arXiv:2003.06242v1 [math.DG] 13 Mar 2020 pc,crauedmnincniin ligconstructi gluing condition, curvature-dimension throu space, Collabor DFG the the through by and Mathematics” and for “RicciBounds” fin Center ERC-AdG acknowledges “Hausdorff the gratefully through KTS Union 396662902. Projektnummer – hw yPrla httedul fa lxnrvsaeo curva of space Alexandrov wher an space of double double the the b nonempty that is with Perelman space construction ind Alexandrov by this same shown w.r.t. the of of understood copies case is two special together boundaries A the between metric. their isometry of The components connected aries. isometric along spaces Alexandrov gi lxnrvo curvature of Alexandrov again .Itouto n ttmn fMi eut 1 1 via bounds curvature of Characterization 4 2.6. 1 functions 2.5. Semi-concave 2.4. Gluing spaces 2.3. Alexandrov condition 2.2. Curvature-dimension condition boundary Dirichlet 2.1. with Preliminaries flow heat Results to 2. Main Application of Statement and 1.1. Introduction 1. 00Mtmtc ujc Classification. Subject Mathmatics 2010 .Apyn 1 Applying 3. .Smcnaefntoso le pcs18 1.1 Theorem of References spaces Proof glued on functions 5. Semiconcave application 4. Second application 3.2. First 3.1. a ocntutAeado pcsi ygun oehrtoo m or two together gluing by is spaces Alexandrov construct to way A NGUN LXNRVSAE IHLWRRICCI LOWER WITH SPACES ALEXANDROV GLUING ON Ki uddb h etceFrcuggmishf DG G (DFG, Forschungsgemeinschaft Deutsche the by funded is CK IAIKPVTH HITA ETRR N ALTEDRST KARL-THEODOR AND KETTERER, CHRISTIAN KAPOVITCH, VITALI ihAeado uvtr one rmblwteRiemannia the below from condition bounded dimension curvature Alexandrov with obigadgun constructions. gluing and doubling Abstract. D oaiaino eeaie ic uvtr ons 11 bounds. curvature Ricci generalized of localisation 1. nrdcinadSaeeto anResults Main of Statement and Introduction D nti ae epoeta ntecaso ercmauespac measure metric of class the in that prove we paper this In localisation RCD UV TR BOUNDS CURVATURE ( ,N K, ≥ k ernnltrsoe Pt7 httelower the that [Pet97] showed later Petrunin . with ) Contents rmr 32,5E5 ewrs ercmeasure metric Keywords: 54E35. 53C21, Primary 1 K ∈ R on. & N ∈ D [1 tv eerhCne 1060. Center Research ative oaiain13 localisation nilspotb h European the by support ancial , ∞ speevdunder preserved is ) ra eerhFoundation) Research erman hteEclec Cluster Excellence the gh curvature- n nrni bound- intrinsic udr.I was It oundary. cdlength uced ture n glues one e e r given ore URM ≥ k 10 23 21 15 14 14 is 8 7 5 5 2 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM curvature bound is preserved in general for any gluing of two possibly different Alexandrov spaces. In this article we study Ricci curvature bounds in the sense of Lott, Sturm and Villani for this setup. More precisely, we consider the class of n-dimensional Alexandrov spaces with some lower curvature bound equipped with a Borel measure of the form Φ n = m for a semi-concave function Φ : X [0, ) such that the correspondingH metric measure space (X, d, m) satisfies a curvature-dimension→ ∞ condition CD(K,N) for K R and N [n, ). Here K does not necessarily coincide with k(n 1). In particular,∈ it’s possible∈ ∞ that k< 0 but K 0. To state our main− theorem we recall the following. The Alexandrov≥ boundary of (X, d) is denoted as ∂X equipped with the induced length metric d∂X . We write Σ for the space of direction at p X that is an Alexandrov space with p ∈ curvature bounded below by 1. We say v Σp for p ∂X is a normal vector at p if ∠(v, w) = π for any w ∂Σ . Here∈ d Φ denotes∈ the differential of the 2 ∈ p x i semi-concave function Φ at some point x Xi. We also refer to the remarks after Definition 2.20. ∈ Our main theorem is
Theorem 1.1 (Glued spaces). For i = 0, 1 let Xi be n-dimensional Alexandrov n spaces with curvature bounded below and let mX =Φi be measures where Φi : i HXi Xi [0, ) are semi-concave functions. Suppose there exists an isometry : ∂X → ∂X∞ such that Φ =Φ . I 0 → 1 0 1 ◦ I If the metric measure spaces (Xi, dXi , mi) satisfy the curvature-dimension con- dition CD∗(K,N) for K R, N [1, ) and if ∈ ∈ ∞ d Φ (v )+ d Φ (v ) 0 p ∂X and any normal vectors v Σ X , i =0, 1, p 0 0 p 1 1 ≤ ∀ ∈ i i ∈ p i then the glued metric measure space (X0 X1, (ι0)# mX +(ι1)# mX )) satisfies ∪I 0 1 the reduced curvature-dimension condition CD∗(K,N). Remark 1.2. If the measures are finite, one can replace in the conclusion of The- orem 1.1 the condition CD∗(K,N) with the full curvature-dimension condition CD(K,N). In this case the two conditions are equivalent [CM16].
Corollary 1.3. For i = 0, 1 let Xi be Alexandrov spaces with curvature bounded below, and let : ∂X0 ∂X1 be an isometry. Assume the metric measure spaces n I → R (Xi, dXi , Xi ) satisfy the condition CD∗(K,N) for K ,N [1, ). H n ∈ ∈ ∞ Then the metric measure space (X0 X1, X I X ) satisfies the condition ∪I H 0∪ 1 CD∗(K,N). Remark 1.4. An Alexandrov space with curvature bounded from below is infinites- imally Hilbertian. Therefore it satisfies the condition CD(K,N) (or CD∗(K,N)) if only if it satisfies the Riemannian curvature-dimension condition RCD(K,N) (or RCD∗(K,N)) (Corollary 2.10).
Remark 1.5. If Xi are convex domains in smooth Riemannian manifolds with lower Ricci curvature bounds, then the statement of Corollary 1.3 is regarded as folklore. A complete proof has been given in [PS18], based on a detailed approximation property derived in [Sch12]. For general noncollapsed RCD spaces there are two natural notions of boundary: one that was introduced by DePhillippis and Gigli in [DPG18] and another by Mondino and the first named author in [KM19]. Conjecturally both notions coincide ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 3 and the boundary (defined either way) is a closed subset in the ambient space. We make the following conjecture
Conjecture 1.6. For i = 0, 1 let Xi be noncollapsed RCD(K,n) spaces with nonempty boundary ∂Xi. Suppose there exists an isometry : ∂X0 ∂X1. n I → Then the glued metric measure space (X0 X1, X I X ) satisfies the condition ∪I H 0∪ 1 RCD(K,n). As a biproduct of our proof Theorem 1.1 we also obtain the following result that also seems to be new.
Theorem 1.7. For i =0, 1 let Xi be n-dimensional Alexandrov spaces with curva- ture bounded below as in the previous theorem, let X0 X1 be the glued Alexandrov ∪I spaces and let Φi : Xi R, i =0, 1, be semi-concave with Φ0 ∂ X0 = Φ1 ∂ X1 such that for any p ∂X it→ holds that | | ∈ i dΦ (v )+ dΦ (v ) 0 normal vectors v Σ X , i =0, 1. 0|p 0 1|p 1 ≤ ∀ i ∈ p i Then Φ0 +Φ1 : X0 X1 R is semiconcave. ∪I → We say that a function Φ : X R on an Alexandrov space X is double semi- → concave if Φ P : Xˆ R is semi-concave in the usual sense where Xˆ denotes the Alexandrov double◦ space→ of X and P : Xˆ X is the canoncial map. We give an alternative characterisation of this condition→ in Lemma 4.1 and Corollary 4.4. As another consequence of our main theorem we also obtain the following. Corollary 1.8 (Doubled spaces). Let X be an n-dimensional Alexandrov space n with curvature bounded below, and let mX =Φ X be a measure for a double semi- concave function Φ: X [0, ). Assume theH metric measure space (X, d , m ) → ∞ X X satisfies the condition CD∗(K,N) for K R and N 1. Then, the double space ˆ ∈ ≥ (X, dXˆ , mXˆ ) satisfies the condition CD∗(K,N). Let us briefly comment on the statement and the proof of Theorem 1.1 and Corollary 1.3. By Petrunin’s glued space theorem one knows that the glued space of two Alexandrov spaces with curvature bounded from below is again an Alexandrov space with the same lower curvature bound. However the intrinsic best lower Ricci bound might be different from the Alexandrov curvature bound. So Petrunin’s theorem does not imply any of the statements above. But we can use the improved regularity of the glued space for our purposes. It im- plies some lower Ricci bound that yields a priori information for transport densities and densities along needles in the Cavalletti-Mondino 1D localisation procedure. At this point a crucial difficulty appears. It is not known whether geodesics cross the boundary set where the spaces are glued together, only finitely many times. This difficulty does not occur for the double space construction. By symmetry in this case it is known that geodesics in the double space only cross once. For general glued spaces we overcome this problem by the following strategy. First, given a 1D localisation we show that the collection of geodesic that cross the boundary infinitely many times has measure 0 w.r.t. to the corresponding quotient measure. Then, we apply a theorem of Cavalletti and Milman on characterization of synthetic Ricci curvature bounds via 1D localisation. Remark 1.9. As pointed out by Rizzi [Riz18], in the previous theorem one cannot re- place the curvature-dimension conditon CD(K,N) for any K R and N (1, ) ∈ ∈ ∞ 4 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM with the measure contraction property MCP (K,N) [Stu06, Oht07]. The MCP is a weaker condition that still characterizes lower Ricci curvature bounds for N- dimensional smooth manifolds and is also consistent with lower Alexandrov curva- ture bounds. For the precise definition we refer to [Stu06, Oht07]. The counterex- ample in [Riz18] is given by the Grushin half-plane which satisfies MCP (0,N) if and only if N 4 while its double satisfies MCP (0,N) if and only if N 5. Another example≥ that is even Alexandrov is provided in the last sectio≥n of this article (Example 5.1). Remark 1.10. Let us mention that one can show that Petrunin’s gluing theorem holds for gluing n-dimensional Alexandrov spaces along isometric extremal subsets of codimension 1 which do not need to be equal to the whole components of their boundaries (see for instance [Mit16]). For example gluing two triangles having a side of equal length with all adjacent angles to it π/2 is again an Alexandrov space (a convex quadrilateral). Our results then generalize≤ to this situation as well. 1.1. Application to heat flow with Dirichlet boundary condition. The con- cept of doubling has recently found significant application in the study of the heat flow with Dirichlet boundary conditions. In particular, it allows the use of opti- mal transportation techniques. As widely known, these techniques are not directly applicable since the Dirichlet heat flow will not preserve masses. As observed in [PS18], this obstacle can be overcome by looking at the heat flow in the doubled space instead. The latter is accessible to optimal transport techniques and to the powerful theory of metric measure spaces with synthetic Ricci bounds. Moreover, it can always be expressed as a linear combination of the Dirichlet heat flow and the Neumann heat flow on the original space – and vice versa, both the Dirichlet and the Neumann heat flow on the original space can be expressed in terms of the heat flow on the doubled space. More precisely now, let X be an n-dimensional Alexandrov space with curvature bounded below, and let m =Φ n be a measure for a double semi-concave func- X HX tion Φ : X [0, ). Let (Pt)t 0 denote the heat semigroup with Neumann bound- → ∞ ≥0 0 ary conditions on X and let (Pt )t 0 denote the heat semigroup on X := X ∂X with Dirichlet boundary conditions≥ with respective generators ∆ and ∆0. \
Theorem 1.11. Assume the metric measure space (X, dX , mX ) satisfies the con- dition CD(K, ) for K R. Then the following gradient estimate of Bakry-Emery type ∞ ∈ 0 Kt 0 (1) P f e− P f a.e. on X ∇ t ≤ t ∇ and the following Bochner inequality hold true
1 2 2 (2) ∆ f f, ∆0f K f 2 ∇ − ∇ ∇ ≥ ∇ 0 weakly on X for all sufficiently smooth f on X. (Note that in the latter estimate, two different Laplacians appear and in the former, two different heat semigroups.) 1,2 0 More precisely, (1) holds for all f W0 (X ), the form domain for the Dirichlet Laplacian. And (2) is rigorously formulated∈ as
1 2 2 ∆ϕ f d m ϕ f, ∆0f d m K ϕ f d m 2 0 ∇ − 0 ∇ ∇ ≥ 0 ∇ ZX ZZX ZX 0 0 1, 2 0 0 for all f D(∆ ) with ∆ f W0 (X ) and all nonnegative ϕ D(∆ ) with 0 ∈ ∈ ∈ ϕ, ∆ ϕ L∞. ∈ ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 5
Proof. Both estimates follow from Corollary 1.8 and [PS18], Thm. 1.26. For the readers’ convenience, let us briefly recall the main argument. The estimates for the Dirichlet heat semigroup and Dirichlet Laplacian are direct consequences of analogous estimates for the heat semigroup (Pˆt)t 0 and Laplacian ∆ˆ on the doubled space ≥
Xˆ := X X′ ∪ ∂X=∂X′ . obtained by gluing X and a copy of it, say X′, along their common boundary ∂X ∂X′. Then Dirichlet and Neumann heat semigroups on X can be expressed ∼ in terms of the heat semigroup on Xˆ as 0 P f = Pˆ (f f ′), P f = Pˆ (f + f ′) t t − t t for any given bounded, measurable f : X R where f is extended to Xˆ by → putting f := 0 on Xˆ X and where f ′ : Xˆ R is defined as f ′(x′) := f(x) if \ → x′ X′ denotes the mirror point of x X. Then the gradient estimate for Pˆ on ∈ ∈ t Xˆ obviously implies that 0 Kt Kt P f = Pˆ (f f ′) e− Pˆ (f f ′) = e− P f |∇ t | |∇ t − |≤ t|∇ − | t|∇ | for every f W 1,2(X0). ∈ 0 Actually, (1) is stated in [PS18] only for functions f W 1,2(X0) which in ad- ∈ 0 dition satisfy f, f L1. But any f W 1,2(X0) can be approximated in W 1,2- |∇ | ∈ ∈ 0 norm by compactly supported Lipschitz functions fn (which in particular satisfy 1 2 fn, fn L ). Hence, Pt f is the L -limit of Pt fn and the claim follows by passing|∇ to| ∈ a suitable subsequence|∇ | which leads to a.e.-convergence.|∇ |
We outline the remaining content of the article. In section 2 we recall preliminaries and basics on optimal transport, Ricci curvature for metric measure spaces, Alexan- drov spaces, gluing of Alexandrov spaces and 1D localisation technique. We also state a new result by Cavalletti and Milman on characterizing the Ricci curvature bounds via 1D localisation. In section 3 we will give two application of the 1D localisation technique. The first application shows that almost all geodesics avoid set of n-measure 0 in the boundary in the glued space. The second application shows thatH givena1D local- isation w.r.t. an arbritrary 1-Lipschitz function, geodesics that are tangential to the boundary have measure 0 w.r.t. the corresponding quotient measure. In section 4 we use the results of the previous section to prove Theorem 1.7. In section 5 we prove the glued space theorem applying the results we obtained in section 3 and section 4.
Acknowledgments. The authors want to thank Anton Petrunin for helpful conver- sations on gluing spaces and other topics.
2. Preliminaries 2.1. Curvature-dimension condition. Let (X, d) be a complete and separable metric space equipped with a locally finite Borel measure m. We call a triple (X, d, m) a metric measure space. A geodesic is a length minimizing curve γ :[a,b] X. We denote the set of → constant speed geodesics γ :[a,b] X with [a,b](X) equipped with the topology → G 6 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM of uniform convergence and set [0,1](X) =: (X). For t [a,b] the evaluation map [a,b] G G ∈ et : (X) X is defined as γ γ(t) and et is continuous. AG set of geodesics→ F (X) is7→ said to be non-branching if ǫ (0, 1) the map ⊂ G ∀ ∈ e[0,ǫ] F is one to one. The| set of (Borel) probability measure is denoted with (X), the subset of probability measures with finite second moment is 2(X), theP set of probability measures in 2(X) that are m-absolutely continuousP is denoted with 2(X, m) and the subsetP of measures in 2(X, m) with bounded support is denotedP with 2 P b (X, m). P 2 2 The space (X) is equipped with the L -Wasserstein distance W2. A dynamical optimal couplingP is a probability measure Π ( (X)) such that t [0, 1] 2 ∈ P G ∈ 7→ (et)#Π is a W2-geodesic in (X). The set of dynamical optimal couplings Π P2 ∈ ( (X)) between µ0,µ1 (X) is denoted with OptGeo(µ0,µ1). P GA metric measure space∈P (X, d, m) is called essentially nonbranching if for any pair 2 µ0,µ1 (X, m) any Π OptGeo(µ0,µ1) is concentrated on a set of nonbranching geodesics.∈P ∈ Definition 2.1. For κ R we define cos : [0, ) R as the solution of ∈ κ ∞ → v′′ + κv =0 v(0)=1 & v′(0) = 0. sinκ is defined as solution of the same ODE with initial value v(0)=0& v′(0) = 1. That is sinh(√ κ x) cosh( κ x) if κ< 0 | | if κ< 0 √ κ | | | | cosκ(x)= 1p if κ =0 sinκ(x)= x if κ =0 cos(√κx) if κ> 0 sin(√κx) √κ if κ> 0 S2 Let πκ be the diameter of a simply connected space form k of constant curvature κ, i.e. if κ 0 πκ = ∞π ≤ ( √κ if κ> 0 For K R, N (0, ) and θ 0 we define the distortion coefficient as ∈ ∈ ∞ ≥ sinK/N (tθ) if θ [0, πK/N ), (t) sinK/N (θ) t [0, 1] σK,N (θ)= ∈ ∈ 7→ ( otherwise. ∞ Note that σ(t) (0) = t. Moreover, for K R, N [1, ) and θ 0 the modified K,N ∈ ∈ ∞ ≥ distortion coefficient is defined as
θ if K > 0 and N =1, (t) t [0, 1] τ (θ)= · ∞ 1 1 K,N 1 (t) − N ∈ 7→ N t σK,N 1(θ) otherwise − h i where our convention is 0 = 0. It holds that · ∞ (3) τ (t) (θ) σ(t) (θ). K,N ≥ K,N Definition 2.2 ([Stu06, LV09, BS10]). A metric measure space (X, d, m) satisfies the curvature-dimension condition CD(K,N) for K R, N [1, ) if for every ∈ ∈ ∞ ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 7
2 2 pair µ0,µ1 b (X, m) there exists an L -Wasserstein geodesic (µt)t [0,1] and an ∈ P ∈ optimal coupling π between µ0 and µ1 such that
(1 t) 1 (t) 1 (4) S (µ m) τ − (θ)ρ (x)− N + τ (θ)ρ (y)− N dπ(x, y) N t| ≤− K,N 0 K,N 1 Z h i where µi = ρid m, i =0, 1, and θ = d(x, y). We say (X, d, m) satisfies the reduced curvature-dimension condition CD∗(K,N) for K R and N (0, ) if we replace the coefficients τ (t) (θ) with σ(t) (θ). ∈ ∈ ∞ K,N K,N Remark 2.3. By the inequality (3) the condition CD(K,N) always implies the condition CD∗(K,N) and the latter is equivalent to a local version of CD(K,N). Under the assumptions that (X, d, m) is essentially nonbranching and m is finite Cavalletti and Milman [CM16] prove that CD(K,N) and CD∗(K,N) are equivalent (compare with Theorem 2.27 below). Definition 2.4. A metric measure space (X, d, m) satisfies the Riemannian curvature- dimension condition RCD(K,N) (or RCD∗(K,N)) if it satisfies the condition CD(K,N) (or CD∗(K,N)) and is infinitesimally Hilbertian, that is the corre- sponding Cheeger energy is quadratic.
Remark 2.5. Since RCD(K,N) and RCD∗(K,N) spaces are essentially non-branching, the two conditions are equivalent provided m is finite (compare with Remark 2.15 in [KK17]. 2.2. Alexandrov spaces. In the following we introduce metric spaces with Alexan- drov curvature bounded from below. For an introduction to this subject we refer to [BBI01]. Definition 2.6. We define md : [0, ) [0, ) as the solution of κ ∞ → ∞ v′′ + κv =1 v(0)=0 & v′(0) = 0. More explicitly
1 κ (1 cosκ x) if κ =0, mdκ(x)= 1 2 − 6 ( 2 x if κ =0. Definition 2.7. Let (X, d) be a complete geodesic metric space. We say (X, d) has curvature bounded below by κ R in the sense of Alexandrov if for any unit speed geodesic γ : [0,l] X such that∈ → (5) d(y,γ(0)) + l + d(γ(l),y) < 2πk, it holds that
(6) [md (d γ)]′′ + md (d γ) 1. κ y ◦ κ y ◦ ≤ If (X, d) has curvature bounded from below for some k R in the sense of Alexan- drov, we say that (X, d) is an Alexandrov space. ∈ Remark 2.8. Alexandrov spaces are non-branching. Theorem 2.9 (Petrunin, [Pet11]). Let (X, d) be an n-dimensional Alexandrov n space with curvature bounded from below by k. Then, (X, d, X ) satisfies the con- dition CD(k(n 1),n). H − 8 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM
Corollary 2.10. Let (X, d) be an n-dimensional Alexandrov space with curvature n bounded from below by κ. Then, the metric measure space (X, d, X ) satisfies the condition RCD(κ(n 1),n). H − Proof. The statement is known. Here, we give a straightforward argument for completeness. It is enough to show that Cheeger energy is quadratic. It is known that a doubling condition and a 1-1 Poincar´einequality hold for Alexandrov spaces. Hence, we can follow the same argument as in [KK17, Section n 6]. It is known [BGP92] that for -a.e. points x X the tangent cone TpX is isometric to Rn, and for a LipschitzH function the differential∈ exists and is linear n-a.e. [Che99, Theorem 8.1]. This implies the Cheeger energy is quadratic by the sameH argument as in [KK17, Section 6].
Let (X, d) be an n-dimensional Alexandrov space. We denote with TpX the unique blow up tangent cone at p X. The tangent cone T X coincides with ∈ p the metric cone C(Σp) where Σp is the space of directions at p equipped with the angle metric. The definition of the angle metric is as follows. The angle ∠(γ1,γ2) between two geodesics γi, i = 1, 2, with γ1(0) = γ2(0) = p and parametrized by arclength is defined by the formula s2 + t2 d(γ1(s),γ2(t)) cos ∠(γ1,γ2) = lim − . s,t 0 → 2st Then, the space of directions ΣpX is given as the metric completion of SpX via ∠ where SpX is the space of geodesics starting in p. We refer to [BBI01] for details. One can show that (ΣpX, ∠) is an (n 1)-dimensional Alexandrov space with − n 1 curvature bounded below by 1. We say p X is a regular point if ΣpX = S − . We denote the set of regular points with Xreg∈ . As was mentioned above n-almost every point p X is regular. A theorem of Petrunin [Pet98] is the nextH statment. ∈ reg If γ :[a,b] X is a geodesic such that there exists t0 [a,b] with γ(t0) = X → reg ∈ then γ(t0) X for all t [a,b]. One can∈ define the boundary∈ ∂X X of X via induction over the dimension. ⊂ One says that p X is a boundary point if ∂ΣpX = . ∂X denotes the set of all boundary points,∈ and we call ∂X the boundary of X6 . ∅ Let p X be a boundary point, that is ∂ΣpX = . We say v ΣpX is a normal vector in∈p if ∠(v, w)= π for any w ∂Σ X. 6 ∅ ∈ 2 ∈ p Theorem 2.11 (Perelman [Per93], [PP93, Lemma 4.3] ). For any point in an n- dimensional Alexandrove space there exists an arbitrary small, closed, geodesically convex neighborhood.
2.3. Gluing. Let (X0, dX0 ) and (X1, dX1 ) be complete, n-dimensional Alexandrov spaces with non-empty boundaries ∂X0 and ∂X1 equipped with their intrinsic dis- tances d and d respectively. Let : ∂X ∂X be an isometry. ∂X0 ∂X1 I 0 → 1 The topological glued space of X0 and X1 along their boundaries w.r.t. is defined as the quotient space X ˙ X /R of the disjoint union X ˙ X where I 0∪ 1 0∪ 1 x y if and only if (x)= y if x ∂X ,y ∂X , and x = y otherwise. ∼R I ∈ 0 ∈ 1 The equivalence relaiton R induces a pseudo distance on X ˙ X = X as follows. 0∪ 1 First, we introduce an extended metric d on X0 ˙ X1 via d(x, y) = dXi (x, y) if x, y X for some i 0, 1 and d(x, y) = otherwise.∪ Then, for x, y X ˙ X ∈ i ∈ { } ∞ ∈ 0∪ 1 ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 9 we define k 1 − dˆ(x, y) = inf d(pi, qi) i=0 X where the infimum runs over all collection of tuples (pi, qi) i=0,...,k 1 X X for some k N such that q p , for all i = 0,...,k{ 1} and x −= p⊂,y =×q . ∈ i ∼R i+1 − 0 k One can show that x y if and only if dˆ(x, y) = 0 if X and X are Alexandrov ∼R 0 1 spaces. The glued space between (X0, dX0 ) and (X1, dX1 ) w.r.t. : ∂X0 ∂X1 is the metric space defined as I →
X0 X1 := (X0 X1/R, dˆ). ∪I ∪ In the following we denote the glued space as (Z, dZ ), and boundary ∂X0 with its intrinsic metric with (Y, d ). In the case when X = X = X and = id , we Y 0 1 I ∂X call X X =: Xˆ the double space of X. ∪I Remark 2.12. For every point p X Y , i = 0, 1, there exists ǫ > 0 such that ∈ i\ Bǫ(p) Xi and dZ B (p) B (p) = dXi B (p) B (p). ⊂ | ǫ × ǫ | ǫ × ǫ
Theorem 2.13 (Petrunin, [Pet97]). Let (X0, dX0 ) and (X1, dX0 ) be n-dimensional Alexandrov spaces with nonempty boundary and curvature bounded from below by k. Let : ∂X ∂X be an isometry w.r.t. the induces intrinsic metrics. Then, I 0 → 1 X0 X1 is an Alexandrov space with curvature bounded from below by k. ∪I Remark 2.14. The special case of a double space was proven first by Perelman [Per]. Remark 2.15. By symmetry of the construction one can see that geodesics in the double space Xˆ of an Alexandrov space X connecting points in Xˆ ∂X intersect with the boundary at most once, and the restriction of the double metric\ to X ∂X \ coincides with dX . This observation was crucial in Perelman’s proof of the double theorem. However, in the general case of glued spaces it’s not clear if geodesics connecting points in Z Y intersect Y at most finitely many times. This creates an extra difficulty in the\ proof of Petrunin’s theorem and also in the proof of Theorem 1.1. Let us recall some additional facts about the glued space Z [Pet97]. Since the boundary Y X0 is an extremal subset in X0, the following holds. Con- ⊂ 1 sider the blow up tangent cone limǫ 0(X0, ǫ dX0 ,p) = TpX0 for p Y . Then, 1 → ∈ limǫ 0(Y, dY ,p)= TpY w.r.t. the intrinsic metric dY on Y is equal to C(∂ΣpX0)= → ǫ ∂C(ΣpX0). It follows that ∂ΣpX0 is isometric to ∂ΣpX1 via an isometry ′ that arises as blow up limit of . I I Then it also follows from Petrunin’s proof of the glued space theorem that TpZ = TpX0 ′ TpX1 and ΣpZ =ΣpX0 ′ ΣpX1. ∪I ∪I n 1 If p Y is a regular point in the glued space Z, that is ΣpZ = S − , it follows by ∈ n 1 maximality of the volume of S − in the class of Alexandrov spaces with curvature Sn 1 Sn 1 Sn 1 bounded below by 1 that ΣpX0 = +− and ΣpX0 = − where +−/ denote the − n 2 − lower and upper half sphere respectively, and ΣpY = ∂ΣpX0 = S − . In particular, Sn 1 the north pole N in +− is the unique normal vector a p Y X0, and the south n 1 ∈ ⊂ pole S S − is the unique normal vector a p Y X1. ∈ − ∈ ⊂ 10 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM
2.4. Semi-concave functions. We recall a few basic facts about concave functions following [Pla02]. A function u :[a,b] R is called concave if the segment between any pair of points lies below the→ graph. If u is concave, u is lower semi continuous and u(s) u(t) continuous on (a,b). The secant slope s−t is a decreasing function in s and t. It follows that the right and left derivative− + d u(r + h) u(r) d− u(r h) u(r) u(r) = lim − & u(r) = lim − − dr h 0 h dr h 0 h ↓ ↓ − exist in R and R respectively for all r [a,b] with values in R − − ∪ {∞} d∪+ {−∞} d d+/ ∈ if r (a,b). Moreover dr u(r) dr u(r) and dr u(r) are decreasing in r. If − d+ ∈ d ≤ dr u(a) < ( dr u(b) > ), u is continuous in a (in b). Let u :[∞a,b] (0, )−∞ satisfy → ∞ (1 t) (t) (7) u γ(t) σ − ( γ˙ )u γ(0) + σ ( γ˙ )u γ(1) ◦ ≥ κ | | ◦ κ | | ◦ for any constant speed geodesic γ : [0, 1] [a,b]. It follows that u is lower semi continuous and continuous on (a,b). → Definition 2.16. Let f :[a,b] R be continuous on (a,b), and let F :[a,b] R → → such that F ′′ = f on (a,b). For a function u :[a,b] R we write u′′ f on (a,b) if u F is concave on (a,b). → ≤ − We say a function u : (0,θ) R is λ-concave if u′′ λ. We say u is semiconcave if for any r (0,θ) we can find→ ǫ > 0 and λ R ≤such that u is λ-concave on (r ǫ, r + ǫ).∈ ∈ −If u satisfies (7) for every constant speed geodesic γ : [0, 1] [a,b], then one can check that →
u′′ + ku 0 on (a,b) ≤ in the sense of the previous definition. We note that (7) implies that u is continuous b on (a,b), and U(t)= a g(s,t)u(s)ds satisfies U ′′ = u on (a,b) where g(s,t) is the Green function of the interval [a,b]. − R On the other hand we have the next lemma. Lemma 2.17. Let u :[a,b] R be lower semi-continuous and continuous on (a,b) → such that u′′ + ku 0 on (a,b) in the sense of the definition above. Then u satisfies≤(7) for every constant speed geodesic γ : [0, 1] [a,b]. → Proof. We sketch the proof. If u′′ + ku 0 then u kU is concave. In particular, it follows for φ C2((a,b)), φ 0, that≤ − ∈ c ≥
0 (u + kU)φ′′dt = uφ′′ + k uφdt ≥ Z Z Z by the distributional characterisation of convexity (see [Sim11]). Hence, u satisfies u′′ + ku 0 in distributional sense, and therefore (7) follows by [EKS15, Lemma 2.8]. ≤
Lemma 2.18. If u satisfies (7) for every constant speed geodesic γ : [0, 1] [a,b] of length less than θ
If u :[a,b] R satifies (7), it is is semi-concave, therefore locally Lipschitz on (a,b) and hence→ differentiable 1-almost everywhere. Moreover, the right and left − L d+ d derivative also exist in this case and satisfy dr u(r) dr u(r) with equality if and − d+/ ≤ only if u is differentiable in r. dr u is continuous from the right/left. Since u is 1 locally semi concave, the second derivative u′′ exists -almost everywhere. The following Lemma can be found in [Pla02] (LemmaL 113).
Lemma 2.19. Consider u : (a,b) R continuous such that u′′ ku on (a,c) → ≤ − and on (c,b) for some c (a,b). Then u′′ ku on (a,b) if and only if ∈ ≤− d d+ − u(c) u(c). dr ≥ dr Definition 2.20. Let (X, d) be an n-dimensional Alexandrov space and Ω X. A function f :Ω R is λ-concave if f is locally Lipschitz and f γ : [0, L(γ⊂)] R is λ-concave for→ every constant speed geodesic γ : [0, L(γ)] ◦ Ω. A function→ f : X R is semi-concave if for every p X there exists a neighborhood→ U p such that→ f is λ-concave for some real λ∈. ∋ |U We say a function f : X R is double semi-concave if the function f P : Xˆ R → ◦ → is semi-concave where the P : Xˆ X is the projection map form the double space → Xˆ to X. If ∂X = , concavity and double concavity coincide. In [Pet07] Petrunin∅ defines concavity as double concavity. Let X be an Alexandrov space and let f : X R be locally Lipschitz. Then, the limit → f γ(r) f γ(0) d+ lim ◦ − ◦ = (f γ)(0) =: dfp(γ ˙ ) =: df(γ ˙ ) R r 0 r dr ◦ ∈ ↓ exists for every geodesic γ : [0,θ] X parametrized by arc length with γ(0) = p, and for every p X. We call df : →T X R the differential of f. ∈ p → The differential dfp on TpX can be equivalently defined as limit of the sequence 1 1 R ǫ (f f(p)) : ( ǫ X,p) . This limit is understood in the sense of Gromov’s Arzela-Ascoli− theorem (see→ for instance [Sor04]. It also makes sense for functions that are just locally Lipschitz but the differential is not unique in this case. Note that since Alexandrov spaces are nonbranching, under GH convergence of Alexan- drov spaces every geodesic in the limit is a limit of geodesics in the sequence. Therefore it follows that dfp : TpX R is Lipschitz for Lipschitz functions, and also concave if f is semiconcave. This→ in turn implies that v = df : Σ R p p → satisfies v′′ + v 0 along geodesics in Σ . ≤ p 2.5. 1D localisation of generalized Ricci curvature bounds. In this section we will recall the localisation technique introduced by Cavalletti and Mondino. The presentation follows Section 3 and 4 in [CM17]. We assume familarity with basic concepts in optimal transport. Let (X, d, m) be a locally compact metric measure space that is essentially non- branching. We assume that supp m = X. Let u : X R be a 1-Lipschitz function. Then → Γ := (x, y) X X : u(x) u(y)= d(x, y) u { ∈ × − } 1 is a d-cyclically monotone set, and one defines Γ− = (x, y) X X : (y, x) Γ . u { ∈ × ∈ u} If γ [a,b](X) for some [a,b] R such that (γ(a),γ(b)) Γ then (γ(t),γ(t)) Γ ∈ G ⊂ ∈ u ∈ u 12 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM for a 1 b By definition = g− ( u ). The map g is Borel measurable, g(γ, )= γ : (aγ ,bγ) V T b · 1 → X is a geodesic, g : u is bijective and its inverse is given by g− (x) = (Q(x), d(x, Q(x))). V → T ± Theorem 2.23. Let (X, d, m) be an essentially non-branching CD∗(K,N) space with supp m = X, m(X) < , K R and N (1, ). Then, for any 1-Lipschitz∞ function∈ u : X∈ ∞R there exists a disintegration →b mγ γ Q of m that is strongly consistent with Ru. { Moreover,} ∈ for q-a.e. γ Q, m is a Radon measure with m = h 1 and ∈ γ γ αH |Xα (Xγ, dXγ , mγ ) verifies the condition CD(K,N). More precisely, for q-a.e. γ Q it holds that ∈ 1 (1 t) 1 (t) 1 N−1 − N−1 N−1 (8) hγ (ct) σK/N 1( c˙ )hγ (γ0) + σK/N 1( c˙ )hγ (γ1) ≥ − | | − | | for every geodesic c : [0, 1] (a ,b ). → γ γ Remark 2.24. The property (8) yields that hγ is locally Lipschitz continuous on (a ,b ) [CM17, Section 4], and that h : R (0, ) satifies γ γ γ → ∞ 2 1 1 d N−1 K N−1 hγ + hγ 0 on (a ,b ) in distributional sense. dr2 N 1 ≤ γ γ − 2.6. Characterization of curvature bounds via 1D localisation. Definition 2.25. Let (X, dX , mX ) be an essentially non-braching metric measure space with m(X) = 1, let K R and N 1, and let u : X R be a 1-Lipschitz ∈ ≥ →1 function. We say that (X, dX , mX ) satisfies the condition CDu(K,N) if there exist subsets X X, γ Q, such that γ ⊂ ∈ (i) There exists a disintegration of m on (Xγ)γ Q: Tu ∈ m = mγ dq(γ) with mγ (Xγ )=1 for q-a.e. γ Q. |Tu ∈ ZXγ (ii) For q-a.e. γ Q the set Xγ is the image Im(γ) of a geodesic γ : Iγ X for an interval∈ I R. → γ ⊂ (iii) The metric measure space (Xγ , dXγ , mγ) satisfies the condition CD(K,N). 1 The metric measure space (X, dX , mX ) satisfies the condition CDLip(K,N) if it satisfies the condition CD1 (K,N) for any 1-Lipschitz function u : X R. u → Remark 2.26. From the previous subsection it is immediatly clear that the condition 1 CD(K,N) implies the condition CDLip(K,N). The following theorem will play an important role in the proof of our gluing result. Theorem 2.27 (Cavalletti-Milman). If an essentially non-branching metric mea- sure space (X, d , m ) satisfies the condition CD1 (K,N) for K R and N X X Lip ∈ ∈ [1, ) then it satisfies the condition CD∗(K,N). If∞m is a finite measure it even satisfies CD(K,N). Remark 2.28. Taking into account a disintegration result for σ-finite measures in [CM18, Section 3.1] it should be possible to prove the previous theorem also for σ-finite measures. Consequently in our main theorem we would then be able to replace the condition CD∗ with the condition CD in general. 14 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM 3. Applying 1D localisation 3.1. First application. Let X0 and X1 be n-dimensional Alexandrov spaces, let : ∂X0 ∂X1 be an isometry as in Theorem 2.13. Let Z be the glued space, I → n set ∂X0 = ∂X1 =: Y and recall that (Z, dZ , ) satisfies CD(k(n 1),n) for some k R. We consider continuous functions ΦH and Φ on X and−X respectively ∈ 0 1 0 1 such that Φ0 ∂X =Φ1 ∂X , and we define ΦZ : X0 X1 R by | 0 | 1 ∪I → Φ0(x) if x X0, ΦZ (x)= ∈ (Φ1(x) otherwise. Lemma 3.1. Let X be an n-dimensional Alexandrov space with Y = ∂X = . 6 ∅ Let Φ: X R, i = 0, 1 be semi-concave. Then Φ Y : Y R is differentiable n 1 → | n→1 − -a.e. meaning that Y contains a subset A such that − (Y A)=0 and for H n 1 H \ every a A it holds that T Y = R − and dΦ: T Y R is linear. ∈ a ∼ a → n 1 Proof. One says a point p Y = ∂X is boundary regular if TpY = R − . Since ∈ n 1 Y is the boundary of an n-dimensional Alexandrov space X, it is − -rectifiable. H n 1 Even stronger, the set of boundary regular points (Y ) in Y has full Y− -measure and for any ǫ > 0 one can cover (Y ) by (1 +Rǫ)-biLipschitz coordinateH maps ǫ Rn 1 R ǫ Fi : Ui Vi − where each Vi is open. The maps Fi are given by standard strainer→ coordinates⊂ centered at boundary regular points. By the metric version of Rademacher’s theorem due to Cheeger [Che99] it follows that Φ Y is differentiable n 1 | − -a.e. . H Let us give another, self-contained argument that does not rely on Cheeger’s theorem. Without loss of generality we can assume that Φ is L-Lipscjhitz for some finite ǫ L > 0. For 0 <ǫ< 1 we consider the maps Fi . Let us drop the superscript ǫ j for a moment. In particular, each coordinate component Fi , j = 1,...,n 1, of j − Fi is a semiconcave function on Ui and admits a differential dFi p in the sense of 1 | Alexandrov spaces at every point p Ui. Since Fi and Fi− are (1+ ǫ)-biLipschitz, j ∈ one has that (1+ǫ) v dFi p(v) (1+ǫ) v for every p Ui and every v TpX. | | ≤1 | | R |≤ | | ∈ ∈ n 1 The function Φ Fi− : Vi is 2L-Lipschitz and therefore differentiable − - a.e. by the standard◦ Rademacher→ theorem. So we can choose a set of full measureL ǫ Wi = Wi in Ui such that p Fi(Wi) Vi the point p Wi is regular and the 1 ∀ ∈ ⊂ ∈ function Φ Fi− is differentiable at Fi(p). The chain◦ rule for Alexandrov space differentials yields 1 (9) dΦ = d(Φ F − ) DF p W |p ◦ i |Fi(p) ◦ i|p ∀ ∈ i 1 n 1 where DF = (dF ,...,dF − ). i|p i |p i |p We obtain by (9) that for all p Wi and for any ǫ> 0 the Alexandrov differential n 1 ∈ ǫ dΦ p : R − R of Φ at p is the composition of a 2L-Lipschitz linear map A = | 1 7→ Rn 1 R ǫ Rn 1 Rn 1 d(Φ Fi− ) Fi(p) : − and 1-homogeneous map B = DFi p : − − that◦ is ǫ-close| to an isometry7→ on a the unit ball around the origin (we| have iden7→ tified n 1 TpY with R − in the above). 1 1 n Let us consider ǫn = n and let p n N Wi . After eventually choosing a 1 ∈ ∈ 1 subsequence A n A for a linear map A and B n B for an isometry B. Hence → T1 S → n n 1 dΦp = A B is linear. Since n N Wi has full − -measure this yields the claim. ◦ ∈ H T S ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 15 Proposition 3.2. Let Z be the glued space, let Y = ∂Xi Z. Let N Y such n 1 ⊂ n ⊂ that − (N)=0. Let (Xγ)γ Q be the 1D localisation of m = Z w.r.t. the 1-LipschitzH function u = d(x , )∈for x B (x ) X and η > 0.H Let (Q, q) and 1 · 1 ∈ η 1 ⊂ 1 Q : b Q be the corresponding quotient space and the quotient map. Then Tu → n 1 (X Q− ( γ Q : t [0,L(γ)) s.t. γ(t) N ))=0. H 0 ∩ { ∈ ∃ ∈ ∈ } n 1 Proof. The property that − (N) = 0 is equivalent to the following statement. H For any ǫ> 0 and any δ (0,η/4) there exist (ri)i N with ri (0,δ) and xi Z, i N, such that ∈ ∈ ∈ ∈ ∈ n 1 (10) N B (x ) and (r ) − ǫ. ⊂ ri i i ≤ i N i N [∈ X∈ i 1 We set Q′ = γ (X): t [0, 1] s.t. γ(t) B (x ) and Q = Q′ − Q . i { ∈ G ∃ ∈ ∈ r i } i i\ j=1 j W.l.o.g. we assume that q(Q ) > 0 for any i N. We will prove that i ∈ S n 1 X Q− (Q ) Cǫ H 0 ∩ i ≤ i N ! [∈ for some constant C = C(k,n). This implies the claim of the proposition. N Let us fix i . There exists Qi† with q(Qi) = q(Qi†) such that mγ admits a ∈ 1 density hγ w.r.t. and (Xγ, mγ ) is CD(k(n 1),n) for all γ Qi†. In particular, if J I and J =Htb + (1 t)J, then the Brunn-Minkowski− inequality∈ implies ⊂ γ t γ − m (J ) C(k,n)tn m (J). γ t ≥ γ 1 N D D We pick J = γ− (X0). Let D = diam Z and choose s such that s 1 ri s . l,s ∈ l,s D − ≤ ≤ We decompose J into intervals (J )l=1,...,s such that J s . Moreover, there l,s 1 | |≤ exists tl (0, 1) such that Jtl γ− (B2ri (xi)). Since ri δ < η/4 and since B (x ) ∈X , we have B (x ) ⊂B (x )= . Therefore t ≤ η . Hence η 1 ⊂ 1 η/2 1 ∩ 2ri i ∅ l ≥ 2D s 1 s l,s n l,s n mγ (B2ri (xi)) mγ (Jtl ) C(k,n)tl mγ (J ) C(k,n,D)η mγ (J). ri ≥ D ≥ ≥ Xl=1 Integration w.r.t. q on Qi yields ˆ n 1 1 n n 1 C(k,n)ri − (B2ri (xi)) C(k,n,D,η) (X0 Q− (Qi)). ≥ ri H ≥ H ∩ After summing up w.r.t. i N together with (10) and since Qi i N are disjoint, we obtain the claim and we∈ proved the proposition. { } ∈ 3.2. Second application. Let u : X R be a 1-Lipschitz function, let (mγ)γ Q → ∈ be the induced disintegration of n. We pick a subset Qˆ of full q measure in Q H such that Ru(x) = Xγ for all x Xγ . By abuse of notation we write Qˆ = Q and 1 ∈ u = Q− (Qˆ). T We say that a unit speed geodesic γ :[a,b] X is tangent to Y if there exists t [a,b] such that γ(t ) Y andγ ˙ (t ) T Y →. We define 0 ∈ 0 ∈ 0 ∈ p 1 Q† := γ Q : #γ− (Y ) < . ∈ ∞ Lemma 3.3. If γ Q Q†, then γ is tangent to Y . ∈ \ 16 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM 1 Proof. If γ Q Q†, then #γ− (Y )= . Hence, after taking a subseqence one can ∈ \ ∞ find a strictly monotone sequence ti [aγ ,bγ] such that γ(ti) Y , ti t0 [aγ,bγ] and γ(t ) γ(t ) Y . In a blow up∈ of Y around γ(t ) the sequence∈ γ→(t ) converges∈ i → 0 ∈ 0 i to the the velocity vector of γ at t0. Hence, we conclude thatγ ˙ (t0) Tγ(t0)Y and γ is tangent to Y . ∈ For U X open we write ⊂ n ( u U)= mγ(U)dq(γ)= hγ(r)dr dq(γ) H T ∩ −1 ⊗ ZQ Zg (U) R b where g : Q u is the ray map defined in Subsection 2.5. We also note that (r, γ)V ⊂ ×h (→r) isT measurable. ∈ V 7→ γ Remark 3.4. Let B R Q be measurable. Then g(B Q) =: X is a ⊂ × × B ⊂ measurable subset since g is a Borel isomorphism. Then Xγ is measurable w.r.t. the induced measurable structure and by Fubini’s theoremB ∩ the map γ Q L(γ γ−1( )) ∈ 7→ | B is measurable. We can apply this for the case when B = ( , 0) Q. It follows −∞ × that γ Q aγ = L(γ γ−1(g(( ,0) Q)) R is measurable. Similar for bγ. ∈ 7→ | −∞ × ∈ Remark 3.5. Consider the map Φ : R Q R Q, Φ (r, q) = (tr, q) for t > 0. t × → × t Then, it is clear that Φt( )= t is a measurable subset of for t (0, 1]. Moreover b V V b V b ∈ g( t)= u,t is a measurable subset of u such that Xγ u,t = tXγ (aγ,bγ). If V T n b b T ∩ T ⊂ t (0, 1), then ( u u,t) > 0. ∈ H T \T b Again by Fubinis theorem U Xγ u,t = U tXγ is measurable in Xγ for q-a.e. γ Q and the map ∩ ∩T ∩ ∈ 1 −1 LU,t : γ Q L(γ (ta ,tb ) γ (U))= 1U tXγ d ∈ 7→ | γ γ ∩ ∩ L Z 1 is measurable. We note that the set (ta ,tb ) γ− (U) might not be an interval. γ γ ∩ Let Y X and consider Uǫ = Bǫ(Y ) for ǫ > 0. For s N and t (0, 1] we define ⊂ ∈ ∈ Cǫ,s,t = γ Q : L(γ γ−1(U ) (ta ,tb )) > ǫs . ∈ | ǫ ∩ γ γ Further, we set Cs,t = Cǫ′,s,t = γ Q : lim inf L(γ γ−1(U ) (ta ,tb ))/ǫ s { ∈ ǫ 0 | ǫ ∩ γ γ ≥ } ǫ>0 ǫ′ ǫ → [ \≤ and Ct = Cs,t = γ Q : lim L(γ γ−1(U ) (ta ,tb ))/ǫ = . { ∈ ǫ 0 | ǫ ∩ γ γ ∞} s N → \∈ Lemma 3.6. Let 0 < t 1 and let γ Q. If γ [taγ ,tbγ ] is tangent to Y , then γ C . ≤ ∈ | ∈ t Proof. For the proof we ignore t (0, 1] and consider γ . ∈ |[aγ ,bγ ] Let γ Q be tangent to Y . Assume γ / C. Then there exists a sequence (ǫi)i N ∈ ∈ ∈ such that limi L(γ γ−1(B (Y )) (a ,b ))/ǫi = C [0, ). By assumption there →∞ ǫi γ γ | ∩ ∈ 1∞ exists t [a ,b ] such that γ(t ) Y . Hence t γ− (B (Y )) [a ,b ]. There 0 ∈ γ γ 0 ∈ 0 ∈ ǫi ∩ γ γ ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 17 ǫi 1 ǫi exists a maximal interval I contained in γ− (Bǫi (Y )) [aγ,bγ] such that t0 I . After taking another subsequence we still have ∩ ∈ − L(γ 1 ) ǫi γ (Bǫi (Y )) (aγ ,bγ ) L(γ I (aγ ,bγ )) > lim | ∩ lim | ∩ =: C′ 0 i ǫ i ǫ ∞ →∞ i ≥ →∞ i ≥ ǫi and L(γ Iǫi (a ,b )) =: Li 0. We set γi = γ Iǫi (a ,b ). Since I is maximal such | ∩ γ γ → | ∩ γ γ that Im(γi) Bǫi (Y ), we have supy Y,t Iǫi d(y,γ(t)) ǫi. In the rescaled space 1 ⊂ ∈ ∈ ≥ (Z, dZ ) the geodesic γi is a geodesic of length 1 and Li 1 sup d(y,γi(t0)) C′/2 y Y,t I Li ≥ ∈ ∈ ǫi for i N suffienciently large. By the proof of Petrunin’s glued space theorem we ∈ 1 know that (Y, L d Y ) converges in GH sense to Tγ(t0)Y . Therefore, for ǫi 0 a i | ∠ → sequence of points γi(ti) converges to v Tγ(t0)Z such that supw T Y (w, v) ∈ ∈ γ(t0) ≥ C′/2. This is a contradiction since the tangent vectorγ ˙ (t0) Tγ(t0)Y and tangent vector of geodesics in Alexandrov spaces are well-defined and∈ unique. Li Hence, for any sequence ǫi 0 it follow that and therefore γ Ct. → ǫi → ∞ ∈ Corollary 3.7. Let γ Q. If γ (aγ ,bγ ) is tangent to Y , then γ C = t (0,1)Ct. ∈ | ∈ ∈ Lemma 3.8. Let q be associated to u. Then q(Q t (0,1) Ct)=0. S ∩ ∈ Proof. It is clearly enough to show that for any t (0S, 1) it holds that q(Q Ct) = 0. Therefore in the following we work with a fixed ∈t. ∩ We recall that aγ < 0 < bγ, γ Q aγ ,bγ are measurable and Q = l N l 1 ∈ 7→ l ∈ { ≥ bγ , aγ l . It is obviously enough to prove the lemma for Q = l aγ , bγ |1 | | |≥ } N {l ≥ | S| | |≥ l for arbitrary l . Therefore we fix l N and replace Q with Q . By abuse of notation} we will drop∈ the superscript l for∈ the rest of the proof. By rescaling the whole space with 4l we can assume that 4 a , b 4l2 for each γ Q. ≤ | γ| | γ |≤ ∈ Let Cǫ,s,t be defined as before for ǫ (0,ǫ0) and s N. 1 ∈ ∈ We pick γ Cǫ,s,t and consider γ− (Bǫ(Y )) (taγ ,tbγ) =: Iγ,ǫ. We set L(γ Iγ,ǫ ) =: Lǫ. ∈ ∩ | We observe that 4l2 (1 t) a (1 t)4, 4l2 (1 t) b (1 t)4. ≥ − | γ |≥ − ≥ − | γ|≥ − We pick r Iγ,ǫ and τ (aγ ,taγ) (tbγ ,bγ). Theorem 2.23 implies that ([aγ ,bγ],hγ dr) satisfies the∈ condition∈CD(k(n ∪1),n). Then, the following estimate holds (c.f. [CM17, Inequality (4.1)]) − n 1 sink− ((r aγ) (bγ r)) hγ(r) k 1 − ∧ − hγ(τ) ≥ sin − ((τ a ) (b τ)) k − γ ∧ γ − n 1 sin − ((1 t)4) k − h (τ)= C(k,n,t,l)h (τ). n 1 2 γ γ ≥ sink− 4l for a universal constant C(k,n,t,l). We take the mean value w.r.t. 1 on both sides and obtain L 1 1 1 1 ǫ hγd C(k,n,t,l) 2 hγd . L I L ≥ 4l (a ,ta ) (tb ,b ) L Z γ,ǫ Z γ γ ∪ γ γ 18 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM Hence, after integrating w.r.t. q on C and taking into account 1 1 by ǫ,s,t ǫs ≥ Lǫ definition of Cǫ,s,t, it follows 1 1 n(B (Y )) m (B (Y ))dq(γ) ǫsH ǫ ≥ sǫ γ ǫ ZCǫ,s,t 1 h d 1dq(γ) ≥ Lǫ γ L ZCǫ,s,t ZIγ,ǫ 1 Cˆ hγ d dq(γ) ≥ C (a ,ta ) (tb ,b ) L Z ǫ,s,t Z γ γ ∪ γ γ Cˆ m ( b b )dq(γ) ≥ γ Tu \Tu,t ZCǫ,s,t ˆ 1 n where C = 2l C(k,n,t,l). It is known that (Bǫ(Y )) ǫM for some constant M > 0 provided ǫ > 0 is sufficiently small.H This follows≤ from semiconcavity of the boundary distance function in Alexandrov spaces, Lipschitz continuity of the induced gradient flow and the coarea formula. Hence M C(K,N,k,t) m ( b b )dq(γ). s ≥ γ Tu \Tu,t ZCǫ,s,t If we take limit for ǫ 0, we obtain → M C(K,N,k,t) m ( b b )dq(γ). s ≥ γ Tu \Tu,t ZCs,t Finally, for s it follows → ∞ 0= m ( b b )dq(γ). γ Tu \Tu,t ZCt But by construction of b we know that m ( b b ) is positive for every γ Q Tu,t γ Tu \Tu,t ∈ if t (0, 1). Therefore, it follows q(C ) = 0. ∈ t Combining the above lemma with Corollary 3.7 gives Corollary 3.9. Let q be associated to u. Then q(γ Q : γ is tangent to ∈ |(aγ ,bγ ) Y )=0. Let us remark here that we do not claim that the set of geodesics in Q which are tangent to Y at one of the endpoints has measure zero. We suspect this is true but this is not needed for the applications. As a first consequence of Proposition 3.2 and Corollary 3.9 we obtain the follow- ing corollary. Corollary 3.10. Let x X Y . Then, for n -a.e. point x X the geodesic 1 ∈ 1\ HZ 0 ∈ 0 that connects x0 and x1 intersects with Y only finitely many times and in any intersection point Φ is differentiable. Z |Y 4. Semiconcave functions on glued spaces Lemma 4.1. Let (X, d) be an n-dimensional Alexandrov space and let Φ: X R be a double semi-concave function. → Then Φ is semi-concave in the usual sense and for any p ∂X it holds that dΦ(v) 0 for any normal vector v Σ . ∈ ≤ ∈ p ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 19 Proof. A function Φ : X R is semi-concave if Φ P is semi-concave on the → ◦ double space Xˆ that is an Alexandrov space without boundary. In particular, for any geodesic γ in Xˆ such that Im(γ) X the composition Φ γ is semi-concave. ⊂ ◦ Moreover, let p X be a boundary point such that there exists v ΣpX ∈ [ ∈ normal to the boundary. Considering p in Xˆ we know that ΣpXˆ = ΣpX. Hence v, v Σ Xˆ and ∠(v, v) = π. Therefore, v and v generate a geodesic line in − ∈ p − − TpXˆ = C(ΣpXˆ). Since Φ is double semi-concave, its differential dΦp : TpXˆ R is concave, and by Lemma 2.19 it follows that dΦ( v) dΦ(v). On the other→ hand we have dΦ( v)= dΦ(v). This implies the claim.− ≥ − − We continue to work with the setup from the previous section. Let Φ : X i i → R, i = 0, 1, be semi-concave such that they agree on the boundaries ∂X0 and ∂X1 respectively identified via an isometry . Let Z be the glued space and let Φ : Z R be the naturally constructed glueingI of Φ , Φ . Z → 0 1 Lemma 4.2. Let γ : [0,L(γ)] Z be a constant speed geodesic with γ(0) → ∈ X0 ∂X0 and γ(L(γ)) X1 ∂X1. Suppose γ intersects Y in a single point p = γ(t0). Suppose\ further that the∈ following\ conditions hold: (i) p is a regular point in Z; (ii) dΦ (v )+ dΦ (v ) 0 normal vectors v Σ X , i =0, 1; 0|p 0 1|p 1 ≤ ∀ i ∈ p i (iii) The restriction Φ is differentiable at p. |Y Then Φ γ : [0,L(γ)] R is semi-concave. In particular Z ◦ → + d− d + (11) dΦ (γ ˙ −)= Φ γ(t ) Φ γ(t )= dΦ (γ ˙ ). − 0 dt 0 ◦ 0 ≥ dt 1 ◦ 0 1 + where t [0,L(γ)] γ−(t)= γ(L(γ) t) and γ = γ. ∈ 7→ − +/ Proof. By Lemma 2.19 we only need to check (11). In the following we write γ − +/ instead ofγ ˙ −. By assumption we have that γ([0,t0]) X0 and γ([t0,L(γ)]) X1. ⊂ n ⊂n 1 By assumption γ(t0) =: p is a regular point in Z, that is TpZ = R and ΣpZ = S − . Moreover ΣpZ is the glued of ΣpX0 and ΣpX1 along their isometric boundary and Sn 1 Sn 1 ∂ΣpX0 = +− and ∂ΣpX1 = − (see the remarks at the end of Subsection 2.3). In this case the north pole N and− the south pole S are the unique normal vectors in ΣpX0 and ΣpX1, respectively. + If γ−(t0) = N ΣpX0, then by symmetry γ (t0) = S and by assumption it follows ∈ + dΦ (γ−) 0 dΦ (γ ). − 0 ≥ ≥ 1 + If γ−(t0) = N, then it also follows γ (t0) = S. There exists a geodesic loop in 6 + 6 n 2 ΣpZ that contains γ−(t0) and γ (t0), and intersects with ∂ΣpX0 = ΣpY = S − + + twice in w− and w such that ∠(w−, w )= π. + Since ∠(w , w−) = π, there exists a geodesic line in TpX of the form s R + ∈ 7→ ( sw−) ⋆ (sw ) passing through 0 where ⋆ denotes the concatenation of curves. − Since we assume ΦZ Y is differentiable in p, dΦZ : TpXi R is linear and therefore +| → dΦZ (w−)+ dΦZ (w ) = 0. 0 1 1 Let σ , (σ )− : [0, π] ΣpX0, ΣpX1 be the geodesics in ΣpX0 and ΣpX1 → + respectively connecting w− and w such that their concatenation is the geodesic + + loop S in Σp through w , w−,γ and γ−. 20 VITALI KAPOVITCH, CHRISTIAN KETTERER, AND KARL-THEODOR STURM Since dΦ : T X R is concave, T X is the metric cone over Σ X and i p i → p i p i dΦ (rv)= rdΦ (v), it follows that dΦ σ = ui : [0, π] R satisfies i i i ◦ i → i i (u )′′ + u 0, i =0, 1. ≤ Moreover v(s)= dΦ σ (s)+ dΦ σ (s) satisfies 0 ◦ 0 1 ◦ 1 + v′′ + v 0, v(0) = v(π)= dΦ (w−)+ dΦ (w )=0. ≤ 0 1 R2 Let r, s be the polar coordinates on + = (x, y): y 0 with x = r cos s,y = r sin s. Then the function f(x, y)= rv(s) is concave.{ Since≥ }f( 1, 0) = f(1, 0)=0 and f(0, 1) 0 concavity of f implies that f 0. Therefore− v 0 and hence ≤ + ≤ ≤ dΦ0(γ−)+ dΦ1(γ ) 0. ≤ Theorem 4.3. Let Φi : Xi R, i = 0, 1, be semi-concave such that for any p ∂X it holds that → ∈ i dΦ (v )+ dΦ (v ) 0 normal vectors v Σ X , i =0, 1. 0|p 0 1|p 1 ≤ ∀ i ∈ p i Then Φ : Z R is semiconcave. Z → Proof. It is obvious that we only need to check semi-concavity of ΦZ near Y . By changing Z to a small convex neighborhood of a point p Y we can assume that ∈ Φi are λ-concave on Xi for some real λ. Let γ : [0,L] Z be a unit speed geodesic. We wish to prove that Φ (γ(t)) is → Z λ-concave. Fix an arbitrary 0 <δ (Xi, dXi , mi), i =0, 1, is given by (X0 X1, mZ ) where mZ = (ι0)# m0 +(ι1)# m1 . ∪I The maps ι : X Z, i = 0, 1, are the canonical inclusion maps. Note that i i → X0 X1 is an n-dimensional Alexandrov space by Petrunin’s glued space theorem. By∪ RemarkI 2.12 it follows that n n X I X Xi = X , i =0, 1 H 0∪ 1 | H i n we can write mZ =ΦZ X I X . H 0∪ 1 ON GLUING ALEXANDROV SPACES WITH LOWER RICCI CURVATURE BOUNDS 21 5. Proof of Theorem 1.1 In this section we present the proof of the glued space theorem (Theorem 1.1). Proof of Theorem 1.1 1. Let Xi, i =0, 1, be Alexandrov spaces with curvature bounded from below by k0 and k1, respectively. By Theorem 2.13 it follows that X X =: Z has curvature bounded from 0 ∪φ 1 below by min k0, k1 =: k. { } n By Theorem 2.9 the metric measure space (Z, dZ , Z ) satisfies the condition CD(k(n 1),n). H − Hence, any 1-Lipschitz function u : (Z, dZ ) R induces a disintegration mγ γ Q → { } ∈ that is strongly consistent with Rb , and for q-a.e. γ Q the metric measure space u ∈ (Xγ, mγ ) satisfies the condition CD(k(n 1),n) and hence CD(k(n 1),N) by monotinicity in N . It follows that m = h− 1 and h :[a ,b ] R− satisfies γ γH |Xγ γ γ γ → 2 1 1 d N−1 N−1 hγ + khγ 0 on(a ,b ) for q-a.e.γ Q. dr2 ≤ γ γ ∈ By Lemma 2.19 it follows 1 + 1 d− N−1 d N−1 hγ hγ everywhere on (a ,b ). dr ≥ dr γ γ 2. Fix 0 Recall q˜ = (Q)# m U . Let A Q be a set such that q(A) = 0. Hence 0 = 1 1 | ⊂ m(Q− (A)) m(Q− (A) U). Hence ˜q(A) = 0. Therefore,≥ there exists∩ a measurable function G : Q [0, ) such that q = Gq˜ → ∞ and Q Q−1(U) Gdq˜ = 0. In particular, it follows that q( ) = 0. \ N n A unique and strongly consistent disintegration of mZ b =Φ Z b is given by R |Tu H |Tu Φ mγ dq ZQ where Φm = (γ) [Φ γh 1]. Then, it follows by uniqueness of the disintegration γ # ◦ γH and since q = Gq that G(γ)h˜γ˜ = (Φ γ)hγ on (aγ˜,bγ˜) for ˜q-a.e.γ ˜. ◦ N 4. We repeat the steps 2. and 3. for any Upi , i . 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