n = σ ( u ′ ) , x i ∈ λ s i ) . Let γ ∆ := S x i − x i denote a broken geodesic in X from σ ( u ) to σ ( u ′ )such that x i ∈ λ s i and L ( γ ∆ ) = e ∆ σ ( π ( u ) , π ( u ′ )). Lemma 3.5.
Under the above situation, we have the following : (1) e σ ( π ( u ) , π ( u ′ )) = sup ∆ e ∆ σ ( π ( u ) , π ( u ′ )) , where ∆ runs over all decompositions of [ s , s ′ ] ;(2) For any sequence ∆ n of decompositions of [ s , s ′ ] satisfying lim n →∞ | ∆ n | = 0 , we have e σ ( π ( u ) , π ( u ′ )) = lim n →∞ e ∆ n σ ( π ( u ) , π ( u ′ )) ;(3) For any sublimit γ of γ ∆ n , γ ∗ := σ − ∗ ( γ ) defines is a shortestcurve on ( R ∗ , e σ ) from π ( u ) to π ( u ′ ) , and has a monotone rep-resentaion. Namely there is a motone curve c in R such that γ ∗ = π ◦ c ( that is, γ = σ ◦ c ) up to monotone parametrization.Proof. Let c ∗ : [0 , → ( R ∗ , e σ ) be a shortest curve from π ( u ) to π ( u ′ ).By Lemma 3.1, there is a monotone curve c such that π ◦ c = c ∗ up tomonotone parametrization. Set γ ( t ) := σ ∗ ◦ c ∗ ( t ). For any decomposi-ton ∆ = { s i } Ni =1 of [ s , s ′ ], take t i ∈ [0 ,
1] such that γ ( t i ) ∈ λ s i . Thenwe have e ∆ σ ( π ( u ) , π ( u ′ )) ≤ N X i =1 | γ ( t i − ) , γ ( t i ) | ≤ L ( γ ) = L ( c ∗ ) = e σ ( π ( u ) , π ( u ′ )) . Thus we have sup ∆ e ∆ σ ( π ( u ) , π ( u ′ )) ≤ e σ ( π ( u ) , π ( u ′ )).Let ∆ be any decomposition of [ s , s ′ ] with lim n →∞ | ∆ n | = 0. Let γ n :[0 , → X be a ∆ n -minimizing broken geodesic in X . Take a subseqe-unce { n k } with lim k →∞ e ∆ nk σ ( π ( u ) , π ( u ′ )) = lim inf n →∞ e ∆ n σ ( π ( u ) , π ( u ′ )).Passing to a subseqeunce, we may assume that γ n k converges to a curve γ : [0 , → X . From | ∆ n | →
0, it follows that γ ([0 , ⊂ S .Next we show that γ has a monotone representation. Let us consider γ ∗ := σ − ∗ ( γ ). From | ∆ n | →
0, it follows that π − ( γ ∗ ) ∩ I s is nonemptyfor each s ∈ [ s , s ′ ]. Let J := { s ∈ [ s , s ′ ] | ♯ ( π − ( γ ∗ ) ∩ I s ) ≥ } . Weshow that K s := π − ( γ ∗ ) ∩ I s is a closed interval in I s for every s ∈ J .Let v = v ∈ K s . Let π ( v ) = γ ∗ ( t ) and π ( v ′ ) = γ ∗ ( t ′ ). Take t k , t ′ k ∈ [0 , γ n k ( t k ) → γ ( t ) and γ n k ( t ′ k ) → γ ( t ′ ), where we may assume t k < t ′ k . From the ∆ k -minimality of γ n k , we have L ( γ n k | [ t k ,t ′ k ] ) ≤ | γ ( t ) , γ ( t ′ ) | + ǫ k , with lim k →∞ ǫ k = 0. This implies that γ n k | [ t k ,t ′ k ] converges to the seg-ment [ γ ( t ) , γ ( t ′ )] in λ s . Thus [ v, v ′ ] ⊂ π − ( γ ∗ ) ∩ I s .By Corollary 3.3, one can introduce a continuous monotone parame-ter c ( s ) of π − ( γ ∗ ) such that σ ◦ c = γ up to monotone parametrization.Now we have e σ ( π ( u ) , π ( u ′ )) ≤ L ( γ ) ≤ lim k →∞ L ( γ n k ) = lim inf n →∞ e ∆ n σ ( π ( u ) , π ( u ′ )) . This completes the proof. (cid:3)
WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 15
In [4, Theorem 2], Alexandrov proved the following result, whichplays a crucial role in the present paper.
Theorem 3.6 ([4]) . Let S be a ruled surface in a CAT( κ ) -space X with parametrization σ : R → X . Then ( R ∗ , e σ ) is a CAT( κ ) -space. The proof of Theorem 3.6 is differed to Appendix A.One might expect to define the induced “metric” d σ on S along σ as d σ ( x, y ) := inf { e σ ( u, v ) | σ ( u ) = x and σ ( v ) = y } . However, d σ does not neessarily satisfy the triangle inequality. SeeRemark 4.3. Even if ( S, d σ ) becomes a metric space, in certain cases,it could be far from the notion of “induced metric”, as described in thefollowing example. Example 3.7.
Let us consider the following curve α : [0 , π ] → C on C = R defined as α ( s ) = e √− s (0 ≤ s ≤ π/ , (0 , s/π ) ( π/ ≤ s ≤ π ) , (0 , − s/π ) ( π ≤ s ≤ π/ ,e √− s − π ) (3 π/ ≤ s ≤ π ) . We define a ruled surface σ : [0 , π ] × [0 , → R by σ ( s, t ) = ( α ( s ) , t ).In this case, d σ is a distance on the image S of σ . Actually d σ coincideswith the interior metric of S defined in Definition 3.11.On the other hand, if we consider the restriction σ ′ of σ to [0 , π ] × [0 , d σ ′ is not the distance on the ruled surface S ′ defined by σ ′ . Lemma 3.8.
Suppose that we have for all u, v ∈ R , (3.1) σ ( u ) = σ ( v ) ⇐⇒ e σ ( u, v ) = 0 . Then ( S, d σ ) is a metric space, and σ ∗ : ( R ∗ , e σ ) → ( S, d σ ) is an isom-etry.Proof. First note that e σ ( u, u ′ ) = 0 implies σ ( u ) = σ ( v ). Suppose (3.1)holds for all u, v ∈ R . Then we have d σ ( x, y ) = e σ ( u, v ) for all x, y ∈ S and u ∈ σ − ( x ), v ∈ σ − ( y ). This implies that d σ is a metric on S . Itis also obvious that σ ∗ : ( R ∗ , e σ ) → ( S, d σ ) is an isometry. (cid:3) Definition 3.9.
We say that S has the induced metric from σ if σ ∗ : R ∗ → S is injective. This is the case when (3.1) holds for all u, v ∈ R ,and therfore σ ∗ : ( R ∗ , e σ ) → ( S, d σ ) is an isometry by Lemma 3.8. Inthis case, d σ is called the induced metric from σ . Corollary 3.10.
Let S be a ruled surface in a CAT( κ ) -space X withparametrization σ : R → X . Then if S has the induced metric from σ ,then ( S, d σ ) is a CAT( κ ) -space. Interior metrics on ruled surfaces
Let S be a ruled surface in X with parametrization σ : R → X . Definition 3.11.
We denote by d S the interior metric on S associatedwith d X defined as d S ( x , x ) := inf { L ( γ ) | γ is a curve in S from x to x } . Due to the Hopf-Rinow theorem for length spaces, (
S, d S ) is a geo-desic space. Note that d S ≤ d σ when S has the induced metric from σ . Theorem 3.12.
Let S be a ruled surface in a CAT( κ ) -space X withparametrization σ : R → X . If S has the induced metric from σ , thenwe have d S = d σ , and ( S, d S ) is also a CAT( κ ) -space.Proof. To see d S = d σ , it suffices to show d S ( x, x ′ ) ≥ d σ ( x, x ′ ) forarbitrary x, x ′ ∈ S . Take a d S -shortest curve γ : [0 , → S from x to x ′ . Let x ∈ λ s and x ′ ∈ λ s ′ , where we may assume s < s ′ . In away similar to the proof of Proposition 3.4, we can show that γ ∩ λ s is nonempty for every s ∈ [ s , s ′ ]. Obviously if γ ( t ) , γ ( t ) ∈ λ s with t < t , then γ ([ t , t ]) ⊂ λ s . Therefore by Corollary 3.3, we have acontinuous monotone curve c in R such that γ = σ ◦ c up to monotoneparametrization. Thus we have d S ( x, x ′ ) = L ( γ ) ≥ d σ ( x, x ′ ). FinallyCorollary 3.10 implies that ( S, d S ) is a CAT( κ )-space. This completesthe proof. (cid:3) Thin ruled surfaces
Let X be a locally compact, geodesically complete two-dimensionalspace with curvature ≤ κ , and fix p ∈ X . It is known that Σ p ( X ) is afinite metric graph without endpoints. For a vertex v of Σ p ( X ), take ν , ν ∈ Σ p ( X ) with equal distance to v such that ∠ ( ν , v ) + ∠ ( v, ν ) = ∠ ( ν , ν ) and v is the unique vertex contained in the shortest geodesicjoining ν and ν in Σ p ( X ). We set δ := ∠ ( ν , v ) = ∠ ( ν , v ) , (4.1)where δ is assumed to be small enough and will be determined lateron in Section 4. Let α i : [0 , ℓ ] → X be geodesics in the directions ν i , i = 1 ,
2. Joining α ( s ) to α ( s ) by the minimal geodesic λ s : [0 , → X ,we have a ruled surface S in X . Let B ( p, r ) be a small ball, and weassume ℓ = 2 r . Set R = [0 , ℓ ] × [0 , σ : R → S be the map thatdefines S : σ ( s, t ) = λ s ( t ) . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 17 p α ( s ) α ( s ) λ s λ ℓ σ ( s, t ) ν ν v We define the boundary and the interior of S as(4.2) ∂S := α ∪ α ∪ λ ℓ , int S := S \ ∂S. The purpose of this section is to prove the following
Theorem 4.1.
There exists an r p > such that for every r ∈ (0 , r p ] , S with length metric is a CAT( κ ) -space homeomorphic to a two-disk. The proof of Theorem 4.1 is completed in Subsection 4.4. As shownin the following example, Theorem 4.1 does not hold for a general ruledsurface even in a two-dimensional ambient space.
Example 4.2.
For any 0 < a < π/
2, let X be the complement of thedomain { ( x, y ) | | y | < (tan a ) x } on the xy -plane. For b with(4.3) a < b < a + π/ , consider the Euclidean cone K ( I ) over a closed interval I of length2 b . Let X be the gluing of X and K ( I ) along their boundaries,where the origin o of X is identified with the vertex of K ( I ). Let ξ be the midpoint of I and γ ξ denote the geodesic ray of X from o in the derection ξ . Next consider the Euclidean cone K ( J ) over aninterval J of length θ with π − ( b − a ) ≤ θ < π . Let X be the gluingof X and K ( J ) in such a way that ∂K ( J ) is identified with γ ξ and L := { ( x, | x ≤ } ⊂ X in an obvious way. It is easy to see that X is a locally compact, geodesically complete, two-dimensional CAT(0)-space. Let p = (0 , − ∈ X ⊂ X , and let σ + (resp. σ − ) be thegeodesic ray starting from o defined by the ray y = (tan a ) x (resp. bythe ray y = − (tan a ) x ). Note that the geodesic in X joining p and σ + (1) intersect σ − \ { o } because of (4.3). Let ℓ := 2 d ( p, σ + (1)), andlet α : [0 , ℓ ] → X be the geodesic starting from p through σ + (1). Let q be the intersection point of α with γ ξ . Let q be the point of L such that d ( p, q ) = d ( p, q ). Letting α : [0 , ℓ ] → X be the geodesicstarting from p through q , consider the ruled surface S = S ( α , α )in X . Let ∆ (resp.∆ ) be the geodesic triangle region in X (resp in K ( J )) with vertices p , α ( ℓ ) and α ( ℓ ) (resp. o , q and q ). Obviously, S is the gluing of ∆ and ∆ along the geodesic segments oq and oq .In particular S is not homeomorphic to a disk. p o q q α α K ( J ) S ( α , α ) X Remark 4.3. (1) Note that in Example 4.2, diam (( ∇ d p )( o )) = π ,which never happens in a small neighborhood of p by Lemma 2.2. Thissuggests the validity of Theorem 4.1, which is verified in the argumentbelow.(2) In Example 4.2, take two points x, y from the distinct componentsof S \ ∆ respectively. Then if x and y are sufficiently close to the point o , then we have d σ ( x, y ) > d σ ( x, o ) + d σ ( o, y ). Thus d σ is not a distancefor Example 4.2.4.1. Behavior of ruling geodesics.
In this subsection, we start thestudy of the behavior of ruling geodesics of S . We begin with twoexamples, which help us to understand the argument in the rest of thepaper. Example 4.4 (Kleiner) . ( cf. [19]) First consider a smooth nonnegativefunction f : R → R + such that { f = 0 } = { /n | n = 1 , , . . . } ∪ [1 , ∞ ) ∪ ( −∞ , { ( x, y ) | | y | ≤ f ( x ) , x ∈ R } , equipped withthe natural length metric induced from that of R . We set I + n := { ( x, + f ( x )) | / ( n + 1) ≤ x ≤ /n } , I − n := { ( x, − f ( x )) | / ( n + 1) ≤ x ≤ /n } , and L + := { ( x, | x ≥ } , L − := { ( x, | x ≤ } . Let ℓ n denote the length of I ± n , and let κ n be the maximum of absolutegeodesic curvature of I ± n . We choose f satisfying X ℓ n < ∞ , X κ n ℓ n < π. (4.4)By these conditions, one can take a closed domain H in R such that(1) ∂H is smooth, connected and concave in the sense that thegeodesic curvature is nonpositive everywhere;(2) there are consecutive points p , p , . . . , on ∂H such that thesubarc K n between p n and p n +1 of ∂H has length equal to ℓ n ;(3) if we denote p ∞ the limit of p n , the closure of the complementof the arc between p and p ∞ in ∂H consists of two geodesicrays, say R + and R − , in R with p ∈ R + and p ∞ ∈ R − ;(4) the absolute geodesic curvature of K n is greater than or equalto κ n everywhere. WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 19
Take four copies H , . . . , H of H , and denote K n , R ± ⊂ ∂H α by K ( α ) n , R ( α ) ± (1 ≤ α ≤
4) respectively. We put ∂ + Ω : = ∞ [ n =1 I + n ! ∪ L + ∪ L − ,∂ − Ω : = ∞ [ n =1 I − n ! ∪ L + ∪ L − . Now glue H , H and Ω along their boundaries ∂H , ∂H , ∂ + Ω in sucha way that I n , L + and L − are glued with K ( α ) n , R ( α )+ and R ( α ) − ( α = 1 , H , H and Ω along theirboundaries ∂H , ∂H and ∂ − Ω.Let X be the result of these gluings equipped with natural lengthmetric, which is a two-dimensional locally compact, geodesically com-plete space. Let ι : Ω → X be the natural inclusion, and let O =(0 , ∈ Ω. Note that no neighborhood of p := ι ( O ) in X has triangu-lation. Approximating f by functions f k ( k = 1 , , . . . ) that are 0 near0, we have polyhedral spaces X k in a similar way which approximate X in the sense of Gromov-Hausdorff distance. Applying a result in [12],we see that X k are CAT(0)-spaces. Thus the limit space X is also aCAT(0)-space. Note that S ( X ) consists of the two curves ι ( ∂ + Ω) and ι ( ∂ − Ω). p Ω I + n I − n L + L − H H H H Example 4.5.
This example is based on Example 4.4. The point is wemake different gluings. This time we glue H , H , H , H and Ω alongtheir boundaries as follows:(1) K (1) n is glued with I + n for all n ;(2) K (2) n is glued with I + n if n I − n if n ≡ K (3) n is glued with I + n if n ≡ , I − n if n ≡ , K (4) n is glued with I + n if n ≡ I − n if n Here, R α + and R α − (1 ≤ α ≤
4) are glued with L + and L − respectivelyin those gluings. The result Y of these gluings equipped with naturallength metric is a two-dimensional locally compact, geodesically com-plete CAT(0)-space. Let ι : ( ∐ i =1 H i ) ∐ Ω → Y be the identificationmap. Note thatfor all 1 ≤ α = β ≤ ι ( K αn ) = ι ( K βn ) for some n .(4.5)Let p := ι ( O ), where O is the origin of Ω, and let v denote the directionat p defined by the union of all I ± n ( n = 1 , , . . . ). For small ǫ >
0, takesufficiently small r > a i ∈ S ( p, r ) ∩ ι ( H i ), 1 ≤ i ≤
4, suchthat ∠ ( γ ′ p,a i (0) , v ) = ǫ . Let S ( a i , a j ) be the ruled surface defined in thebeginning of thes section. Then it follows from (4.5) that S ( a i , a j ) arenot convex in Y for ll i = j . Remark 4.6.
In Example 4.4, if we take a i in a way similar to Example4.5, then S ( a i , a j ) for i = 1 , j = 3 , X while S ( a , a ) and S ( a , a ) are not convex. Considering the other vertex ofΣ p ( X ), it is possible to fill a neighborhood of the singular set B ( p, r ) ∩S via those convex ruled surfaces. This is not the case of Example 4.5.For x ∈ S with x = λ s ( t ), from Lemma 2.2, we have | ∠ ( ± ˙ λ s ( t ) , ( ∇ d p )( x )) − π/ | < τ p ( | p, x | , δ ) . (4.6)For x ∈ S , let Σ x ( S ) denote the set of all directions ξ ∈ Σ x ( X ) suchthat ξ = lim i →∞ ↑ x i x for some sequence x i ∈ S with | x, x i | X →
0, as inSection 2. We call Σ x ( S ) the extrinsic space of directions of S at x . p xλ s (1) λ s (0) ↑ px ( ∇ d p )( x )Σ x ( X )In this paper, we use the following terminology. We call a direction ξ ∈ Σ x ( X ) horizontal if ∠ ( ξ, ± ( ∇ d p )( x )) ≤ π/ , vertical if ∠ ( ξ, ± ( ∇ d p )( x )) ≥ π/ , medial if it is horizontal and vertical . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 21
We also call a direction ξ ∈ Σ x ( X ) ( negative if ∠ ( ξ, − ( ∇ d p )( x )) < π/ positive if ∠ ( ξ, ( ∇ d p )( x )) < π/ . We say that an open subset Ω ⊂ Σ x ( X ) is in the positive side (resp. negative side ) of Σ x ( X ) if every element of Ω is positive (resp. nega-tive).Assume that a Lipschitz curve c : [ a, b ] → B ( p, r ) \ { p } has theright and left directions ˙ c + ( t ) and ˙ c − ( t ) respectively at every t ∈ [ a, b ].We say that such a curve c is vertical (resp. horizontal or medial ) ifboth ˙ c + ( t ) and ˙ c − ( t ) are vertical (resp. horizontal or medial) for every t ∈ [ a, b ]. Definition 4.7.
For every x ∈ int S , we set s ( x ) := { s ∈ [0 , ℓ ] | x ∈ λ s } ,s max ( x ) := max s ( x ) , s min ( x ) := min s ( x ) , + ˙ λ ( x ) := { ↑ λ s (1) x | s ∈ s ( x ) } , − ˙ λ ( x ) := { ↑ λ s (0) x | s ∈ s ( x ) } . Note that s ( x ) is a closed subset of [0 , ℓ ]. We show that s ( x ) is aclosed interval later in Lemma 4.28. Lemma 4.8.
For every x ∈ int S , we have (1) diam ( s ( x )) < τ p ( | p, x | );(2) diam ( ± ˙ λ ( x )) < τ p ( | p, x | )) . Proof.
Suppose that the conclusion does not hold. Then we have asequence x i ∈ int S and a positive constant c such that one of thefollowing holds:( i ) diam s ( x i ) / | p, x i | ≥ c ;( ii ) diam ( ± ˙ λ ( x i )) ≥ c .Let x i = λ s i ( t i ). Note that 0 < t i < ℓ .We may assume t i = min { t i , − t i } since the other case is similar. Forany other s ′ i ∈ s ( x i ), from | x i , p | →
0, we havelim i →∞ ∠ x i λ s i (0) p = π/ − δ, lim i →∞ ∠ x i λ s ′ i (0) p = π/ − δ. We may assume that s ′ i < s i without loss of generality. Note also thatlim i →∞ ∠ x i λ s ′ i (0) λ s i (0) = π/ δ, lim i →∞ ( ˜ ∠ λ s i (0) x i λ s ′ i (0) + ˜ ∠ x i λ s i (0) λ s ′ i (0) + ˜ ∠ λ s i (0) λ s ′ i (0) x i ) = π. It follows thatlim i →∞ ∠ λ s i (0) x i λ s ′ i (0) ≤ lim i →∞ ˜ ∠ λ s i (0) x i λ s ′ i (0)= lim i →∞ ( π − ˜ ∠ x i λ s i (0) λ s ′ i (0) − ˜ ∠ λ s i (0) λ s ′ i (0) x i )= 0 . Thus we conclude that diam ( − ˙ λ ( x i )) →
0. Therefore the assumption( ii ) does not hold. Note that | s i − s ′ i | ≤ | p, x i | t i ˜ ∠ λ s i (0) x i λ s ′ i (0) , Therefore, from lim i →∞ ˜ ∠ λ s i (0) x i λ s ′ i (0) = 0, we see that the assump-tion ( i ) does not hold either. (cid:3) Now, for any x ∈ S and s ∈ (0 , ℓ ) with x / ∈ λ s , let y ∈ λ s be suchthat | x, y | = | x, λ s | . By Lemma 2.2, we have either ∠ ( ↑ xy , −∇ d p ) < τ p ( δ, r ) or ∠ ( ↑ xy , ∇ d p ) < τ p ( δ, r ) . Lemma 4.9.
Under the above situation, if ∠ ( ↑ xy , −∇ d p ) < τ p ( δ, r )( resp. ∠ ( ↑ xy , ∇ d p ) < τ p ( δ, r )) , then we have ∠ ( ↑ yx , −∇ d p ) < τ p ( δ, r ) ( resp . ∠ ( ↑ yx , −∇ d p ) < τ p ( δ, r )) . .Proof. We assume ∠ ( ↑ xy , −∇ d p ) < τ p ( δ, r ). The other case is simi-lar. Let µ be a small positive number. First consider the case when | x, y | / | x, p | < µ . Then by Lemma 2.3, we have(4.7) ˜ ∠ pyx < ∠ pyx + τ p ( r ) + τ ( µ ) < τ p ( δ, r ) + τ ( µ ) . Since ˜ ∠ xpy < τ ( µ ), we have ˜ ∠ pxy > π − τ p ( δ, r ) − τ ( µ ). We fix µ = µ such that τ ( µ ) < τ p ( δ, r ). By Lemma 2.3 again, we have ∠ pxy > π − τ p ( δ, r ) − τ ( µ ) ≥ π − τ p ( δ, r ), which implies the conclusion.Next consider the case when | x, y | / | x, p | ≥ µ . If the conclusion doesnot hold in this case, we have sequences x n , y n ∈ S converging to p such that • | x n , y n | / | x n , p | ≥ µ ; • y n ∈ λ s n and | x n , y n | = | x n , λ s n | ; • ↑ x n p → v , ↑ y n p → v ; • ∠ px n y n < π − c for some positive constant c independent of n .Now consider the convergence (cid:18) | p, x n | X, p (cid:19) → ( K p ( X ) , o p ) , as i → ∞ . Let x ∞ and y ∞ be the limits of x n and y n under this convergence.Obviously x ∞ = v and y ∞ is on the geodesic ray in K p from o p in thedirection v . Let x n = λ u n ( t n ), and let λ ∞ be the limit of λ u n under WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 23 the above convergence. From the assumption ∠ px n y n < π − c , we haveeither ∠ ( ˙ λ u n ( t n ) , ↑ y n x n ) > π/ c/ ∠ ( − ˙ λ u n ( t n ) , ↑ y n x n ) > π/ c/ ∠ ( ˙ λ ∞ , ↑ y ∞ x ∞ ) = ∠ ( − ˙ λ ∞ , ↑ y ∞ x ∞ ) = π/ . This contradicts Lemma 2.1. (cid:3)
Lemma 4.10.
For x ∈ int S , fix s and t with x = λ s ( t ) . Then forevery u ∈ Σ x ( S ) with (4.8) ∠ ( u, ± ˙ λ s ( t )) ≥ π/ , there exists a shortest path ξ ∞ : [ − , → Σ x ( X ) satisfying (4.9) u ∈ ξ ∞ ([ − , , ∠ X ( ξ ∞ ( ± , ± ˙ λ s ( t )) < τ p ( r ) ,ξ ∞ ([ − , ⊂ Σ x ( S ) .xλ s (1) λ s (0) ↑ px u ∈ Σ x ( S ) ξ ∞ ξ ∞ (1) ξ ∞ ( − x ( X )We need a sublemma. Sublemma 4.11.
For any x ∈ S and any horizontal direction ξ ∈ Σ x ( S ) , let y i ∈ S be a sequence such that y i → x and ↑ y i x → ξ . Thenthere is an i such that if λ s i meets y i for some i ≥ i , then we have | x, λ s i | ≥ c | x, y i | , for some uniform positive constant c .Proof. Note that λ s i does not pass through x for any large enough i since ξ would be a direction tangent to λ s i at x otherwise, which is acontradiction. Take z i ∈ λ s i such that | x, z i | = | x, λ s i | . By Lemma 2.2and 2.10, we have | ∠ xz i y i − π/ | < τ p ( δ, r ) . It follows that ∠ ( ↑ xz i , ( −∇ d p )) < τ p ( δ, r ) or ∠ ( ↑ xz i , ∇ d p ) < τ p ( δ, r ). Weassume the former since the latter case is similar. It follows fromLemma 4.9 that ∠ ( ↑ z i x , ∇ d p ) < τ p ( δ, r ). Lemma 2.3 implies that(4.10) | ˜ ∠ pz i y i − π/ | < τ p ( δ, r ) . By (4.7), we have(4.11) ˜ ∠ pz i x < τ p ( δ, r ) . Let x i be the point on the geodesic z i p satisfying | z i , x i | = | z i , x | . By(4.10), we have | ˜ ∠ x i z i y i − π/ | < τ p ( δ, r ) . It follows from (4.11) that(4.12) | ˜ ∠ xz i y i − π/ | < τ p ( δ, r ) . Now let us consider the convergence (cid:18) | x, z i | X, x (cid:19) → ( K x ( X ) , o x ) , as i → ∞ . Let z ∞ ∈ Σ x ⊂ K x ( X ) be the limit of z i under thisconvergence. Since ∠ ( z ∞ , ∇ d p ) < τ p ( δ, r ) and since ↑ y i x → v , the limit y ∞ of y i under the above convergence certainly exists, and we have(4.13) ∠ y ∞ o x z ∞ < π/ − π/ τ p ( δ, r ) < π/ . By (4.12), we also have(4.14) | ∠ o x z ∞ y ∞ − π/ | < τ p ( δ, r ) , (4.13) and (4.14) imply that | o x , z ∞ | ≥ c | o x , y ∞ | for some uniform con-stant c >
0. This yields the conclusion of the lemma via contradic-tion. (cid:3)
Proof of Lemma 4.10.
Let y i ∈ S be such that | x, y i | X → ↑ y i x converges to u . Take s i ∈ (0 , ℓ ), t i ∈ (0 ,
1) such that y i = λ s i ( t i ). Let ǫ i := | x, y i | X , and consider the convergence (cid:18) ǫ i X, x (cid:19) → ( K x ( X ) , o x ) , as i → ∞ . Note that the minimal geodesic ˆ λ s i ( t ) := λ s i ( t i + ǫ i t ),( − t i /ǫ i < t < (1 − t i ) /ǫ i ), has a uniformly bounded speed for ǫ i X independent of i . Therefore passing to a subsequence, we may assumethat ˆ λ s i ( t ) converges to a minimal geodesic ˆ λ ∞ ( t ) in K x ( X ) defined on( −∞ , ∞ ), where this convergence is uniform on every bounded interval.Note that ˆ λ ∞ (0) = u . From Sublemma 4.11 and (4.8), the geodesic ˆ λ ∞ does not pass through o x . Consider the curve(4.15) ˆ ξ ∞ ( t ) := ˆ λ ∞ ( t ) / | ˆ λ ∞ ( t ) | . Obviously, ˆ ξ ∞ is a shortest path in Σ x ( X ) and ˆ ξ (( −∞ , ∞ )) ⊂ Σ x ( S ).Let ξ ∞ : [ − , → Σ x ( S ) be a reparametrization of the extensionˇ ξ ∞ : [ −∞ , ∞ ] → Σ x ( S ) of ˆ ξ ∞ .Take an arbitrary w + ∈ ( ∇ d p )( x ) and set w − = − ( ∇ d p )( x ). Considerthe two sets { w + , ˙ λ s ( t ) , w − , − ˙ λ s ( t ) } and { w + , ξ ∞ (1) , w − , ξ ∞ ( − } .They are on a circle C in Σ x ( X ) in these orders, where(4.16) | L ( C ) − π | < τ p ( r ) . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 25
Since | ∠ ( ± w, ξ ∞ ( ± − π/ | < τ p ( δ, r ) and | ∠ ( ± w, ± ˙ λ s ( t )) − π/ | <τ p ( δ, r ), we have the conclusion (4.9).This completes the proof. (cid:3) A direction ξ ∈ Σ x ( X ) is called regular if ξ / ∈ S (Σ x ( X )). Lemma 4.12.
For x ∈ int S , let ξ , ξ ∈ Σ x ( S ) be positively horizontal(resp. negatively horizontal). Assume that ξ is regular. Take an X -geodesic γ such that ˙ γ (0) = ξ , and a sequence x i ∈ S such that | x, x i | X → and ↑ x i x → ξ . Then for s with λ s ∋ x , there exists an ǫ > such that if some ruling geodesic λ s with | s − s | < ǫ passesthrough x i for a sufficiently large i , then it passes through γ , too. xλ s λ s ↑ px ξ ξ γ Σ x ( X ) x i Proof.
Suppose that the conclusion does not hold. Then there existsa sequence s i → s such that λ s i meets x i while λ s i does not passthrough γ . Applying Lemma 4.10 to x = λ s ( t ) and ξ , we have ashortest arc ξ ∞ : [ − , → Σ x ( X ) joining two points close to ± ˙ λ s ( t )such that ξ ∈ ξ ∞ ([ − , ξ , wehave a shortest arc ¯ ξ ∞ : ([ − , → Σ x ( X ) joining two points close to ± ˙ λ s ( t ) such that ξ ∈ ¯ ξ ∞ ([ − , ξ ∞ and ¯ ξ ∞ pass through the positive side of Σ x ( X )and connect points close to ± ˙ λ s ( t ). From construction, ξ ∞ and ¯ ξ ∞ pass through horizontal directions ξ and ξ respectively. Furthermoreboth intersections ξ ∞ ([ − , − ǫ ]) ∩ ¯ ξ ∞ ([ − , − ǫ ]) and ξ ∞ ([1 − ǫ , ∩ ¯ ξ ∞ ([1 − ¯ ǫ , ǫ i , ¯ ǫ i > i =1 ,
2) since they are in the regular parts of Σ x ( X ) by Corollary 2.11.Therefore by the uniqueness of geodesics in the CAT(1)-space Σ x ( X ),we conclude that ξ ∞ ([ − ǫ , − ǫ ]) = ¯ ξ ∞ ([ − ǫ , − ¯ ǫ ]) some small ǫ , ¯ ǫ >
0, and in particular ξ ∞ and ¯ ξ ∞ pass through both ξ and ξ .Take ξ , ξ ∈ Σ x ( X ) close to ξ such that every element of the arc[ ξ , ξ ] in Σ x ( X ) is regular and ξ is the midpoint of [ ξ , ξ ]. Let γ i :[0 , ǫ ] → X be X -geodesics with ˙ γ (0) = ξ i ( i = 3 , γ the X -minimal geodesic joining γ ( ǫ ) to γ ( ǫ ). If ǫ > △ ( γ , γ , γ ) bounds a domain in X homeomorphic to atwo-disk D . Let int D denote the interior of the disk D . Note thatint D is open in X and that γ ([0 , ǫ ]) ⊂ D for a small ǫ < ǫ . Sinceint D is open in X and since ξ ∞ is constructed by (4.15), λ s i really xλ s ↑ px U + ( x, ǫ ) U − ( x, ǫ ) B ( x, ǫ ) Figure 1.
A canonical ball around x passes through γ for large i , which is a contradiction. This completesthe proof. (cid:3) Canonical balls.
In this subsection, we introduce the notion ofcanonical balls, which turns out to be useful to have better understand-ing of the behavior of ruling geodesics of S .We denote by R ( X ) the set of topological regular points, R ( X ) = X \ S ( X ). Definition 4.13.
For x ∈ B ( p, r ), a ball B ( x, ǫ ) is called canonical iffor every y ∈ B ( x, ǫ ) \ { x } with vertical ↑ yx , we have y ∈ R ( X ). Lemma 4.14.
There exists an r = r p > such that there is a canonicalball around every point in B ( p, r ) \ { p } . Lemma 4.14 is a direct consequence of the following Lemma 4.15,which is immediate from Corollary 2.11.
Lemma 4.15.
For every x ∈ S ( X ) ∩ B ( p, r ) \ { p } , we have sup { ∠ ( ξ, ∇ d p ) | ξ ∈ Σ x ( S ( X )) is positive } < τ p ( | p, x | ) , sup { ∠ ( ξ, −∇ d p ) | ξ ∈ Σ x ( S ( X )) is negative } < τ p ( | p, x | ) . Definition 4.16.
For x ∈ int S , let B ( x, ǫ ) be a canonical ball. We set U + ( x, ǫ ) := { y ∈ B ( x, ǫ ) | ∠ ( ↑ yx , ˙ λ ( x )) < π/ } ,U − ( x, ǫ ) := { y ∈ B ( x, ǫ ) | ∠ ( ↑ yx , − ˙ λ ( x )) < π/ } . Note that both U + ( x, ǫ ) and U − ( x, ǫ ) are convex in X for small ǫ > U ± ( x, ǫ ) ⊂ S for a small ǫ > | A | the cardinality of a set A . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 27 λ s λ s i λ s i ( u i ) λ s ( t i ) Figure 2. ↑ λ si ( u i ) λ s ( t i ) is positively horizontal Lemma 4.17.
Let γ : [0 , → X be a vertical X -geodesic in B ( p, r ) .Then we have | γ ∩ S ( X ) | < ∞ Proof.
Suppose that the lemma does not hold. Then we would have anaccumulation point x = γ ( t ) of γ ∩S ( X ). It turns out that either ˙ γ ( t )or − ˙ γ ( t ) is in Σ x ( S ( X )), which is a contradiction to the existence ofa canonical ball around x . (cid:3) Remark 4.18.
At this stage, we do not know yet a uniform boundon | γ ∩ S ( X ) | for all the vertical geodesics γ . In Section 6, we give auniform bound (see (6.3)).The following is a key lemma. Lemma 4.19. (no-return lemma)
For every s , there exists an ǫ > such that for any s ∈ ( s − ǫ, s ) ( resp. s ∈ ( s , s + ǫ )) , there areno t , t ∈ [0 , satisfying that ↑ λ s ( t ) λ s ( t ) is positively horizontal ( resp.negatively horizontal ) of Σ λ s ( t ) ( X ) .Proof. Suppose that the conclusion does not hold. Then we have somesequence s i < s with lim i →∞ s i = s such that ↑ λ si ( u i ) λ s ( t i ) is positively horizontal for some t i , u i ∈ (0 , . (4.17)(see Figure 2). We show that both λ s i ((0 , u i )) and λ s i (( u i , λ s ,which yields a contradiction to the minimality of λ s .From Lemmas 4.14 and 4.17, it is possible to cover λ s by finitelymany canonical balls B ( x α , ǫ α ), 1 ≤ α ≤ N , where x α = λ s ( t α ) and t α < t α +1 . Taking smaller ǫ α if necessary, we may further assume thatfor any large i (1) λ s i ⊂ S Nα =1 B ( x α , ǫ α );(2) B ( x α , ǫ α ) ∩ B ( x α +1 , ǫ α +1 ) ∩ λ s i ⊂ U + ( x α , ǫ α ) ∩ U − ( x α +1 , ǫ α +1 )for each α .Note that λ s ∩ S X ⊂ { x α } Nα =1 , and that U + ( x α , ǫ α ) ∩ U − ( x α +1 , ǫ α +1 )is convex in X and homeomorphic to a disk. Suppose that λ s i ((0 , u i )) does not meet λ s . Take a maximal interval I iα in [0 ,
1] such that λ s ( I iα ) ⊂ B ( x α , ǫ α ), and set ξ iα ( t ) := ↑ λ si ( t ) x α ( t ∈ I iα ) . From the assumption, ξ iα ( I iα ) is in either the negative side or the positiveside of Σ x α ( X ). Note that ξ i ( I i ) is in the negative side of Σ x ( X ). Let α = α ( i ) be such that λ s ( t i ) ∈ B ( x α , ǫ α ). From (4.17), ξ iα ( I iα ) isin the positive side of Σ x α ( X ). Therefore for some α ≤ α , ξ iα − ( I iα − )is in the negative side of Σ x α − ( X ) and ξ iα ( I iα ) is in the positive side ofΣ x α ( X ). Now λ s divides the disk domain U + ( x α − , ǫ α − ) ∩ U − ( x α , ǫ α )into two disk domains D − and D + , where we may assume that λ s i ( t − ) ∈ D − and λ s i ( t + ) ∈ D + for some t − ∈ I α − and t + ∈ I α . Thus λ s i ([ t − , t + ])must meet λ s .Similarly, we would have another intersection point of λ s i (( u i , λ s . This completes the proof. (cid:3) The following lemma is a global version of Lemma 4.19.
Lemma 4.20.
For arbitrary s < s , there are no t , t ∈ [0 , satis-fying that ↑ λ s ( t ) λ s ( t ) ( resp. ↑ λ s ( t ) λ s ( t ) ) is negatively horizontal in Σ λ s ( t ) ( X )( resp. positively horizontal in Σ λ s ( t ) ( X )) .Proof. Let I ( s ) be the set of all s ∈ ( s , s ] such that there are no t , t ∈ [0 ,
1] satisfying that ↑ λ s ( t ) λ s ( t ) is negatively horizontal in Σ λ s ( t ) ( X ). ByLemma 4.19, ( s , s ) ⊂ I ( s ) for some s ∈ ( s , s ). Let u be thesupremum of those s . From the continuity of the map σ : R → S ,( s , s ] \ I ( s ) is open in ( s , s ]. It follows that u ∈ I ( s ). Supposethat u < s . Then we have a sequence of positive numbers ǫ i with ǫ i → u i := u + ǫ i / ∈ I ( s ). Namely we have sequences t i and t ′ i satifsying that ↑ λ ui ( t ′ i ) λ s ( t i ) is negatively horizontal in Σ λ s ( t i ) ( X ). Set x i := λ u i ( t ′ i ), and let y i := λ s ( t i ) . Take z i ∈ λ u i and w i ∈ λ u such that | y i , z i | = | y i , λ u i | , | z i , w i | = | z i , λ u | . Since ↑ x i y i is horizontal, we have y i = z i . By (4.6), we obtain ∠ ( ↑ y i z i , ∇ d p ) < τ p ( δ, r ) or ∠ ( ↑ y i z i , −∇ d p ) < τ p ( δ, r ) . We show that(4.18) ∠ ( ↑ y i z i , ∇ d p ) < τ p ( δ, r ) . Otherwise, we have ∠ ( ↑ y i z i , −∇ d p ) < τ p ( δ, r ). In view of Lemma 4.9, itturns out that ∠ x i y i z i > π/ − τ p ( δ, r ) , and hence ∠ x i z i y i < π − ∠ x i y i z i − ∠ y i x i z i + τ p ( r ) < π/ τ p ( δ, r ) . This is a contradiction to the choice of z i . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 29
Next note that w i = z i . Because if w i = z i , then ↑ w i y i must benegatively horizontal by (4.18), which contradicts u ∈ I ( s ). Now byLemma 4.19, ↑ z i w i is positively horizontal. In view of Lemma 4.9, wehave(4.19) ∠ ( ↑ w i z i , −∇ d p ) < τ p ( δ, r ) , It follows from (4.18) and (4.19) that ∠ y i z i w i > π − τ p ( δ, r ), which im-plies ∠ ( ↑ w i y i , −∇ d p ) < τ p ( δ, r ). In particular ↑ w i y i is negatively horizontal.This contradicts u ∈ I ( s ). Thus we conclude u = s .Similarly we see that there are no t , t satisfying that ↑ λ s ( t ) λ s ( t ) ispositively horizontal. This completes the proof. (cid:3) Lemma 4.21.
For every x ∈ int S , there exists an ǫ > such that U + ( x, ǫ ) ⊂ S, U − ( x, ǫ ) ⊂ S. Proof.
Let B ( x, ǫ ) be a canonical ball. Take the positively horizon-tal v + ∈ Σ x ( X ) (resp. negatively horizontal v − ∈ Σ x ( X )) such that ∠ ( v ± , ˙ λ ( x )) = π/
4. Let γ ± be X -geodesics starting from x with ˙ γ ± (0) = v ± . Sublemma 4.22.
For any < ǫ < ǫ , there are s − ∈ (0 , s min ( x )) and s + ∈ ( s max ( x ) , r ) such that λ s ± pass through γ ± ((0 , ǫ ]) respctively. x λ s + λ s − ↑ px γ + γ − Proof.
Suppose that there is no such s − < s min ( x ). Then we have asequence s i < s min ( x ) converging to s min ( x ) such that λ s i does not passthrough γ − ((0 , ǫ ]) for some ǫ > i . As in the proof of Lemma4.10 together with Lemma 4.19, the curves ξ i ( t ) = ↑ λ si ( t ) x ( t ∈ [0 , x ( X ) pass through ˙ γ − (0) for all i . This shows in particular that˙ γ − (0) ∈ Σ x ( S ). Since ˙ γ − (0) is regular, it follows from Lemma 4.12 that λ s i meets γ − ((0 , ǫ ]) for every large enough i . This is a contradiction.Similarly, we see that λ s meets γ + ((0 , ǫ ]) for any s > s + sufficientlyclose to s + . (cid:3) Take a sufficiently small 0 < ǫ < ǫ such that(4.20) the triangle ∆ γ + ( ǫ ) xγ − ( ǫ ) spans a disk domain in X .Let s ± be as in Sublemma 4.22 for ǫ , and set I := [ s − , s + ] and γ ǫ = γ + ([0 , ǫ ]) ∪ γ − ([0 , ǫ ]). It follows from the contiuity of σ , Lemma 4.19and (4.20) that λ s meets γ ǫ for all s ∈ I . Now define ϕ : I → γ ǫ by ϕ ( s ) = λ s ∩ γ ǫ . Let γ ± ( ǫ ± ) := ϕ ( s ± ). Since ϕ is continuous, theintermediate-value theorem implies that γ ǫ ⊂ Im ϕ ⊂ S, where ǫ := min { ǫ + , ǫ − } .For any 0 < ǫ ≤ ǫ , let µ ǫ be the X -geodesic joining γ − ( ǫ ) to γ + ( ǫ ).Put ˆ γ ǫ,ǫ := γ − ([ ǫ, ǫ ]) ∪ µ ǫ ∪ γ + ([ ǫ, ǫ ]) . Similarly, we can define the map ψ : I → ˆ γ ǫ,ǫ by ψ ( s ) = λ s ∩ ˆ γ ǫ,ǫ .Again, since ψ is continuous, the intermediate-value theorem impliesthat ψ is surfjective, and hence µ ǫ ⊂ S . Now we can take ǫ +3 > U + ( x, ǫ +3 ) ⊂ [ ≤ ǫ ≤ e µ ǫ ⊂ S. Similarly we have U − ( x, ǫ − ) ⊂ S for some ǫ − >
0. This completes theproof of Lemma 4.21. (cid:3)
Spaces of directions.
In this subsection, we determine the struc-ture of the space of directions of S at each point of S . Lemma 4.23.
For every x ∈ S , let ξ ∈ Σ x ( S ) be regular in Σ x ( X ) ,and let γ be an X -geodesic with ˙ γ (0) = ξ . Then γ ([0 , ǫ ]) ⊂ S for asmall ǫ > . Furthermore, ǫ can be taken locally uniformly for ξ .Proof. First assume x ∈ int S , and let x = λ s ( t ). From Lemma 4.21,we may assume that ∠ ( ξ, ± ˙ λ s ( t )) ≥ π/
3. If ξ is positive, Lemmas 4.12and 4.19 imply that for some s > s max ( x ), λ s meets γ at, say γ ( t ( s ))for every s ∈ [ s max ( x ) , s ]. Since ξ is horizontal, t ( s ) is unique andcontinuous in s , and t ( s ) > s > s max ( x ). Therefore γ ([0 , t ( s )]) ⊂ S .From this argument, the local uniformness of t ( s ) for ξ is clear. Thecase when ξ is negative is similar. If x ∈ ∂S \ { p } , the proof is similar.Finally we consider the case x = p . For small enough ǫ >
0, let ξ ± ∈ Σ p ( X ) be such that ∠ ( ξ + , ξ − ) = ∠ ( ξ + , ξ ) + ∠ ( ξ, ξ − ) = 2 ∠ ( ξ + , ξ ) = 2 ǫ, and let γ ± be X -geodesics with ˙ γ ± (0) = ξ ± . For a small η >
0, let U ( η )denote the domain bounded by γ ± and S ( p, η ). If η is small enough,then U ( η ) is homeomorphic to a disk. Take x i ∈ S with x i → p suchthat ↑ x i p → ξ . Then for a large N , we have x i ∈ U ( η ) for all i ≥ N . If x i = λ s i ( t i ), λ s i must meet γ ± . The intermediate-value theorem thenyields that the subdomain of U ( η ) bounded by γ ± and λ s N is containedin S . In particular γ ([0 , ǫ ]) ⊂ S for small ǫ > (cid:3) Remark 4.24. If ξ ∈ Σ x ( S ) is singular in Σ x ( X ), Lemma 4.23 doesnot hold in general. See Examples 4.4 and 4.5. WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 31
Lemma 4.25.
Let x ∈ S . (1) If x ∈ int S , then Σ x ( S ) is a circle of length < π + τ p ( r ) ;(2) If x ∈ ∂S , then Σ x ( S ) is an arc.Proof. (1) First we show that Σ x ( S ) contains a circle C . Take an s ∈ s ( x ) and t with x = λ s ( t ). Obviously C := { ξ ∈ Σ x ( X ) | ∠ ( ± ˙ λ s ( t ) , ξ ) ≤ π/ } consists of two arcs in the regular part of Σ x ( X ). It follows fromLemma 4.21 that C is contained in Σ x ( S ). For a positively horizontaldirection v + ∈ Σ x ( S ), we apply Lemma 4.10 to obtain a minimal arc C + in Σ x ( S ) joining two points close to ± ˙ λ s ( t ) and containing v + .Similarly, for a negatively horizontal direction v − ∈ Σ x ( S ), we applyLemma 4.10 to obtain a minimal arc C − in Σ x ( S ) joining two pointsclose to ± ˙ λ s ( t ) and containing v − . Obviously the union of C , C + and C − forms a circle C in Σ x ( S ). It follows from Lemma 4.8 and (4.9)that | L ( C ± ) − π | < τ p ( r ) , L ( C \ ( C + ∪ C − )) < τ p ( r ) , which implies | L ( C ) − π | < τ p ( r ).Suppose next that Σ x ( S ) \ C is not empty, and take a w in Σ x ( S ) \ C . Since ∠ ( w, ± ˙ λ s ( t )) ≥ π/
3, we can apply Lemma 4.10 to obtaina minimal arc C in Σ x ( S ) joining two points close to ± ˙ λ s ( t ) andcontaining w . Note that the complement C ′ of a small neighborhoodof ± ˙ λ s ( t ) in C is contained in C , and w must be contained in C ′ ,which is a contradiction.(2) If x = λ s (0) with 0 < s < ℓ (resp. s = ℓ ), then Σ x ( S ) is anarc with endpoints ± ˙ α ( s ) (resp. − ˙ α ( ℓ ) and ˙ λ ℓ (0)) through ˙ λ s (0)(recall (4.2)). The case x = λ s (1) with 0 < s ≤ ℓ is similar. Nextconsider the case x = p . Let v and ν , ν be as in (4.1). We showthat Σ p ( S ) coincides with the arc [ ν , ν ] in Σ p ( X ). Let η i be anyinterior point of [ ν i , v ], and let σ i be X -geodesics with ˙ σ i (0) = η i . If s > λ s meets both σ and σ . This impliesthat [ ν , η ] ∪ [ η , ν ] is contained in Σ p ( S ). Letting η , η → v , weobtain that [ ν , ν ] ⊂ Σ p ( S ). Conversely, for any ξ ∈ Σ p ( S ), take x i ∈ S with | p, x i | → ↑ x i p → ξ . Since x i can be written as x i = λ s i ( t i ) with s i →
0, it is obvious that ξ ∈ [ ν , ν ]. Thus we haveΣ p ( S ) = [ ν , ν ]. (cid:3) Definition 4.26.
For x ∈ S , let Σ x ( S ) int denote the intrinsic space ofdirections of S at x , which is defined as the completion of the set of allequivalence classes of S -geodesics starting from x equipped with theupper angle ∠ S for the induced interior metric of S . Lemma 4.27. Σ x ( S ) is isometric to Σ x ( S ) int .Proof. First assume x ∈ int S . LetΩ := Σ x ( S ) ∩ S (Σ x ( X )) . Note that | Ω | < ∞ . We first show that each component Σ of Σ x ( S ) \ Ωis isometrically embedded in Σ x ( S ) int . Take ξ and ξ from Σ with | ξ , ξ | < π . Let µ i : [0 , ǫ ] → X be an X -geodesic with ˙ µ i (0) = ξ i .Then for small ǫ , we have from Lemma 4.23(1) µ i ⊂ S (2) every X -geodesic joining µ ( t ) and µ ( t ) is contained in S forevery t ∈ [0 , ǫ ].Thus we conclude that ∠ X ( ξ , ξ ) = ∠ S ( ξ , ξ ).Next, for any v ∈ Ω, take ξ , ξ ∈ Σ x ( S ) \ Ω close to v such that ξ , v and ξ are in this order on the circle Σ x ( S ). Take X -geodesics γ i ( i = 3 ,
4) in the direction ξ i . By Lemma 4.12, we can find a sequence s i such that s i → s ∈ s ( x ) and λ s i meets both γ and γ . This impliesthat ∠ X ( ξ , ξ ) = ∠ S ( ξ , ξ ). This completes the proof.The case x ∈ ∂S is similar, and hence we omit the proof. (cid:3) Proof of Theorem 4.1.
In this subsection, we first prove Theo-rem 4.1. Then we controll the defference between the geometries of S and X . Lemma 4.28.
For every x ∈ S , we have (1) s ( x ) is either a point or a closed interval ;(2) σ − ( x ) is a strictly monotone arc in R . x ↑ px α ( s ( x )) α ( s ( x )) α ( s min ( x )) α ( s max ( x )) α ( s min ( x )) α ( s max ( x )) Proof.
Suppose that the conclusion (1) does not hold. Then we wouldhave s − < s + such that s ± ∈ s ( x ) and ( s − , s + ) does not meet s ( x ).Take a horizontal broken S -geodesic γ : [ − , → S with γ (0) = x andthe positive direction ˙ γ + ( t ) at every t ∈ [ − ,
1] such that | γ ∩ S ( X ) | < WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 33 ∞ . Let us denote I + = { s ∈ ( s − , s + ) | λ s passes through γ + \ { x } } ,I − = { s ∈ ( s − , s + ) | λ s passes through γ − \ { x } } , where γ + = γ ([0 , γ − = γ ([ − , ǫ − > ǫ + > s − , s − + ǫ − ) ⊂ I + and ( s + − ǫ + , s + ) ⊂ I − .Moreover extending γ in both direction if necessary, we may assumethat I + and I − are open and ( s + , s − ) = I + ∪ I − . This is a contradiction.(2) follows immediately from (1) and the injectivity of σ | I s for each s ∈ (0 , ℓ ]. (cid:3) Proof of Theorem 4.1.
By Lemma 4.28, we have (3.1) for all u, v ∈ R .Thus S has the induced metric from σ . Theorem 3.12 then implies that( S, d S ) is a CAT( κ )-space.We set S int := ( S, d S ). Lemma 4.29. int S int is locally geodesically complete.Proof. For each x ∈ S , take a canonical ball B ( x, ǫ ). If y ∈ B ( x, ǫ ) ∩ S with ↑ yx being vertical, then the geodesic xy is easily extended in theboth directions. If ↑ yx is horizontal, take z, w ∈ ∂B ( x, ǫ ) ∩ S such that ∠ ( ↑ zx , ↑ yx ) + ∠ ( ↑ yx , ↑ wx ) = ∠ ( ↑ zx , ↑ wx ). Let γ : [0 , → S be an S -geodesicjoining z and w . Considering an S -geodesic joining x and γ ( t ), we seethat there is an S -geodesic in any direction in Σ x ( S ). Applying this tothe direction − ↑ xy , we conclude that the geodesic xy can be extendedto the direction − ↑ xy . (cid:3) Lemmas 4.25, 4.27 and 4.29 imply that S int is a topological two-manifold with boundary, and hence homeomorphic to a disk, since itis contractible. This completes the proof of Theorem 4.1. (cid:3) In the rest of this section, we present a few results that control thedifference between the geometries of X and S . These will be needed inSections 5 and 6. Lemma 4.30.
For every x ∈ S \ { p } , let ( −∇ S d p )( x ) denote ˙ γ Sx,p (0) ,where γ Sx,p is the S -geodesic from x to p . Then we have (1) ∠ ( ˙ γ Sp,x (0) , ˙ γ Xp,x (0)) < τ p ( | p, x | X ) ;(2) ∠ (( −∇ S d p )( x ) , ( −∇ d p )( x )) < τ p ( | p, x | X ) . For the proof, we need a sublemma.
Sublemma 4.31.
For every x ∈ S , we have sup x ∈ S \{ p } | p, x | S | p, x | X < τ p ( r ) . Proof.
If the sublemma does not hold, there would exist a sequence x n in S converging to p such that(4.21) | p, x n | S | p, x n | X > c > , for some constant c > i . Passing to a subsequence, wemay assume that ↑ x n p converges to a direction v ∈ Σ p ( X ). It is easilyseen from (4.21) that v is the vertex of Σ p ( X ). Take a small enough ǫ > c and an s n ∈ s ( x n ). Let y n be an element of λ s n with | x n , y n | = ǫ | p, x n | X . From Lemma 4.25, the X -geodesic joining p and y n is contained in S for any large i . It follows from triangleinequality that | p, x n | S | p, x n | X ≤ | p, y n | S + | y n , x n | S | p, x n | X ≤ | p, x n | X + 2 | y n , x n | X | p, x n | X = 1 + 2 ǫ < c, which is a contradiction. (cid:3) Proof of Lemma 4.30.
If Lemma 4.30 does not hold, there would be asequence x i of S converging to p such that ∠ ( ˙ γ Sp,x i (0) , ˙ γ Xp,x i (0)) > c > , or(4.22) ∠ (( −∇ S d p )( x i ) , ( −∇ d p )( x i )) > c > , (4.23)where c is a uniform positive constant. Let Σ := Σ p ( S ) = [ ν , ν ]. Let-ting t Xi := | x i , p | X , consider the congengence ( t Xi X, x i ) → ( K p ( X ) , ξ X ),where ξ X ∈ Σ is the limit of ˙ γ Xp,x i (0) in Σ p ( X ). Similarly, letting t Si := | x i , p | S , from Lemma 4.27, we have the congengence ( t Si S, x i ) → ( K (Σ) , ξ S ), where ξ S ∈ Σ is the limit of ˙ γ Sp,x i (0) in Σ. From (4.22), wemay assume ξ X / ∈ { ν , ν } . Take any s i ∈ s ( x i ), and set y i = λ s i (0), z i = λ s i (1), u i := | x i , y i | X = | x i , y i | S . Since s i = | p, y i | X = | p, y i | S , Itfollows from Sublemma 4.31 thatlim i →∞ cos( | ν , ˙ γ Xp,x i (0) | ) = cos( | ν , ξ X | ) = lim i →∞ ( t Xi ) + s i − u i t Xi ℓ i = lim i →∞ ( t Si ) + s i − u i t Si ℓ i = cos( | ν , ξ S | ) = lim i →∞ cos( | ν , ˙ γ Sp,x i (0) | ) . It follows that ξ X = ξ S and lim i →∞ ˙ γ Xp,x i (0) = lim i →∞ ˙ γ Sp,x i (0), whichshows (1).We set ξ := ξ X = ξ S . Let y ∞ ∈ K (Σ) and z ∞ ∈ K (Σ) be the limitof y i and z i under the above rescaling limit respectively. By Lemma2.1, we havelim sup i →∞ ∠ X y i x i p ≤ ∠ y ∞ ξo p , lim sup i →∞ ∠ X z i x i p ≤ ∠ z ∞ ξo p , lim sup i →∞ ∠ S y i x i p ≤ ∠ y ∞ ξo p , lim sup i →∞ ∠ S z i x i p ≤ ∠ z ∞ ξo p . WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 35
It follows from ∠ X y i x i p + ∠ X z i x i p ≥ π, ∠ S y i x i p + ∠ S z i x i p ≥ π, ∠ y ∞ ξo p + ∠ z ∞ ξo p = π that(4.24) lim i →∞ ∠ X y i x i p = ∠ y ∞ ξo p = lim i →∞ ∠ S y i x i p, lim i →∞ ∠ X z i x i p = ∠ z ∞ ξo p = lim i →∞ ∠ S z i x i p. Now let w i ∈ Σ x i ( S ) be the nearest point of Σ x i ( S ) from ˙ γ Xx i ,p (0). If˙ γ Xx i ,p (0) ∈ Σ x i ( S ), then (4.24) implies that ∠ ( ˙ γ Xx i ,p i (0) , ˙ γ Sx i ,p i (0)) → i → ∞ . This is a contradiction to (4.23). Suppose ˙ γ Xx i ,p (0) / ∈ Σ x i ( S ). Then w i ∈ V (Σ x i ( X )). It follows from Lemma 2.10 that ∠ ( ˙ γ Xx i ,p (0) , w i ) < τ p ( | p, x i | X ). In the below, we may assume that ∠ ( ˙ γ x i ,y i (0) , w i ) ≥ ∠ ( ˙ γ x i ,y i (0) , ˙ γ Sx i ,p (0)) without loss of generality by replacing y i by z i ifneccesary. Then using Lemma 4.25 and (4.24), we obtain ∠ ( ˙ γ Xx i ,p (0) , ˙ γ Sx i ,p (0)) = ∠ ( ˙ γ Xx i ,p (0) , w i ) + ∠ ( w i , ˙ γ Sx i ,p (0))= ∠ ( ˙ γ Xx i ,p (0) , w i ) + ∠ ( w i , ˙ γ x i ,y i (0)) − ∠ ( ˙ γ Sx i ,p (0) , ˙ γ x i ,y i (0)) ≤ ∠ ( ˙ γ Xx i ,p (0) , w i ) + ∠ ( ˙ γ Xx i ,p (0) , ˙ γ x i ,y i (0)) − ∠ ( ˙ γ Sx i ,p (0) , ˙ γ x i ,y i (0)) < τ p ( | p, x i | X ) + o i , where lim i →∞ o i = 0. This is a contradiction to (4.23). (cid:3) Lemma 4.32.
For x, y ∈ S with x ∈ σ and y ∈ σ satisfying || p, x | X − | p, y | X | < | x, y | X / , the S -geodesic joining x and y is an X -geodesic.Proof. It follows from the assumption that || p, x | S − | p, y | S | < | x, y | X / ≤ | x, y | S / . Using Lemma 2.3 in S , we have | ∠ pγ Sx,y ( t ) x − π/ | < π/ , | ∠ pγ Sx,y ( t ) y − π/ | < π/ , for all t ∈ (0 , γ Sx,y : [0 , → S is the S -geodesic joining x to y . This implies that γ Sx,y is vertical and therefore is an X -geodesic. (cid:3) In a way similar to Lemma 4.32, we have the following.
Lemma 4.33.
For arbitrary x, y ∈ S such that || p, x | S − | p, y | S | < | x, y | S / , the S -geodesic joining x and y is an X -geodesic. Filling via
CAT( κ ) -disks Let X and p ∈ X be as in Theorem 1.1. We assume p ∈ S ( X ). Lemma 5.1. Σ p ( S ( X )) coincides with the set V (Σ p ( X )) of all verticesof the graph Σ p ( X ) .Proof. In view of Lemma 2.8, it suffices to show that V (Σ p ( X )) ⊂ Σ p ( S ( X )). Suppose that there is a vertex v of Σ p ( X ) not contained inΣ p ( S ( X )). Then there exist positive numbers ǫ and r such that(1) the domain C ( v ; ǫ , r ) := { x ∈ B ( p, r ) | ∠ ( γ ′ p,x (0) , v ) < ǫ } contains no topological singular points of X except p ;(2) for three distinct directions ξ , ξ , ξ at p with ∠ ( ξ i , v ) = ǫ / ≤ i ≤ x i := γ ξ i ( r ). Then the ruled surfaces S ij := S ( γ p,x i , γ p,x j ) spanned by γ p,x i and γ p,x j ( i = j ) are CAT( κ )-spaces and homeomorphic to a disk.We consider the ruled surfaces S i ( i = 1 , ξ of v and ξ , let γ be a geodesic in S starting from p with γ ′ (0) = ξ andset y := γ ( r ) for an r with 0 < r ≪ r . If r is small enough, then y isnot contained in S and a nearest point z of S from y lies in int R .From Theorem 4.1, it turns out that z is a topological singular pointof X , which is a contradiction. (cid:3) Let v be a vertex of Σ p of order N , and let ν , . . . , ν N be the setof all points of Σ p with d ( ν i , v ) = δ for a sufficiently small positivenumber δ . Take a small enough r p > a , . . . , a N of S ( p, r )with γ ′ p,a i (0) = ν i and r ≤ r p . For simplicity, we denote by S ij theruled surface S ( γ p,a i , γ p,a j ) spanned by γ p,a i and γ p,a j . We may assumethat all S ij satisfy the conclusion of Theorem 4.1. Then obviously S ( X ) ∩ B ( p, r ) is contained in the union of all S ij when v runs over theset of all vertices of Σ p ( X ). Sector correspondence . We fix S := S ij for a moment, and setΩ( S, r ) X := B X ( p, r ) ∩ S, C X := S X ( p, r ) ∩ S. From here on, we use the symbols B X ( p, r ) and S X ( p, r ) to emphasizethe metric ball and the metric circle in X . Note that Ω( S, r ) X isbounded by the two geodesics γ , γ and C X . Lemma 5.2.
For any small enough r ≤ r p , the sector Ω( S, r ) X is τ p ( r ) -almost isometric to a Euclidean sector.Proof. By Lemma 4.30, for every x ∈ C X , we have ∠ ( ˙ γ Xx,p (0) , ˙ γ Sx,p (0)) <τ p ( r ). Since ∠ ( ˙ γ Xx,p (0) , ˙ C X ) = π/
2, it follows that(5.1) | ∠ ( ˙ γ Sx,p (0) , ˙ C X ) − π/ | < τ p ( r ) . Consider the rescaling limit of the CAT( κ )-space: ( r S, p ) → ( K p ( S ) , o p )as r →
0. By Theorem 2.4, we have a τ p ( r )-almost isometry ϕ : WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 37 Ω( S, r ) X → image( ϕ ) ⊂ R . It suffices to show that image( ϕ ) is bi-Lipschitz homeomorphic to a Euclidean sector. We may assume that ϕ ( p ) = (0 ,
0) = O .Let L k be the line segment from O to ϕ ( γ k ( r )) ( k = 1 , L k in the polar coordinates as L k ( x ) = ( x, θ k ) (0 ≤ x ≤ x k ( r )) , where θ k is a constant and x k ( r ) := | ϕ ( γ k ( r )) , O | . Let θ be the di-rection representing the midpoint of ˙ L (0) and ˙ L (0). We may assumethat θ < θ = 0 < θ , and let L be the line segment from 0 in thedirection θ : L ( x ) = ( x, ϕ ( C X ) intersect L with ( r , q k := L k ( x k ( r )). Let C k denote the arc from q k to q ,k of circle of radius x k ( r ).Let D k (resp. U k ) be the domain bounded by L , L k and ϕ ( C X )(resp. by ϕ ( γ k ), L and ϕ ( C X )). Let p k ∈ L be the point such that ∠ Oq k p k = π/
4. Note that [ q k , p k ] ⊂ U k . Let J k denote the union[ O, p k ] ∪ [ p k , q k ]. Let ˆ D k (resp. ˆ U k ) be the domain bounded by L , L k and [ q k , p k ] (resp. by ϕ ( γ k ), L and [ q k , p k ]).Let J k ( x ) (0 ≤ x ≤ L ( J k )) be the arc-length parameter of J k with J k (0) = O . For every x ∈ [0 , L ( J k )], let ζ k ( x, s ) (0 ≤ s ≤
2) be thesegment such that • ζ k ( x,
0) = J k ( x ), ζ k ( x, ∈ L k ; • | O, ζ k ( x, | = | O, ζ k ( x, | ; • s ζ k ( x, s ) is proportional to the arc-length.Then ζ k ( x, s ) (0 ≤ x ≤ L ( J k ) , ≤ s ≤
1) defines a parametrizationof ˆ D k , and differentiable except x = x k , where J k ( x k ) = p k . Take aunique t k ( x ) ∈ (0 ,
2) such that ζ k ( x, t k ( x )) ∈ Im ( ϕ k ◦ γ k ) . Now, we define ψ k : ˆ U k → ˆ D k ( k = 1 ,
2) by ψ k ( ζ k ( x, s )) := ζ k (cid:18) x, st k ( x ) (cid:19) . Obviously t k ( x ) is Lipschitz, and hence differentiable on a set Ω ⊂ [0 , L ( J k )] with full measure since ζ k ( x, s ) defines a bi-Lipschitz embed-ding. O = ϕ ( p ) ϕ ◦ γ ϕ ◦ γ L L L ϕ ( C X ) ζ ζ q = ϕ ( γ ( r )) q = ϕ ( γ ( r )) r p p ˆ D ˆ D Sublemma 5.3.
Each ψ k : ˆ U k → ˆ D k is a τ p ( r ) -almost isometry.Proof. In the expression ψ k ( x, s ) := ψ k ◦ ζ k ( x, s ) = ζ k ( x, s/t k ( x )), wehave on Ω × [0 , ∂ψ k ∂s = 1 t k ( x ) ∂ζ k ∂s , ∂ψ k ∂x = ∂ζ k ∂x − (cid:18) − st ′ k ( x ) t k ( x ) (cid:19) ∂ζ k ∂s . (5.2)It is easily checked that(5.3) < c < (cid:12)(cid:12)(cid:12)(cid:12) ∂ζ k ∂x (cid:12)(cid:12)(cid:12)(cid:12) < c , < c < ∠ (cid:18) ∂ζ k ∂s , ∂ζ k ∂x (cid:19) < π − c , for some uniform positive constants c , . . . , c . Note also that(5.4) | t k ( x ) − | < τ p ( r ).Consider the curve η k ( x ) = ζ k ( x, t k ( x )) parametrizing ϕ ◦ γ k . Since dη k dx ( x ) = ∂ζ k ∂x ( x, t k ( x )) + ∂ζ k ∂s ( x, t k ( x )) t ′ k ( x )is τ p ( r )-almost parallel to ∂ζ k ∂x ( x, t k ( x )), we have(5.5) (cid:12)(cid:12)(cid:12)(cid:12) ∂ζ k ∂s ( x, s ) t ′ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < τ p ( r ) . Let v := ∂ζ k ∂x , V := dψ k ( v ) ,w := ∂ζ k ∂s / (cid:12)(cid:12)(cid:12)(cid:12) ∂ζ k ∂s (cid:12)(cid:12)(cid:12)(cid:12) , W := dψ k ( w ) . Combining (5.2), (5.3), (5.4) and (5.5), we have || V | − | v || < τ p ( r ) , || W | − | w || < τ p ( r ) , |h V, W i − h v, w i| < τ p ( r ) . Together with (5.3), this implies || dψ k ( u ) | − | < τ p ( r ) for each unittangent vector u on Ω × [0 , (cid:3) WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 39
Obviously, the τ p ( r )-almost isometry ψ k : ˆ U k → ˆ D k extends to a τ p ( r )-almost isometry ψ k : U k → D k . Thus we obtain a τ p ( r )-almostisometry between the images of ϕ and ψ : ψ : Im( ϕ ) → Im( ψ ) ⊂ R . Note that image( ψ ) is bounded by L , L and the arc ϕ ( C X ) whichis unchanged under ψ . Finally we deform image( ψ ) to the Euclideansector Ω( L , L ; r ) bounded by L k and the circle of radius r as follows.Let ϕ ( C X ) be parametrized as ϕ ( C X ) = ( r ( t ) , θ ( t )) 0 ≤ t ≤
1. Forevery 0 ≤ r ′ ≤ r ( t ), let us define φ ( r ′ , θ ( t )) = (cid:18) rr ( t ) r ′ , θ ( t ) (cid:19) , which defines a τ p ( r )-almost isometry φ : Im( ψ ) → Ω( L , L ; r ) . Thus the composition φ ◦ ψ ◦ ϕ : Ω( S, r ) X → Ω( L , L ; r ) is a τ p ( r )-almost isometry. This completes the proof of Lemma 5.2. (cid:3) Example 5.4.
Consider a smooth function f : R → R such that f = 0on { /n | n = 1 , , . . . } ∪ [1 , ∞ ) ∪ ( −∞ ,
0] and f > f <
0) on(1 / n, / (2 n − / (2 n + 1) , / n )), n = 1 , , . . . . We set I n := { ( x, f ( x )) | / ( n + 1) ≤ x ≤ /n } , and L + := { ( x, | x ≥ } , L − := { ( x, | x ≤ } . Let ℓ n denote the length of I n , and κ n themaximum of absolute geodesic curvature of I n . We choose f satisfying(4.4). By (4.4), one can take a closed domain H in R satisfying (1)-(4) in Example 4.4. Let L be the union of all I n , L + and L − . Nowmake a gluing of R and H along L and ∂H in such a way that I n , L + and L − are glued with K n , R + and R − respectively in an obvious way,where K n , R + and R − are subset of ∂H as defined in Example 4.4.The result Z of this gluing equipped with natural length metric is atwo-dimensional locally compact, geodesically complete CAT(0)-space.Let ι : R ∐ H → Z be the identification map, and let p := ι ( O ),where O is the origin of R . Let v denote the direction at p definedby P n ∈ N I n . For small ǫ >
0, take sufficiently small r > a , a , a ∈ S ( p, r ) such that a , a ∈ ι ( R ), a ∈ ι ( H ) and ∠ ( γ ′ p,a i (0) , v ) = ǫ . Then S ( a , a ) is convex, however S ( a , a ) and S ( a , a ) are not convex in Z . It is impossible to fill B ( p, r ) via properlyembedded convex disks for any r > Example 5.5.
This is also a deformation of Example 4.4. For thesmooth nonnegative function f : R → R + in Example 4.4, let Ω ′ := { ( x, y ) | | y | ≤ f ( x ) , x ≥ } . Take a closed concave domain H in R asin Example 4.4 and four copies H , . . . , H of H . Define ∂ + Ω ′ and ∂ − Ω ′ in a way similar to Example 4.4. Now glue H , H and Ω ′ along theirboundaries ∂H , ∂H , ∂ + Ω ′ in a way similar to Example 4.4. However,this time, R (1) − ⊂ ∂H is glued only with R (2) − ⊂ ∂H . The gluing of other part is similar to Example 4.4. Similarly glue H , H and Ω ′ along their boundaries ∂H , ∂H , ∂ − Ω ′ . The result W of these glu-ings equipped with natural length metric is a two-dimensional locallycompact, geodesically complete CAT(0)-space.Example 5.5 shows that one cannot take properly embedded disksto fill B ( p, r ) in Theorem 1.1. Thus we really need properly immersedbranched disks. Lemma 5.6. S ∩ B ( p, r ) is a CAT( κ ) -space with respect to the interiormetric.Proof. It suffices to show that every point q ∈ S ∩ S ( p, r ) has a neigh-borhood U in S ∩ B ( p, r ) such that any S -geodesic triangle region whosevertices are in U is contained in S ∩ B ( p, r ). To achieve this, we onlyhave to show that S ∩ B ( p, r ) is boundary convex, in the sense thatfor arbitrary x, y ∈ S ∩ S ( p, r ), any S -minimal geodesic γ Sx,y joiningthem is contained in S ∩ B ( p, r ). We may assume that γ Sx,y is vertical,and therefore it is an X -geodesic (see also Lemma 6.4). Hence theconclusion follows from the X -convexity of B ( p, r ). (cid:3) Filling ball . Now we fill the ball B ( p, r ) via properly embedded/branchedimmersed CAT( κ )-disks. For a vertex v of Σ p of order N , let ν , . . . , ν N and a , . . . , a N be as in the beginning of Section 5. For every pair ( i, j )with 1 ≤ i < j ≤ N , we want to take a simple loop in Σ p ( X ) passingthrough ν i , v and ν j . Since this is not possible in general, we considerthe two cases.Case I. There is a simple loop ζ in Σ p ( X ) through ν i , v and ν j .Consider the ruled surface S ( a i , a j ) as well as the other ruled surfacesdefined around other points of S which are vertices of Σ p ( X ) (if theyexist). By Lemma 5.2, considering the regular part of ζ as well, wecan define a proper Lipschitz embedding f vij : D ( ℓ ; r ) → B ( p, r ) with f vij (0) = p , where ℓ is the length of ζ . Lemma 5.6 implies that Im( f vij )is a CAT( κ )-space. Note that f vij has bi-Lipschitz constant < τ p ( r ).Case II. There are no simple loops in Σ p ( X ) containing ν i , v and ν j .Find ν k and ν ℓ , 1 ≤ k, ℓ ≤ N together with two simple loops ζ , ζ in Σ p ( X ) such that ζ contains ν i , v and ν k and ζ contains ν j , v and ν ℓ . We may assume k = ℓ and ζ ∩ ζ = { v } , because we could reduceto Case I otherwise. Chasing on ζ ∪ ζ in the order v → ν i → ν k → v → ν ℓ → ν j → v , we have a loop at v of length, say ℓ . Considerthe ruled surfaces S ( a i , a j ), S ( a k , a ℓ ) as well as the other ruled surfacesdefined around other points of ζ ∪ ζ which are vertices of Σ p ( X ) (ifthey exist). By Lemma 5.2, considering the regular part of ζ ∪ ζ WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 41 as well, we can define a proper Lipschitz immersion f vij : D ( ℓ ; r ) → B ( p, r ) with ( f vij ) − ( p ) = { } in a way similar to Case I. Note thatany multiple point q ∈ Im f vij is contained in S ( a i , a j ) ∩ S ( a k , a ℓ ), andthat ( f vij ) − ( q ) consists of two points, which are contained in sectors in D ( ℓ ; r ) corresponding to S ( a i , a j ) and S ( a k , a ℓ ) respectively.Note that f vij has bi-Lipschitz constant (resp. local bi-Lipschitz con-stant except the origin) < τ p ( r ) in Case I (resp. in Case II). Lemma 5.7. B ( p, r ) = [ v ∈ V (Σ p ( X )) [ ≤ i First note that from construction, Σ p ( X ) coincides with all theunion of Σ p (Im f vij ). Suppose there is a point x ∈ B ( p, r ) which is notcontained in any image Im f vij . Let ξ := ↑ xp . Take some Im f vij such that ξ ∈ Σ p (Im f vij ). We may assume that ξ is close to the vertex v , sinceif ξ is far from any vertex of Σ p ( X ), then x = γ ξ ( | p, x | X ) is certainlycontained in the union of all the images Im f vij , which is a contradiction.Let γ be a geodesic in Im f vij starting from p in the direction ξ . Notethat γ reaches the metric sphere S ( p, r ) (see also Sublemma 4.31). Let x ′ be the point of γ ξ such that | p, x ′ | X = | p, x | X . Consider the geodesic γ Xx,x ′ . If we extend γ Xx,x ′ through x ′ , it meets γ p,a k for some k . Similarly,if we extend γ Xx,x ′ through x , it meets γ p,a ℓ for some ℓ . Lemma 4.32yields that x ∈ S ( a k , a j ). This is a contradiction. (cid:3) Combinning Lemma 5.7 and the above discussion, we complete theproof of the first half of Theorem 1.1.6. Graph structure of singular set Our next step is to characterize S ( X ) ∩ B ( p, r ) as a union of finitelymany Lipschitz curves.For a closed subset A of X , we denote by ∂A the complement ofthe set of all points a of A such that there is a neighborhood of a homeomorphic to a disk and contained in A . From Theorem 4.1 andLemma 5.7, we immediately have the following corollary. Corollary 6.1. S ( X ) ∩ B ( p, r ) is the union of all ∂ ( S ( a i , a j ) ∩ S ( a k , a ℓ )) ∩ B ( p, r ) when the vertex v , a i and a j run over all the possibilities. For distinct 1 ≤ i, j, k ≤ N , we set C ij ; k := ( ∂ ( S ( a i , a j ) − S ( a j , a k )) − ∂S ( a i , a j )) ∩ B ( p, r ) . Lemma 6.2. C ij ; k is a simple Lipschitz arc in S ( X ) such that (1) it starts from p and reaches a point of ∂B ( p, r ) ; (2) its length is less than (1 + τ p ( r )) r ; (3) each point of Σ x ( C ij ; k ) is a vertex of Σ x ( X ) for every x ∈ C ij ; k .In particular, C ij ; k has definite directions everywhere.Proof. For each s ∈ [0 , r ], consider the ruling geodesic λ s ( t ) (0 ≤ t ≤ 1) of S ( a k , a j ) joining γ p,a k ( s ) to γ p,a j ( s ) in X . Let t ∈ (0 , 1) be thefirst parameter at which λ s meet S ( a i , a j ).We claim that λ s ([ t , ⊂ S ( a i , a j ) . (6.1)Since z s := λ s ( t ) is a topological singular point of X , by Lemma 4.25,we can take a direction ξ ∈ Σ z s ( S ( a i , a j )) with ∠ ( ξ , λ ′ s ( t )) = π .A geodesic γ ξ in S ( a i , a j ) with direction ξ reaches γ p,a i . Take ξ ∈ Σ z s ( S ( a i , a j )) with ∠ ( ξ , ξ ) = π . Similarly a geodesic γ ξ in S ( a i , a j )with direction ξ reaches γ p,a j . It follows from Lemma 4.33 that γ ξ and γ ξ form a geodesic in X . In particular, γ ξ is a geodesic in X , andtherefore γ ξ and λ s ([ t , γ , in X , Lemma 4.32implies that γ is contained in S ( a i , a j ), and so is λ s ([ t , C ( s ) := z s is continuous, its image coincides with C ij ; k . By Corollary 2.11, we have ∠ (( ∇ d p )( c ( s )) , Σ + C ( s ) ( C )) < τ p ( r ) , ∠ (( −∇ d p )( c ( s )) , Σ − C ( s ) ( C )) < τ p ( r ) , where Σ + C ( s ) ( C ) (resp. Σ − C ( s ) ( C )) denote the space of directions of C at C ( s ) in the positive direction (resp. negative direction). Now we takeanother parametrization ϕ ( s ) of C ij ; k defined as ϕ ( s ) = C ij ; k ∩ S ( p, s ),where S ( p, s ) denotes the metric circle of radius s with respect to d X .If s ′ is close enough to s , then we have ∠ ( ↑ ϕ ( s ′ ) ϕ ( s ) , ∇ d p ( ϕ ( s ))) < τ p ( r ) , which implies that lim s ′ → s | ϕ ( s ) , ϕ ( s ′ ) | X | s − s ′ | ≤ τ p ( r ). Thus ϕ is Lipschitzwith Lipschitz constant ≤ τ p ( r ), and therefore having length L ( ϕ ) = L ( C ij ; k ) ≤ (1 + τ p ( r )) r . (cid:3) Lemma 6.2 claims that the closure of S ( a j , a k ) − S ( a i , a j ) “transver-sally” intersects S ( a i , a j ) with the Lipschitz curve C ij ; k . In particularwe have Lemma 6.3. C ij ; k = C ji ; k = C jk ; i . In view of Lemma 6.3, we use the notation C ijk := C ij ; k . Using the discussion in the proof of Lemma 6.2, we show a refinedversion of Lemma 4.32: Lemma 6.4. For arbitrary x, y ∈ S = S ( a i , a j ) such that || p, x | X − | p, y | X | < | x, y | X / , the X -geodesic joining x and y is an S -geodesic. WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 43 Proof. Consider the geodesic γ Xx,y and extend it in the both directionsuntil it reaches γ p,a k and γ p,a ℓ for some k, ℓ at w k ∈ γ p,a k and w ℓ ∈ γ p,a ℓ respectively. That is,[ w k , w ℓ ] X = [ w k , x ] X ∪ [ x, y ] X ∪ [ y, w ℓ ] X . Let z (resp. u ) be the first point at which [ w k , x ] (resp. [ w ℓ , y ]) meets R ( a i , a j ). As in the proof of Lemma 6.2, we have points w i ∈ γ p,a i and w j ∈ γ p,a j such that[ w i , w j ] X = [ w i , z ] X ∪ [ z, x ] X ∪ [ x, y ] X ∪ [ y, w j ] X . From the hypothesis, we have || p, w i |−| p, w j || < | w i , w j | X / w i , w j ] X is an S -geodesic. Thus we conclude that[ x, y ] X is an S -geodesic as required. (cid:3) For a vertex v of Σ p ( X ), let a , . . . , a N ∈ S ( p, r ) be as in Section5, where N = N v . Let S ( a , . . . , a N ; r ) be the closed domain of B ( p, r )bounded by γ p,a i (1 ≤ i ≤ N ), and S ( p, r ). Note that S ( a , . . . , a N ; r )is the union of all the ruled surfaces S ( a i , a j ) and B ( p, r ). Corollary 6.5. For a vertex v of Σ p ( X ) , the union of all C ijk as abovecoincides with S ( X ) ∩ S ( a , . . . , a N ; r ) .Proof. Since every element of S ( X ) ∩ S ( a , . . . , a N ; r ) comes from theintersection of some S ( a i , a j ) and S ( a k , a ℓ ), it suffices to show that ∂ ( S ( a i , a j ) ∩ S ( a k , a ℓ )) \ S ( p, r ) is contained in C ijk ∪ C ijℓ . For every x ∈ ∂ ( S ( a i , a j ) ∩ S ( a k , a ℓ )), take an s such that the ruling geodesic λ s joining γ p,a i ( s ) to γ p,a j ( s ) goes through x , say at λ s ( t ) = x . Since x ∈ S ( X ), Lemma 2.2(2), Theorem 4.1 and Corollary 2.11 imply theexistence of a direction ξ ∈ Σ x ( S ( a k , a ℓ )) such that ∠ ( ξ, λ ′ s ( t )) = π .Then a geodesic γ ξ in S ( a k , a ℓ ) with direction ξ must reach γ p,a k or γ p,a ℓ . Suppose it reaches γ p,a k for instance: γ ξ ( t ) ∈ γ p,a k for some t > 0. An argument similar to that in the proof of Lemma 6.2 thenimplies that γ ξ ([0 , t ]) does not meet S ( a i , a j ) except for x , and that theunion γ ξ ([0 , t ]) ∪ λ s ([ t , S ( a k , a j ). This shows x ∈ C ijk . (cid:3) Using Corollary 6.5 together with the above discussion, we completethe proof of the second half of Theorem 1.1. Remark 6.6. Each C α contained in S ( a , . . . , a N ; r ) has direction at p equal to v . For each interior point q ∈ C α , C α has directions at q equal to two vertices of Σ q ( X ) , one is near ( −∇ d p )( q ) and the other isnear ( ∇ d p )( q ) . Structure of metric circles. Next we discuss the structure of S ( p, r ).Let b i := γ p,a i ( r ). For 0 < t ≤ r , set S ( v ; t ) := [ ≤ i For each < t ≤ r , S ( v ; t ) is a tree with endpoints γ p,a i ( t ) (1 ≤ i ≤ N ) .Proof. For 3 ≤ k ≤ N , put S k ( v ; t ) := [ ≤ i Let r p ≥ r > S ( a i , a j ), 1 ≤ i, j ≤ N , contained in R ( v ) for avertex v of Σ p ( X ). By Lemma 6.7, S ( v ; r ) is a tree with endpoints b i (1 ≤ i ≤ N ). Therefore S ( p, r ) has the same homotopy type as Σ p ( X ).Let L be any non-contractible simple closed loop in S ( p, r ). Fromthe discussion of the proof of Theorem 1.1, there is a non-contractiblesimple closed loop Ω in Σ p ( X ) of length, say ℓ ≥ π , and a properlyemmbedded CAT( κ )-disk f : D ( ℓ ; r ) → B ( p, r ) associated with Ωsuch that f ( ∂D ( ℓ ; r )) = L . For v ∈ Ω ∩ V (Σ p ( X )), let ξ i , ξ j ∈ Ω bepoints nearby v such that v is the midpoint of the arc [ ξ i , ξ j ]. Let S ij bethe ruled surface defined by ξ i , ξ j . Note that B S ij ( p, r ) ⊂ S ij ∩ B X ( p, r )and the nearest point map S ij ∩ S X ( p, r ) → S S ij ( p, r ) is distance non-increasing. Since L ( S S ij ( p, r )) ≥ L ([ ξ , ξ ]) / p µ ( κ, r ), we have L ( S ij ∩ S X ( p, r )) ≥ L ([ ξ , ξ ]) / p µ ( κ, r ), It follows that the length of L is atleast 2 π/ p µ ( κ, r ). (cid:3) Now we define a metric graph structure of S ( X ) in a generalizedsense as follows. Definition 6.8. We consider the relative topology of S ( X ) with lengthmetric. Let I be an open set of S ( X ). We call I an open arc in S ( X ) ifit is open in S ( X ) and is isometric to an open interval. A maximal openarc I with respect to the inclusion is called an open edge of S ( X ). Wedenote by E ( S ( X )) (resp. | E ( S ( X )) | ) the set (resp. the union) of allopen edges in S ( X ). We call each element of S ( X ) \| E ( S ( X )) | a vertex of S ( X ). We denote by V ( S ( X )) the set of all vertices of S ( X ). Let us WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 45 denote by V ∗ ( S ( X )) ⊂ V ( S ( X )) the set of all accumulation points of V ( S ( X )). The case V ∗ ( S ( X )) = V ( S ( X )) or H ( V ∗ ( S ( X ))) > v and v of S ( X )are adjacent if there is at least one open edge joining them. The order of a vertex v is deined as the limit of the number of components of B S ( X ) ( v, ǫ ) \ { v } as ǫ → • S ( X ) is locally path-connected. • V ( S ( X )) has locally finite order , in the sense that for every com-pact set K of S ( X ), the set of the orders of vertices containedin K is bounded.We exhibit the following example, which is another version of Exam-ple 4.4. Here we use the notion of ǫ -Cantor set (cf.[7]) to produce a two-dimensional CAT(0)-space X such that V ∗ ( S ( X )) is one-dimensional.Similar construction for a boundary singular set of a limit space ofmanifolds with boundary was made in [28]. Example 6.9. For any 0 < ǫ < 1, set δ := 1 − ǫ . We define the so-called ǫ -Cantor set of [0 , 1] inductively as follows: We start with I := [0 , I the open interval of length δ/ I . We denote by I be the rusult of this removing. Note that I consists of 2 disjoint closed intervals I ,j ( j = 1 , 2) having the samelength and that L ( I ) = 1 − δ/ 2. Suppose that we have constructed I k consisting of 2 k disjoint closed intervals I k,j (1 ≤ j ≤ k ) of thesame length such that L ( I k ) = 1 − δ/ − · · · − δ/ k . Remove from each I k,j the open interval of length δ/ k +1 around the center of I k,j . Wedenote by I k +1 the result of this removing. Thus, inductively we haveconstructed I n for every n . Finally we set I ∞ := ∞ \ n =0 I n , J n := [0 , \ I n , J ∞ := ∞ [ n =0 J n = [0 , \ I ∞ . Note that H ( I ∞ ) = lim n →∞ L ( I n ) = 1 − δ = ǫ . The set I ∞ is calledan ǫ -Cantor set.Next, inductively we define smooth functions f n : R → [0 , 1] ( n ∈ N )such that • supp( f n ) = J n ; • f n = f n − on J n − ; • if we set ˆ J ± n := { ( x, ± f n ( x )) | x ∈ J n } , then the length ℓ n andthe maximum κ n of absolute geodesic curvature of ˆ J ± n satisfy(4.4).Now we define the limit f := lim n →∞ f n : R → [0 , f ) = J ∞ . Using f , we define the closed subset Ω of R byΩ := { ( x, y ) | | y | ≤ f ( x ) , x ∈ R } , equipped with the length metric. Set ∂ ± Ω := { ( x, y ) | y = ± f ( x ) , x ∈ R } . Take closed concave domains H ± in R homeomorphic to the half planesuch that for certain isometries g ± : ∂ ± Ω → ∂H ± the absolute geodesiccurvature of g ± ( J ± n ) is greater than κ n . Take two copies H ± , H ± of H ± ,and make a gluing of H , H , H − , H − and Ω along their boundaries via g ± as in Example 4.4 to get a two-dimensional locally compact, geodesi-cally complete CAT(0)-space X . Note that V ∗ ( S ( X )) = V ( S ( X )) = I ∞ and therefore H ( V ∗ ( S ( X ))) = ǫ > Proof of Corollary 1.3. From the argument so far, we see that S ( X ) ∩ B ( p, r ) has the structure of a (possibly infinite) graph. Thus with itslength structure, S ( X ) carries the structure of a metric graph in thesense of Definition 6.8.Below, we give an estimate of the order of a vertex of the graphΓ := S ( X ) ∩ B ( p, r ). For a vertex v of Σ p ( X ), let a , . . . , a N ∈ S ( p, r )( N = N v ) and S ( a , . . . , a N ; r ) be as just before Corollary 6.5. Let µ ( v ) denote the minimal number of the curves C α starting from p suchthat • the direction of C α at p coincides with v ; • C α reaches S ( p, r ) ; • Γ = S α ∈ A C α .We first claim(6.3) µ ( v ) ≤ N − . Let γ be a geodesic starting from q := γ p,a ( r/ 2) in the directionperpendicular to −∇ d p and going into S ( a , . . . , a N ; r ). Then γ mustmeet some C α , say C α , for the first time, at say z := γ ( t ). Let n be the maximal number such that there exists ξ , . . . , ξ n satisfying ∠ ( γ ′ z ,q (0) , ξ i ) = π and ∠ ( ξ i , ξ j ) = π for all 1 ≤ i = j ≤ n . Extend γ in the direction ξ i for each 1 ≤ i ≤ n . Each extension either meetssome C α , say C α , again, or reaches some γ p,a j without meeting any C α . If it meets C α , extend it in several directions in a way similar tothe previous extension. Repeating this procedure finitely many times,we get a geodesic tree T such that(1) the set { z , . . . , z k } of the branching points of T coincides with T ∩ S ( X ) ;(2) the number of endpoints of T is equal to N .See also Lemma 6.7.For each 1 ≤ i ≤ k , let m ( z i ) denote the branching number of T at z i , and set b ( z i ) := m ( z i ) − WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 47 Sublemma 6.10. b ( z ) + · · · + b ( z k ) = N − . Proof. We proceed by induction on k . The case of k = 1 is trivial.Suppose it is true for k − 1. Let T ∗ be the smallest subtree of T containing z , . . . , z k . We may assume that z k is an endpoint of T ∗ and z k − is the vertex adjacent to z k . Let T ′ be the component of T \ { y } containing z , . . . , z k − , where y is the midpoint of [ z k − , z k ]. Note thatthe number N ′ of the endpoints of T ′ is determined as N = N ′ + m ( z k ) − . On the other hand, we have by indection b ( z ) + · · · + b ( z k − ) = N ′ − . Combining these equations, we obtain the sublemma. (cid:3) Sublemma 6.11. For each ≤ i ≤ k , there are at most b ( z i ) singularcurves C i, , . . . , C i,b ( z i ) such that they cover S ( X ) ∩ B ( z i , ǫ ) for small ǫ > .Proof. Let { x i , y i , u , . . . , u b ( z i ) } be the set of all points of T havingdistance ǫ from z i . It follows from the argument in the proof of Lemma6.2 that for each 1 ≤ j ≤ b ( z i ), the set { x i , y i , u j } defines a singularcurve, say C i,j . Conversely, it is similarly checked that S ( X ) ∩ B ( z i , ǫ )is covered by C i, ∪ · · · ∪ C i,b ( z i ) for small enough ǫ > (cid:3) Sublemma 6.11 shows that S ( X ) consists of b ( z i ) singular curvesnear the vertex z i . It follows from an obvious continuation argumenttogether with Sublemma 6.10 that M ( v ) ≤ N − 2. In particular, wesee that the order of Γ at p is bounded by X v ∈ V (Σ p ) ( N v − . Now let w ( = p ) be a vertex of Γ contained in B ( p, r ) arising froman intersection of singular curves tangent to v at p . Considering bothdownward and upward in Γ at w , we obtain that the order at w is atmost 2( N − (cid:3) Appendix A. Alexandrov’s result on ruled surfaces Following the ideas of Alexandrov in [4], we prove Theorem 3.6. Asmentioned in Section 1, it also follows from [22] in the CAT(0)-setting.We denote by D κ the diameter of M κ . Recall that a CAT( κ )-space isdefined as a D κ -geodesic space in which every triangle with perimeter < D κ is not thicker than its comparison triangle in M κ with the sameside lengths, where a D κ -geodesic space means a metric space in whichany two points with distance < D κ can be joined by a minimal geodesic.Throughout this appendix, let X be a CAT( κ )-space. A.1. Finite sequences of ruling geodesics. Let S be a ruled surfacein X with parametrization σ : R → X , where R = [0 , ℓ ] × [0 , π : R → R ∗ and p : R → [0 , ℓ ] be as in Section 3.For u, u ′ ∈ R with π ( u ) = π ( u ′ ), choose a finite decomposition ∆ = { s i } i =0 , ,...,n of [ p ( u ) , p ( u ′ )], so that s = p ( u ), s n = p ( u ′ ), and s i − ≤ s i for all i ∈ { , . . . , n } . Considering e ∆ σ ( π ( u ) , π ( u ′ )), we have asequence { x i } i =0 , ,...,n in X such that x = σ ( u ), x n = σ ( u ′ ), x i ∈ λ s i for all i ∈ { , . . . , n − } , and e ∆ σ ( π ( u ) , π ( u ′ )) = n X i =1 | x i − , x i | . We call such a sequence { x i } i =0 , ,...,n a ∆ -minimizing chain along S from σ ( u ) to σ ( u ′ ). Notice that possibly we have x i − = x i for some i ∈ { , . . . , n } .From the choice of a ∆-minimizing chain along S , we derive thefollowing: Lemma A.1. In the setting discussed above, let { x i } i =0 , ,...,n be a ∆ -minimizing chain along S from σ ( u ) to σ ( u ′ ) . Then for each i ∈{ , . . . , n − } and for each t ∈ { , } we have ∠ x i − x i λ s i ( t ) + ∠ λ s i ( t ) x i x i +1 ≥ π, whenever | x i − , x i | , | x i , x i +1 | < D κ , and the angles ∠ x i − x i λ s i ( t ) and ∠ λ s i ( t ) x i x i +1 can be defined.Proof. First we show the conclusion in the case t = 0. Set θ − i := ∠ x i − x i λ s i (0) and θ + i := ∠ λ s i (0) x i x i +1 . Take t i ∈ (0 , 1] with x i = λ s i ( t i ), where we may assume t i = 0. If we put h ( ǫ ) := | λ s i ( t i − ǫ ) , x i − | + | λ s i ( t i − ǫ ) , x i +1 | for small ǫ > 0, then by the first variationformula (see e.g., [10, Corollary II.3.6]) together with the ∆-minimizingproperty of { x i } i =0 , ,...,n , we have0 ≤ lim ǫ → h ( ǫ ) − h (0) ǫ = − (cos θ − i + cos θ + i ) . This implies θ − i + θ + i ≥ π . Similarly, we see the inequality for t = 1. (cid:3) Let u ∗ := π ( u ) , v ∗ := π ( v ) , w ∗ := π ( w ) be distinct points in R ∗ .Assume for a while that p ( u ) ≤ p ( v ) ≤ p ( w ) , and choose a decomposition ∆ = { s i } i =0 , ,...,n of [ p ( u ) , p ( w )] such thatfor some m ∈ { , . . . , n − } we have p ( v ) = s m . Let ∆ ′ := { s i } i =0 , ,...,m be the decomposition of [ p ( u ) , p ( v )], and ∆ ′′ := { s m + i } i =0 , ,...,n − m thedecomposition of [ p ( v ) , p ( w )]. Take a ∆ ′ -minimizing chain { y i } i =0 , ,...,m along S from σ ( u ) to σ ( v ), and a ∆ ′′ -minimizing chain { y m + i } i =0 , ,...,n − m WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 49 along S from σ ( v ) to σ ( w ), and a ∆-minimizing chain { z i } i =0 , ,...,n along S from σ ( u ) to σ ( w ). Assume in addition that we have e ∆ ′ σ ( u ∗ , v ∗ ) + e ∆ ′′ σ ( v ∗ , w ∗ ) + e ∆ σ ( w ∗ , u ∗ ) < D κ . Set x := σ ( u ), y := σ ( v ) and z := σ ( w ). Let B ∆ ( xy ) be the brokengeodesic S mi =1 y i − y i in X joining x and y , B ∆ ( yz ) the broken geodesic S n − mi =1 y m + i − y m + i in X joining y and z , B ∆ ( zx ) the broken geodesic S ni =1 z i − z i in X joining z and x . We denote by P ∆ ( xyz ) the polygonin X defined by P ∆ ( xyz ) := B ∆ ( xy ) ∪ B ∆ ( yz ) ∪ B ∆ ( zx ) , and we call P ∆ ( xyz ) the ∆ -minimizing chain triple along S . We de-note by θ ∆ x ( y, z ) the angle at x in X between B ∆ ( xy ) and B ∆ ( zx ), by θ ∆ y ( z, x ) the angle at y in X between B ∆ ( yz ) and B ∆ ( xy ), by θ ∆ z ( x, y )the angle at z in X between B ∆ ( zx ) and B ∆ ( yz ). λ s λ s λ s i λ s m λ s n x y zy z y i z i y i +1 z i +1 y n − z n − z m In the model surface M κ , we define a comparison polygon ˜ P ∆ ( xyz )for P ∆ ( xyz ) as follows: Let △ ˜ x ˜ y ˜ z and △ ˜ y n − ˜ z n − ˜ z be comparisontriangles in M κ for △ xy z and for △ y n − z n − z , respectively. For each i ∈ { , . . . , n − } , take comparison triangles △ ˜ y i ˜ y i +1 ˜ z i and △ ˜ y i +1 ˜ z i ˜ z i +1 in M κ for △ y i y i +1 z i and for △ y i +1 z i z i +1 , respectively, and then glue allthe comparison triangles in M κ along ˜ y i ˜ z i , and along ˜ y i +1 ˜ z i , for all i ∈ { , . . . , n − } . Let ˜ B ∆ ( xy ) be the broken geodesic S mi =1 ˜ y i − ˜ y i in M κ joining ˜ x and ˜ y , ˜ B ∆ ( yz ) the broken geodesic S n − mi =1 ˜ y m + i − ˜ y m + i in M κ joining ˜ y and ˜ z , ˜ B ∆ ( zx ) the broken geodesic S ni =1 ˜ z i − ˜ z i in M κ joining ˜ z and ˜ x . Then we put˜ P ∆ ( xyz ) := ˜ B ∆ ( xy ) ∪ ˜ B ∆ ( yz ) ∪ ˜ B ∆ ( zx ) , and we call ˜ P ∆ ( xyz ) a comparison ∆ -minimizing chain triple in M κ for P ∆ ( xyz ). We denote by ˜ θ ∆ x ( y, z ) the angle at ˜ x in M κ between˜ B ∆ ( xy ) and ˜ B ∆ ( zx ), by ˜ θ ∆ y ( z, x ) the angle at ˜ y in M κ between ˜ B ∆ ( yz ) and ˜ B ∆ ( xy ), by ˜ θ ∆ z ( x, y ) the angle at ˜ z in M κ between ˜ B ∆ ( zx ) and˜ B ∆ ( yz ). Note that θ ∆ x ( y, z ) ≤ ˜ θ ∆ x ( y, z ) , θ ∆ y ( z, x ) ≤ ˜ θ ∆ y ( z, x ) , θ ∆ z ( x, y ) ≤ ˜ θ ∆ z ( x, y ) . From Lemma A.1 we derive the following concavity of ˜ P ∆ ( xyz ) exceptthe vertices ˜ x, ˜ y, ˜ z : Namely, for each i ∈ { , . . . , n − } with i = m the inner angle at ˜ y i in ˜ P ∆ ( xyz ) is at least π ; moreover, for each i ∈ { , . . . , n − } the inner angle at ˜ z i in ˜ P ∆ ( xyz ) is at least π .By stretching the comparison ∆-minimizing chain triple ˜ P ∆ ( xyz ) atthe concave vertices, we obtain a triangle △ ¯ x ¯ y ¯ z in M κ whose side-lengths satisfy | ¯ x, ¯ y | = e ∆ ′ σ ( u ∗ , v ∗ ) , | ¯ y, ¯ z | = e ∆ ′′ σ ( v ∗ , w ∗ ) , | ¯ z, ¯ x | = e ∆ σ ( w ∗ , u ∗ ) . We call △ ¯ x ¯ y ¯ z a comparison ∆ -minimizing stretched triangle in M κ for P ∆ ( xyz ), and we denote it by ¯ P ∆ ( xyz ). We denote by ¯ θ ∆ x ( y, z ) theangle ∠ ¯ y ¯ x ¯ z at ¯ x in M κ between ¯ x ¯ y and ¯ z ¯ x , by ¯ θ ∆ y ( z, x ) the angle ∠ ¯ z ¯ y ¯ x at ¯ y in M κ between ¯ y ¯ z and ¯ x ¯ y , and by ¯ θ ∆ z ( x, y ) the angle ∠ ¯ x ¯ z ¯ y at ¯ z in M κ between ¯ z ¯ x and ¯ y ¯ z . Let ¯ y i ∈ ¯ x ¯ y and ¯ z i ∈ ¯ x ¯ z , i ∈ { , . . . , n − } , bethe points corresponding to ˜ y i and to ˜ z i , respectively. Since ˜ P ∆ ( xyz )is concave except the vertices, the Alexandrov stretching lemma (seee.g., [10, Lemma I.2.16]) leads to the following: Lemma A.2. Under the setting discussed above, we have ˜ θ ∆ x ( y, z ) ≤ ¯ θ ∆ x ( y, z ) , ˜ θ ∆ y ( z, x ) ≤ ¯ θ ∆ y ( z, x ) , ˜ θ ∆ z ( x, y ) ≤ ¯ θ ∆ z ( x, y ) . Moreover, for all i ∈ { , . . . , n − } we have | y i , z i | ≤ | ¯ y i , ¯ z i | . Let y j ∈ B ∆ ( xy ) \ { x, y } be a broken point for j ∈ { , . . . , m − } , y k ∈ B ∆ ( yz ) \ { y, z } a broken point for k ∈ { m + 1 , . . . , n − } , and z l ∈ B ∆ ( zx ) \ { z, x } a broken point for l ∈ { , . . . , n − } . Assumethat the broken points y j , y k , and z l are distinct to each other. Choosefour ∆-minimizing chain triples P ∆ ( xy j z l ), P ∆ ( y j yy k ), P ∆ ( z l y k z ), and P ∆ ( y j y k z l ) along S , and take comparison ∆-minimizing stretched tri-angles ¯ P ∆ ( xy j z l ), ¯ P ∆ ( y j yy k ), ¯ P ∆ ( z l y k z ), and ¯ P ∆ ( y j y k z l ) in M κ for P ∆ ( xy j z l ), P ∆ ( y j yy k ), P ∆ ( z l y k z ), and P ∆ ( y j y k z l ), respectively.From Lemma A.2 we derive the following monotonicity: Lemma A.3. Under the setting discussed above, we have ¯ θ ∆ x ( y j , z l ) ≤ ¯ θ ∆ x ( y, z ) , ¯ θ ∆ y ( y k , y j ) ≤ ¯ θ ∆ y ( z, x ) , ¯ θ ∆ z ( z l , y k ) ≤ ¯ θ ∆ z ( x, y ) . Proof. Gluing the triangles ¯ P ∆ ( xy j z l ) = △ ¯ x ¯ y j ¯ z l , ¯ P ∆ ( y j yy k ) = △ ¯ y j ¯ y ¯ y k ,¯ P ∆ ( z l y k z ) = △ ¯ z l ¯ y k ¯ z , and ¯ P ∆ ( y j y k z l ) = △ ¯ y j ¯ y k ¯ z l in M κ along the edges¯ y j ¯ y k , ¯ y k ¯ z l , and ¯ z l ¯ y j , we obtain a hexagon ¯ x ¯ y j ¯ y ¯ y k ¯ z ¯ z l in M κ whose side-lengths satisfy | ¯ x, ¯ y j | + | ¯ y j , ¯ y | = e ∆ ′ σ ( u ∗ , v ∗ ), | ¯ y, ¯ y k | + | ¯ y k , ¯ z | = e ∆ ′′ σ ( v ∗ , w ∗ ), WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 51 and | ¯ z, ¯ z l | + | ¯ z l , ¯ x | = e ∆ σ ( w ∗ , u ∗ ). By Lemmas A.1 and A.2, we have π ≤ θ ∆ y j ( x, z l ) + θ ∆ y j ( z l , y k ) + θ ∆ y j ( y k , y ) ≤ ¯ θ ∆ y j ( x, z l ) + ¯ θ ∆ y j ( z l , y k ) + ¯ θ ∆ y j ( y k , y ) . Similarly, we have π ≤ ¯ θ ∆ y k ( y, y j ) + ¯ θ ∆ y k ( y j , z l ) + ¯ θ ∆ y k ( z l , z ) ,π ≤ ¯ θ ∆ z l ( z, y k ) + ¯ θ ∆ z l ( y k , y j ) + ¯ θ ∆ z l ( y j , x ) . By stretching the hexagon ¯ x ¯ y j ¯ y ¯ y k ¯ z ¯ z l at the concave vertices ¯ y j , ¯ y k , and¯ z l , we obtain a comparison ∆-minimizing stretched triangle ¯ P ∆ ( xyz )in M κ for P ∆ ( xyz ). The Alexandrov stretching lemma (see e.g., [10,Lemma I.2.16]) leads to the desired inequalities. (cid:3) From Lemma A.2 we also derive the following: Lemma A.4. Let u ∗ , u ′∗ ∈ R ∗ be distinct points. Assuming p ( u ) ≤ p ( u ′ ) , we choose a decomposition ∆ = { s i } i =0 , ,...,n of [ p ( u ) , p ( u ′ )] . If e ∆ σ ( u ∗ , u ′∗ ) < D κ , then a ∆ -minimizing chain { x i } i =0 , ,...,n along S from σ ( u ) to σ ( u ′ ) is uniquely determined.Proof. Let x := σ ( u ) , x ′ := σ ( u ′ ), and suppose that two distinct ∆-minimizing chains { x i } i =0 , ,...,n and { y i } i =0 , ,...,n along S from x to x ′ satisfy x m = y m for m ∈ { , . . . , n − } . Then for the ∆-minimizingchain triple P ∆ ( xy m x ′ ) along S we see that a comparison ∆-minimizingstretched triangle ¯ P ∆ ( xy m x ′ ) degenerates in M κ . Hence we have ¯ x m =¯ y m . On the other hand, Lemma A.2 implies | x m , y m | ≤ | ¯ x m , ¯ y m | . Thisis a contradiction. (cid:3) A.2. Curvature bounds on ruled surfaces. Let ˆ △ u ∗ v ∗ w ∗ be a ge-odesic triangle in ( R ∗ , e σ ) with distinct vertices and with perimeter < D κ determined by ˆ △ u ∗ v ∗ w ∗ = d u ∗ v ∗ ∪ [ v ∗ w ∗ ∪ [ w ∗ u ∗ , where d u ∗ v ∗ , [ v ∗ w ∗ , and [ w ∗ u ∗ are the edges of ˆ △ u ∗ v ∗ w ∗ .We denote by △ ˜ u ∗ ˜ v ∗ ˜ w ∗ a comparison triangle in M κ for ˆ △ u ∗ v ∗ w ∗ with the same side-lengths, and by ˜ θ u ∗ ( v ∗ , w ∗ ) the angle ∠ ˜ v ∗ ˜ u ∗ ˜ w ∗ at ˜ u ∗ between ˜ u ∗ ˜ v ∗ and ˜ u ∗ ˜ w ∗ .In order to complete the proof of Theorem 3.6, it suffices to showthe following (see e.g., [10, Proposition II.1.7]). Lemma A.5. Every geodesic triangle ˆ △ u ∗ v ∗ w ∗ in ( R ∗ , e σ ) as abovesatisfies the convexity of angle κ -comparison : Namely for all w ′∗ ∈ d u ∗ v ∗ \{ u ∗ , v ∗ } , u ′∗ ∈ d v ∗ w ∗ \ { v ∗ , w ∗ } , and v ′∗ ∈ [ w ∗ u ∗ \ { w ∗ , u ∗ } , we have thefollowing monotonicity :˜ θ u ∗ ( v ′∗ , w ′∗ ) ≤ ˜ θ u ∗ ( v ∗ , w ∗ ) , ˜ θ v ∗ ( w ′∗ , u ′∗ ) ≤ ˜ θ v ∗ ( w ∗ , u ∗ ) , ˜ θ w ∗ ( u ′∗ , v ′∗ ) ≤ ˜ θ w ∗ ( u ∗ , v ∗ ) . Before proving Lemma A.5, we show the following sublemma. ByProposition 3.4, for every minimal geodesic c ∗ in ( R ∗ , e σ ) there exists amonotone curve c in R with π ◦ c = c ∗ up to monotone parametrization. Sublemma A.6. In the same setting as in Lemma A.5, let u ∗ , u ′∗ ∈ ( R ∗ , e σ ) be distinct points with e σ ( u ∗ , u ′∗ ) < D κ , and let c ∗ be a min-imal geodesic in ( R ∗ , e σ ) from u ∗ to u ′∗ . Assume p ( u ) ≤ p ( u ′ ) andchoose a sequence { ∆ n } n ∈ N of decompositions ∆ n = { s i } i =0 , ,...,n of [ p ( u ) , p ( u ′ )] with lim n →∞ | ∆ n | = 0 . For n ∈ N , let { x i } i =0 , ,...,n be the ∆ n -minimizing chain along S from x := σ ( u ) to x ′ := σ ( u ′ ) , and takea sequence { y i } i =0 , ,...,n in the image of γ := σ ∗ ◦ c ∗ in such a way that y = x , y n = x ′ , and y i ∈ λ s i for all i ∈ { , . . . , n − } . Then we havethe following :(1) e σ ( u ∗ , u ′∗ ) = lim n →∞ n X i =1 | y i − , y i | ;(2) For every s ∈ [ p ( u ) , p ( u ′ )] , and for every sequence { s i n } n ∈ N converging to s with s i n ∈ ∆ n , we have lim n →∞ | x i n , y i n | = 0 . Proof. (1) From Lemma 3.8, we derive e σ ( u ∗ , u ′∗ ) = lim n →∞ P ni =1 | x i − , x i | ;moreover, e σ ( u ∗ , u ′∗ ) = lim n →∞ P ni =1 | y i − , y i | . Indeed, we have e σ ( u ∗ , u ′∗ ) = lim n →∞ n X i =1 | x i − , x i | ≤ lim inf n →∞ n X i =1 | y i − , y i |≤ lim sup n →∞ n X i =1 | y i − , y i | ≤ lim sup n →∞ n X i =1 e σ ( y i − , y i ) = e σ ( u ∗ , u ′∗ ) . (2) For n ∈ N , let P n = ( S ni =1 x i − x i ) ∪ ( S ni =1 y i − y i ) be the polygonin X . In the model surface M κ , we construct a comparison ( n + 1)-gon ¯ P n for P n as follows: Let △ ˜ x ˜ x ˜ y and △ ˜ x n − ˜ y n − ˜ x ′ be comparisontriangles in M κ for △ xx y and △ x n − y n − x ′ , respectively. For each i ∈ { , . . . , n − } , take comparison triangles △ ˜ x i ˜ x i +1 ˜ y i and △ ˜ x i +1 ˜ y i ˜ y i +1 in M κ for △ x i x i +1 y i and △ x i +1 y i y i +1 , respectively, and then glue allthe comparison triangles in M κ along ˜ x i ˜ y i , and along ˜ x i +1 ˜ y i , for all i ∈ { , . . . , n − } . Then we put ˜ P n := ( S ni =1 ˜ x i − ˜ x i ) ∪ ( S ni =1 ˜ y i − ˜ y i ).From Lemma A.1 it follows that for each i ∈ { , . . . , n − } the innerangle at ˜ x i in ˜ P n is at least π . By stretching the polygon ˜ P n at theconcave vertices, we obtain an ( n +1)-gon ¯ P n = ¯ x ¯ x ′ ∪ ( S ni =1 ¯ y i − ¯ y i ) in M κ whose side-lengths satisfy | ¯ x, ¯ x ′ | = e ∆ n σ ( u ∗ , u ′∗ ) and | ¯ y i − , ¯ y i | = | y i − , y i | for all i ∈ { , . . . , n } . Let ¯ x i ∈ ¯ x ¯ x ′ , i ∈ { , . . . , n − } , be the pointscorresponding to ˜ x i . The Alexandrov stretching lemma (see e.g., [10,Lemma I.2.16]) leads to that | x i , y i | ≤ | ¯ x i , ¯ y i | for all i ∈ { , . . . , n − } .Suppose that the second half of the sublemma is false. Then we find s ∈ ( p ( u ) , p ( u ′ )), and a sequence { s i n } n ∈ N converging to s such that WO-DIMENSIONAL SPACES WITH CURVATURE BOUNDED ABOVE 53 for all n ∈ N we have s i n ∈ ∆ n , and we have | x i n , y i n | ≥ C for some C > 0. Then for the points ¯ x i n , ¯ y i n on the comparison ( n + 1)-gon ¯ P n for P n we have C ≤ lim inf n →∞ | x i n , y i n | ≤ lim inf n →∞ | ¯ x i n , ¯ y i n | . On the other hand, since e σ ( u ∗ , u ′∗ ) = lim n →∞ P ni =1 | x i − , x i | , and since e σ ( u ∗ , u ′∗ ) = lim n →∞ P ni =1 | y i − , y i | , the comparison ( n + 1)-gon ¯ P n de-generates in M κ as n → ∞ . This yields a contradiction. (cid:3) Proof of Lemma A.5. Without loss of generality, we may assume that p ( u ) ≤ p ( v ) ≤ p ( w ) . For each n ∈ N , let us choose a decomposition ∆ n = { s i } i =0 , ,...,n of[ p ( u ) , p ( w )] with lim n →∞ | ∆ n | = 0 such that p ( v ) = s m for some m ∈ { , . . . , n − } . Let ∆ ′ n := { s i } i =0 , ,...,m be the decomposition of[ p ( u ) , p ( v )], and let ∆ ′′ n := { s m + i } i =0 , ,...,n − m be the decompositionof [ p ( v ) , p ( w )]. Set x := σ ( u ), y := σ ( v ), z := σ ( w ) and take the(unique) ∆ ′ n -minimizing chain { y i } i =0 , ,...,m along S from x to y , andthe ∆ ′′ n -minimizing chain { y m + i } i =0 , ,...,n − m along S from y to z , andthe ∆ n -minimizing chain { z i } i =0 , ,...,n along S from x to z .Let P ∆ n ( xyz ) be the ∆ n -minimizing chain triple along S defined by P ∆ n ( xyz ) := B ∆ n ( xy ) ∪ B ∆ n ( yz ) ∪ B ∆ n ( zx ) , where B ∆ n ( xy ) is the broken geodesic S mi =1 y i − y i in X joining x and y , B ∆ n ( yz ) is the broken geodesic S n − mi =1 y m + i − y m + i in X joining y and z , and B ∆ n ( zx ) is the broken geodesic S ni =1 z i − z i in X joining z and x . Set x ′ := σ ( u ′ ), y ′ := σ ( v ′ ) and z ′ := σ ( w ′ ). By Sublemma A.6, wecan take sequences { y j n } n ∈ N , { y k n } n ∈ N , { z l n } n ∈ N of broken points on P ∆ n ( xyz ) \ { x, y, z } satisfyinglim n →∞ | y j n , z ′ | = 0 , lim n →∞ | y k n , x ′ | = 0 , lim n →∞ | z l n , y ′ | = 0 , where j n ∈ { , . . . , m − } , k n ∈ { m + 1 , . . . , n − } , l n ∈ { , . . . , n − } .Let ¯ P ∆ n ( xyz ) = △ ¯ x ¯ y ¯ z be a comparison ∆ n -minimizing stretchedtriangle in M κ for P ∆ n ( xyz ) whose side-lengths satisfy | ¯ x, ¯ y | = e ∆ ′ n σ ( u ∗ , v ∗ ) , | ¯ y, ¯ z | = e ∆ ′′ n σ ( v ∗ , w ∗ ) , | ¯ z, ¯ x | = e ∆ n σ ( w ∗ , u ∗ ) . Set ¯ θ ∆ n x ( y, z ) := ∠ ¯ y ¯ x ¯ z , ¯ θ ∆ n y ( z, x ) := ∠ ¯ z ¯ y ¯ x , and ¯ θ ∆ n z ( x, y ) := ∠ ¯ x ¯ z ¯ y .Choose the three ∆ n -minimizing chain triples P ∆ n ( xy j z l ), P ∆ n ( y j yy k ),and P ∆ n ( z l y k z ) along S , and take comparison ∆ n -minimizing chain tri-angles ¯ P ∆ n ( xy j z l ), ¯ P ∆ n ( y j yy k ), and ¯ P ∆ n ( z l y k z ) in M κ for P ∆ n ( xy j z l ), P ∆ n ( y j yy k ), and P ∆ n ( z l y k z ), respectively. As shown in Lemma A.3, wehave the monotonicity ¯ θ ∆ n x ( y j , z l ) ≤ ¯ θ ∆ n x ( y, z ), ¯ θ ∆ n y ( y k , y j ) ≤ ¯ θ ∆ n y ( z, x ),and ¯ θ ∆ n z ( z l , y k ) ≤ ¯ θ ∆ n z ( x, y ). From the choices of the sequences { y j n } n ∈ N , { y k n } n ∈ N , and { z l n } n ∈ N , it follows that ¯ P ∆ n ( xy j n z l n ), ¯ P ∆ n ( y j n yy k n ),and ¯ P ∆ n ( z l n y k n z ) converge to comparison triangles in M κ for trianglesˆ △ u ∗ w ′∗ v ′∗ , ˆ △ w ′∗ v ∗ u ′∗ , and ˆ △ v ′∗ u ′∗ w ∗ in ( R ∗ , e σ ), respectively. Notice that ¯ P ∆ n ( xyz ) converges to a comparison triangle in M κ for the triangleˆ △ u ∗ v ∗ w ∗ . Therefore we obtain˜ θ u ∗ ( w ′∗ , v ′∗ ) = lim n →∞ ¯ θ ∆ n x ( y j n , z l n ) ≤ lim n →∞ ¯ θ ∆ n x ( y, z ) = ˜ θ u ∗ ( v ∗ , w ∗ ) . Similarly, we see ˜ θ v ∗ ( u ′∗ , w ′∗ ) ≤ ˜ θ v ∗ ( w ∗ , u ∗ ), and ˜ θ w ∗ ( v ′∗ , u ′∗ ) ≤ ˜ θ w ∗ ( u ∗ , v ∗ ).Thus ˆ △ u ∗ v ∗ w ∗ satisfies the convexity of angle κ -comparison. (cid:3) From Lemma A.5 we conclude that ( R ∗ , e σ ) is a CAT( κ )-space. Thiscompletes the proof of Theorem 3.6. (cid:3) References [1] S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov geometry: prelim-inary version no. 1, arXiv:1903.08539[2] I. Arshinova and S. Buyalo, Metrics with upper-bounded curvature on 2 -polyhedra. (Russian) Algebra i Analiz 8 (1996),163–188; translation in St.Petersburg Math. J. 8 (1997), 825–844.[3] A. D. Alexandrow, ¨Uber eine Verallgemeinerung der Riemannschen Geome-trie. (German) Schr. Forschungsinst. 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Stadler, The structure of minimal surfaces in CAT(0) spaces, preprint,2018; arXiv:1808.06410; to appear in J. Eur. Math. Soc.[26] T. Yamaguchi, A convergence theorem in the geometry of Alexandrov spaces.Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), Semin.Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 601–642.[27] T. Yamaguchi, Upper curvature bounds and singularities, Noncommutativityand singularities , 161–171, Adv. Stud. Pure Math., 55, Math. Soc. Japan,Tokyo, 2009[28] T. Yamaguchi and Z. Zhang, Limits of manifolds with boundary, in prepa-ration. Koichi Nagano, Institute of Mathematics, University of Tsukuba,Tsukuba 305-8571, Japan Email address : [email protected] Takashi Shioya, Mathematical Institute, Tohoku University, Sendai980-8578, Japan Email address : [email protected] Takao Yamaguchi, Department of Mathematics, Kyoto University,Kyoto 606-8502, Japan Email address ::