Ricci curvature in dimension 2

We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.

Theorem 1.1. Let (X, d) be a metric space and let H 2 be the 2dimensional Hausdorff measure on X. If (X, d, H 2 ) is an RCD(κ, 2) space then X is an Alexandrov space of curvature at least κ.
The converse of our theorem is due to Anton Petrunin, [Pet11]. As a consequence of Theorem 1.1, all RCD(κ, 2) spaces (X, d, H 2 ) are topological surfaces, possibly with boundary.
1.2. Theory of RCD spaces. The synthetic theory of spaces with lower Ricci curvature bounds, initiated in [Stu06] and [LV09], has experienced a tremendous growth over the last 15 years. We refer the reader to the surveys [Vil16] and [Amb18] for an overview of the current state of the theory and the huge bibliography.
The analytic aspects of the theory are highly developed. Most results, previously known on Riemannian manifolds with lower bounds on Ricci curvature, now possess natural generalizations to the synthetic framework, see for instance [AGS14], [Gig15], [CM15]. Moreover, there is a sophisticated understanding of measure theoretic, or rather "almost everywhere" properties of such spaces, [MN17], [BS18].
On the other hand, gaining information on the "everywhere" metric structure, or even less ambitious, on the topological structure of RCD spaces, seems to be very difficult. Only a few results in this direction are known, all of which require either a rigid setting, [Gig13], [DPG16], or strong a priori assumptions on the geometry, [KK17], [KL16].
1.3. Central steps in the proof. The proof is a combination of analytic RCD-techniques, some basic results about Alexandrov spaces, the canonical uniformization of singular surfaces introduced in [LW17b] and Reshetnyak's analytic theory of surfaces with integral curvature bounds, [Res93].
In the first step, we use the Bishop-Gromov comparison and the cone-rigidity theorem, [DPG16], to study the infinitesimal structure of a space X as in Theorem 1.1. More precisely, it is shown that unique tangent spaces exist at all points and are Euclidean cones over either intervals or circles. In analogy with the theory of Alexandrov spaces, we call the set of points, at which the tangent space is homeomorphic to a half-plane, the boundary of X and denote it by ∂X. Another application of the Bishop-Gromov comparison implies that ∂X is a closed subset of X. This is the content of Section 2.
In Section 3, we consider, for any δ > 0, the set of points X δ at which the density of X is δ-close to the density of the Euclidean plane. The set X δ is open, dense, disjoint from ∂X and, by the theorem of Cheeger-Colding-Reifenberg, [CC97], X δ is a topological surface without boundary, once δ is small enough. Due to [CM18], X δ satisfies an isoperimetric inequality of Euclidean type.
The next two sections are the central pieces of the proof. In Section 4, we use [LW17b], to find for any point x ∈ X δ a small closed neighborhoodΩ and a canonical parametrization φ :D →Ω obtained by solving a Plateau problem. This parametrization is a quasisymmetric map, which in addition is infinitesimally conformal. This implies, in particular, that there exists a function f ∈ L 2 (D), the conformal factor, such that the lengths of almost all curves inΩ are obtained by integrating f along the corresponding curve inD. In other words, the metric inΩ appears to be obtained from the flat metric inD by a "singular conformal change with factor f ".
In Section 5, we identify the Laplace operator of Ω in terms of D and f and use the Bakry-Émery property of RCD spaces to derive an analytic condition on f . This condition implies log-subharmonicity in the case κ = 0 and a related property for general κ.
Using [Res93], we deduce in Section 6 that the metric on Ω is everywhere controlled by f , and has curvature bounded below by κ in the sense of Alexandrov.
In order to finish the proof of the main theorem we show that X δ is a convex subset of X, a statement which is expected to be true in much greater generality, [CN12]. Once the convexity is verified, Toponogov's globalization theorem, in the form of [Pet16], provides that last step. The proof of this convexity in Section 7, relies on the non-branching property of RCD spaces, [RS14]. Assuming that a geodesic between two non-boundary points passes through a point x on the boundary ∂X, we easily obtain many branching geodesics, once we know that X is a topological surface with boundary near x. While in general, topological control in RCD spaces is very difficult to achieve, here we obtain the required statement by a twofold application of the Cheeger-Colding-Reifenberg theorem.
1.4. Acknowledgment. The paper arose as a follow-up project to a series of joint papers of the first author and Stefan Wenger. For this and inspiring discussions the authors express their gratitude to Stefan Wenger. We are grateful to Vitali Kapovitch  In a metric space X we denote by d the distance and by B r (x) the open ball of radius r around the point x. By ℓ(γ) = ℓ X (γ) we will denote the length of a curve γ in a metric space X.
2.2. Setting. We assume some familiarity with the synthetic theory of lower Ricci curvature bounds. In particular, we assume that the reader is familiar with the notion of RCD(κ, N) spaces, which we will not define.
In the rest of the paper we fix a space X satisfying the assumptions of Theorem 1.1. Thus, X is a geodesic, locally compact metric space. By assumption, the space X is RCD(κ, 2) with respect to the reference measure H 2 , whose support is X. In the terminology introduced in [DPG17], this means that X is a non-collapsed RCD(κ, 2) space.
For x ∈ X and r > 0 we set Recall that the balls satisfy the Bishop-Gromov property, thus, for all s < r, the quotient b(x, r)/b(x, s) is bounded from above by the corresponding quotient in the 2-dimensional simply connected Riemannian manifold M 2 κ of constant curvature κ, [Stu06]. Therefore, for any x ∈ X we have a well-defined positive density Again by the Bishop-Gromov property, the space X is locally 2-Ahlfors regular. Thus, for every compact subset K of X there exists some L > 0, such that all x ∈ K and all 0 < r < 1 we have 1 L · r 2 ≤ H 2 (B r (x)) ≤ L · r 2 .
2.3. Convergence and blow-ups. Any sequence (Y i , y i ) of non-collapsed RCD(κ, 2)-spaces has a subsequence converging in the pointed Gromov-Hausdorff topology to a limit space (Y, y). This limit space Y is an RCD(κ, 2)-space with respect to a limiting measure µ, see [AGS14], [GMS15]. If there is ǫ > 0 such that H 2 (B 1 (y i )) ≥ ǫ, for all i, then the limit space Y is non-collapsed and µ = H 2 , [DPG17, Theorem 1.2]. Moreover, in this situation the densities behave semi-continuously: In particular, the density function b : X → (0, 1] is lower semi-continuous. As a special case of [DPG17, Theorem 1.2], for any sequence x i ∈ X converging to a point x ∈ X and any sequence r i of positive numbers converging to 0, there is a subsequence, such that the rescaled spaces ( 1 r i X, x i ) converge to a non-collapsed RCD(2, 0) space (Y, y). We call any such space (Y, y) a blow-up of the space X. By the continuity of H 2 with respect to the convergence and by the Bishop-Gromov inequalities in X, we have, for all z ∈ Y and all s ≥ 0, 2.4. Tangent cones. A blow-up for a constant sequence of base points x i = x is called a tangent cone of X at the point x. Relying on the volume rigidity of metric cones, [DPG16], it has been shown in [DPG17, Proposition 2.7], that any such tangent cone T of X at the point x is isometric to a Euclidean cone C(Z) over a compact space Z. Moreover, Z is an RCD(0, 1) space of diameter at most π, [Ket15].
As has been proved in [KL16], such a space Z is a closed interval or a circle. Due to the stability of H 2 and the definition of b(x), the density b(x) coincides with the density of CZ at the vertex 0 x of the cone CZ. Therefore, Thus, at any point x ∈ X at most two different tangent cones can exist.
However, the following lemma, valid for all doubling metric measure spaces and probably well-known to experts shows that there cannot exist exactly two different tangent cones.
Lemma 2.1. The space T x of all metric measure tangent cones (Y, y, H 2 ) at the point x ∈ X is a connected subset of the space X of isomorphism classes of all pointed, proper metric measure spaces, [GMS15].
Proof. Consider the map F : (0, 1] → X which sends a number r > 0 onto the pointed metric measure space ( 1 r X, x, H 2 ). The map is continuous and the image of F has compact closure in X.
The set T x consists of all limit points lim r i →0 F (r i ) of this continuous curve. Thus, T x is an intersection of a nested sequence of compact connected sets, the closures of the tails of F . Hence, T x is connected. Therefore, we have shown: Corollary 2.2. At any x ∈ X there exists a unique tangent cone. This tangent cone is a Euclidean cone CZ, where Z is a circle or an interval. Moreover, H 1 (Z) = b(x) · 2π and the diameter of Z is at most π.
We denote the unique tangent cone at the point x ∈ X by T x = T x X. By above T x is homeomorphic to a plane or to a half-plane.
2.5. The boundary. We define the boundary ∂X of X to be the set of all points x ∈ X for which the tangent space T x X is homeomorphic to a half-plane.
Lemma 2.3. The boundary ∂X is a closed subset of X which contains only points x with b(x) ≤ 1 2 . Proof. By definition, x ∈ ∂X if and only if T x is isometric to CZ, where Z is an interval. Since Z has length b(x) · 2π and diameter at most π, this implies b(x) ≤ 1 2 . Let x ∈ X \ ∂X be arbitrary. Then T x is a cone over a circle, thus for any point z in T x but the vertex 0 x , the density of T x at z is 1. For any sequence x i ∈ X \ {x} converging to x, we choose r i = d(x, x i ). Then (possibly after choosing a subsequence) under the convergence of ( 1 r i X, x) to (T x , 0 x ) the points x i converge to a point z at distance 1 from 0 x . The semi-continuity of densities implies that lim i→∞ b(x i ) = 1. In particular, x i is not in ∂X, for i large enough.
Hence, X \ ∂X must be open.
As a consequences of the splitting theorem we obtain: Lemma 2.4. Let x ∈ X be a point which is an interior point of a geodesic. If x ∈ ∂X then T x is isometric to the Euclidean half-plane. If x ∈ X \ ∂X then T x X is isometric to R 2 .
Proof. The assumption implies that T x X contains a line (the tangent space to the geodesic through x). By the splitting theorem [Gig13], the space T x splits off a line. This implies the claim, since T x is a cone over an interval or a circle.

Almost regular parts
For any δ > 0, we call a point x ∈ X a δ-regular point if b(x) > 1 −δ. By X δ we denote the set of all δ-regular points in X. We have the following discreteness statement: Lemma 3.1. The set X δ is open in X, for any δ > 0. For any δ < 1 2 , the set X δ is disjoint from ∂X and the complement (X \ ∂X) \ X δ is discrete in X \ ∂X.
Proof. The semi-continuity of the density function shows that X δ is open. Due to Lemma 2.3, the set X δ is disjoint from ∂X, for δ < 1 2 . Finally, the last argument in the proof of Lemma 2.3 implies that any point in X \ ∂X has a punctured neighborhood completely contained in X δ . This implies the last claim.
The following observation is a very special and rough case of the results obtained in [CC97] and [CM18].
Lemma 3.2. There exists some δ, C > 0 with the following property. Every point x ∈ X δ has a neighborhood U x homeomorphic to D, such that for any subset K of U x homeomorphic toD, Proof. Due to [DPG17, Theorem 1.2] and the Cheeger-Colding-Reifenberg theorem, [CC97, Theorem A.1], there exists some δ > 0, such that any point x ∈ X δ has a neighborhood U x homeomorphic to D.
It remains to prove, that, for sufficiently small δ, we have the isoperimetric inequality (3.1), for any closed topological disc K in some neighborhood of any point in X δ .
This statement is proved in [CM18, Corollary 1.6] (in a much more general and precise form) with H 1 (∂K) 2 on the right hand side of (3.1) replaced by m 2 (K), where m(K) is the outer Minkowski content Here B r (K) denotes the set of points with distance at most r to K. It suffices to prove 4π · H 1 (∂K) ≥ m(K) for any subset K of X homeomorphic to a closed disc, compare [Fed69, Theorem 3.2.39].
If ∂K is not rectifiable there is nothing to prove. Otherwise, for any natural n, we set r = 1 2n · H 1 (∂K) and find an r-dense subset A n in ∂K with n points. Then B r (∂K) ⊂ B 2r (A n ).
By the Bishop-Gromov property, we deduce, for r small enough, This implies m(K) ≤ 4π · H 1 (∂K) and finishes the proof.

Conformal parametrization
4.1. Choice of a domain. Let δ, C > 0 be as in Lemma 3.2, let x 0 ∈ X δ be arbitrary and let U x 0 be an open neighborhood of x 0 in X provided by Lemma 3.2. Any Jordan curve Γ in U x 0 determines a Jordan domain Ω ⊂ U x 0 homeomorphic to D, such thatΩ = Ω ∪ Γ is homeomorphic toD.
Starting with any Jordan curve Γ in U x 0 whose Jordan domain Ω contains x 0 , we can replace Γ by a nearby curve and assume that Γ is biLipschitz to the round circle S 1 , see [LW17b,Lemma 4.3].
We fix this curve Γ and domain Ω for the rest of the section.

4.2.
Metric properties of Ω. Consider the setΩ with its intrinsic metric d Ω . Since Γ is a biLipschitz embedding of a circle, the metric d Ω is biLipschitz to the induced metric d X onΩ. Moreover, d Ω and d X locally coincide on Ω.
In order to apply the parametrization results of [LW17b], we need to make sure thatΩ is Ahlfors 2-regular and is linearly locally connected in the following sense.
A continuum is a compact connected space. A metric space Y is called linearly locally connected if there exists a positive constant 0 < C < 1, such that for all y ∈ Y and all 0 < r < diam(Y ) the following holds true. Any pair of points z 1 , z 2 ∈ B C·r (y) is contained in a continuum P ⊂ B r (y); any pair of points If the space Y is geodesic, the first condition is always satisfied. Linear local connectedness is preserved under biLipschitz transformations. Proof. Since the H 2 -measures with respect to d X and d Ω coincide on Ω, a quadratic upper bound on the H 2 -area of balls in Ω follows from the corresponding upper bound on the area of balls in X.
The existence of a lower quadratic bound on the area of balls in Ω is essentially proved in [LW18a, Theorem 9.4], as a consequence of the quadratic isoperimetric inequality (3.1). We provide a simplified version of the argument here.
Relying on the lower bound on the area of balls in X, it is sufficient to find a constant C 0 < 1 with the following property: For all small r and any z ∈ Γ the ball B 2r (z) contains a point y ∈ Ω with distance at least C 0 · r to Γ.
By topological reasons, the distance sphere of radius r around z must contain a continuum P , joining two points on Γ, locally separated on Γ by z. From the biLipschitz property of the boundary curve Γ, we now deduce the existence of C 0 < 1, such that the continuum P contains a point y as claimed above.
This finishes the proof of the Ahlfors 2-regularity ofΩ.
In order to prove thatΩ is linearly locally connected, recall first from [LW18a, Theorem 8.6], that the isoperimetric property (3.1) implies the following statement. There exists a constant C 1 > 1 such that, for any y ∈Ω and any r > 0, the ball B r (y) is contractible inside B C 1 ·r (y).
The spaceΩ is geodesic and the boundary Γ is linearly locally connected. Thus it suffices to prove the following claim. There exists some C 2 < 1, such that, for every y ∈Ω, every r < diam(Ω) and every z ∈Ω \ B r (y) there is a curve connecting z with Γ outside of B C 2 ·r (y).
Assume the contrary. By the Jordan curve theorem, there exists a Jordan curve T in B C 2 ·r (y), whose Jordan domain contains the point z. But this implies that B C 2 ·r (y) is not contractible in B r (y), which provides a contradiction to the result [LW18a, Theorem 8.6], mentioned above, once C 2 is sufficiently small.
This finishes the proof of the Lemma.
where the infimum is taken over all Borel functions admissible for C. A statement holds for almost every curve in Y , if the family C of all curves in Y for which the statement does not hold has modulus 0.
and the following statement holds true. There exists a function ρ ∈ L 2 (Y ), called a weak upper gradient of u, such that for almost any curve γ in Y the composition u • γ is absolutely continuous and There is a unique weak minimal upper gradient ρ u ∈ L 2 (Y ) of u such that ρ u ≤ ρ almost everywhere, for any weak upper gradient ρ of u.
4.4. Canonical parametrization. We will not recall the definition of quasisymmetric maps, see [HK98]. Instead we will use the theory of quasisymmetric maps as a black-box, providing references for each required statement. From [LW17b, Theorem 1.1] and Lemma 4.1 above we deduce: Corollary 4.2. Among all homeomorphisms u :D →Ω there exist a homeomorphism φ ∈ N 1,2 (D,Ω) with minimal Cheeger energy ChΩ(φ). This homeomorphism φ is quasisymmetric.
The map φ and its inverse have Lusin's property N, thus φ and φ −1 preserve the class of sets of H 2 -measure 0, [HKST01,Theorem 8.12].
for almost all curves γ in D. Here, as before, ρ φ denotes a weak minimal upper gradient of φ.
For any Borel subset E ⊂D we have, [LW18a, Lemma 3.3] Since the inverse φ −1 has Lusin's property N, the weak minimal upper gradient ρ φ must be positive almost everywhere. From (4.2), [LW18a, Lemma 3.1] and the absolute continuity on almost all curves of the Sobolev maps φ, φ −1 , [HKST15, Proposition 6.3.2], we deduce that for any non-negative Borel function h :D → R and almost every curve γ inD Therefore, the Borel function for almost every curve η inΩ. This implies that g is a weak minimal upper gradient of φ −1 ∈ N 1,2 (Ω,D).

Conformal factor
5.1. Setting and aim. We continue to use the notations from the previous section. Thus, we have a domain Ω ⊂ X and a conformal homeomorphism φ :D →Ω, which is contained in N 1,2 (D,Ω). We let f := ρ φ :D → [0, ∞] be a weak minimal upper gradient of φ.
The aim of this section is the following result and its consequence, Corollary 5.2.
is a Radon measure on D and satisfies as a measure The proof will be a direct consequence of the Bakry-Émery inequality on the RCD(κ, 2) space X, once we have identified via the conformal homeomorphism φ the Sobolev spaces and Laplacians on D with the corresponding objects on X.

5.2.
Identifications. Whenever no confusion is possible we will use the homeomorphism φ −1 to identifyΩ withD.
From This shows that a map v : D → R is contained in N 1,2 (D) if and only if u := v • φ −1 is contained in N 1,2 (Ω). Moreover, in this case the weak minimal upper gradients satisfy Combining with (5.2) and identifying D and Ω this shows (5.4) |∇v| 2 · H 2 D = ρ 2 u · H 2 Ω and Ch Ω (u) = Ch D (v).

5.
5. An analytic conclusion. By a combination of a smoothing argument and a pointwise computation we are going to conclude: Corollary 5.2. The function log(f 2 ) is contained in L 1 loc (D). The distributional Laplacian ∆(log(f 2 )) is a Radon measure on D and satisfies ∆(h · |∇v| 2 ) ≥ 2κ · |∇v| 2 · H 2 D . We fix a ball O = B 1−δ (0) ⊂ D for the rest of the proof. For all δ > 2 · ǫ > 0 consider the mollifications h ǫ : O → R obtained by a convolution of h with the usual smooth mollifiers ρ ǫ : R 2 → [0, ∞) whose support is in B ǫ (0) ⊂ R 2 . Since the function h is positive almost everywhere, the smooth functions h ǫ are positive on O. By a direct computation (or using the observation that (5.9) is a system of linear inequalities on the function h, which is moreover, equivariant with respect to translations), we see that h ǫ satisfies (5.9) on O for all harmonic functions v on R 2 .
The function h is bounded from above on the ǫ-neighborhood of O, hence so is log(h). On the other hand, − log(h) = log( 1 h ) ≤ 1 h . Thus, the integrability of f 2 = 1 h shows log(h) ∈ L 1 (O). The convexity of the function t → 1 t and Jensen's inequality imply where on the right side we have the mollifications of the function 1 h . Since ( 1 h ) ǫ converge in L 1 (O) to 1 h as ǫ goes to 0, we deduce that 1 hǫ converges to 1 h is L 1 (O) as well.
Similarly, t → − log(t) = log( 1 t ) is convex and arguing as above with Jensen's inequality we see that log(h ǫ ) converges in L 1 (O) to log(h) as ǫ goes to 0.
For all n > 2 δ consider the smooth function f n on O such that f −2 n = h 1 n . Like above, log f n = −2 log h 1 n converge to log f in L 1 (O). The remaining statements follow by a continuity argument, once we have verified ∆(log(f 2 n )) ≤ −2κ · f 2 n on O. As we have seen above, the smooth positive functions h 1 n satisfy (5.9) on O, for all harmonic functions v : R 2 → R. Therefore, we only need to verify that for a smooth positive function h on a domain U in R 2 the validity of (5.9) for all harmonic functions v : R 2 → R implies It is sufficient to verify this pointwise statement at a single point z ∈ U which we may assume to be z = 0. Set e := ∇h(0). If e = 0 we choose v to be a linear non-zero function on R 2 . Then (5.9) implies ∆(h)(0) ≥ 2 · κ, hence, (5.10).
6. Curvature bound in the regular part 6.1. Preliminaries from Alexandrov geometry. We refer to [BBI01] for basics about Alexandrov geometry and just agree on notation here, following [Pet16]. Let κ be a fixed real number as before. For points p, x 1 , x 2 in a metric space Y , we denote by∠ κ (p x 1 x 2 ) the κ-comparison angle, whenever it exists. Thus,∠ κ (p x 1 x 2 ) is the angle in the constant curvature surface M 2 κ at the vertexp of a trianglepx 1x2 with the same side-lengths as px 1 x 2 .
A subset O of a metric space Y satisfies the (1+3)-points comparison if for any quadruple of points p; x 1 , x 2 , x 3 ∈ O the inequalitỹ x 1 ) ≤ 2π holds true or one of the κ-comparison angles is not defined.
A metric space Y has curvature ≥ κ if every point y ∈ Y has a neighborhood O which satisfies the (1 + 3)-points comparison and such that every pair of points in O is connected in Y by a geodesic.
A complete geodesic metric space of curvature ≥ κ is called an Alexandrov space of curvature ≥ κ.
If Y has curvature ≥ κ, the ballB 2R (y) ⊂ Y is compact and any pair of points in B R (y) is connected in Y by a geodesic, then B R (y) satisfies the (1 + 3)-point comparison, [Pet16,p. 3].
Note finally that the (1 + 3)-point comparison property (of subsets) is stable under Gromov-Hausdorff convergence.
The aim of this section is the following Proposition 6.1. The subspace X δ of δ-regular points in X has curvature bounded from below by κ.

6.2.
Reshetnyak's theory. We continue to use the notation from Section 4 and Section 5. For a homeomorphism φ : For any RCD(κ, 2) space X, any open subset W of X and any u ∈ N 1,2 (W ) for which the constant function ρ = 1 is a weak upper gradient, the function u has a locally 1-Lipschitz representative.
In fact, the Sobolev-to-Lipschitz property is defined and verified in [Gig13,p. 48] only in the global case W = X, but the proof presented there covers the local version formulated above. Therefore, for any point y ∈ O f , the composition ψ y ∈ N 1,2 (W ) of ψ with the distance function in O f to the point y has the constant function 1 a weak upper gradient. By the Sobolev-to-Lipschitz property, this implies that ψ y is locally 1-Lipschitz. Since X is a geodesic space, and y was arbitrary, this implies that ψ is locally 1-Lipschitz.
In order to prove that φ is locally 1-Lipschitz, we apply the same argument. (Alternatively, this can be seen directly, as in [LW18b,Lemma 9.3]). Firstly, H 2 O f = f 2 · H 2 O and the same computation as in Section 4 shows that any family of curves of modulus 0 in O has modulus 0 in O f . Thus, the map φ preserves the lengths of almost all curves in O f . Therefore, φ ∈ N 1,2 (O f , X) and the constant function 1 is a weak upper gradient of φ. Arguing as above we deduce that φ is locally 1-Lipschitz, once the Sobolev-to-Lipschitz property has been verified locally in O f . But any point in O f has a compact neighborhood isometric to an Alexandrov space, by [Pet07,Theorem 7.1.3]. This implies the Sobolevto-Lipschitz property, as a consequence of [Pet11] and [Gig13,p. 40].
For any δ-regular point x 0 ∈ X, we choose a domain Ω containing x 0 as in Section 4. Lemma 6.3 and Lemma 6.2 imply the existence of a neighborhood of x 0 in Ω which has curvature bounded below by κ. This finishes the proof of Proposition 6.1.
7. Extension to the singular points 7.1. Topological statement. Most of this rather long section is devoted to the proof of the following topological statement.
Proposition 7.1. Any point z ∈ ∂X with b(z) = 1 2 has an open neighborhood U in X homeomorphic to the closed half-plane H, such that z lies in the boundary line ∂H.
Before we embark on the proof of this proposition, we explain why this statement is sufficient to finish the proof of our main theorem.
Lemma 7.2. The validity of Proposition 7.1 implies that the set X δ is strongly convex in X. Thus any geodesic γ with endpoints x, y ∈ X δ is completely contained in X δ .
Proof. Assume the contrary and consider a point z ∈ X \ X δ on γ closest to the point x.
Due to Lemma 2.4, the point z is contained in ∂X and satisfies b(z) = 1 2 . Applying Proposition 7.1 we find a neighborhood U of z homeomorphic to H, such that z lies on the boundary line ∂H.
The part γ + of the geodesic γ between x and z is contained in X δ (up to the point z) hence it intersects ∂H only in the point z. By choosing the neighborhood U smaller we can therefore assume that γ + separates U into two components.
Fix a point q on the part of γ between z and y sufficiently close to z and let U + be the component of U \γ + which does not contain q. Thus, for any point m ∈ U + , sufficiently close to z, any geodesic between q and m intersects γ + .
In particular, the point z lies on a geodesic between m and q. This statement, valid for the fixed point q and an open set of points m, contradicts the essentially non-branching property of the RCD space X, [GRS16, Theorem 1.2, Corollary 1.4].
Therefore, the validity of Proposition 7.1 implies that the space X δ is a geodesic space. Since X is the completion of X δ , we would deduce from [Pet16] that X is an Alexandrov space with curvature ≥ κ and finish the proof of Theorem 1.1. 7.2. Interior points. Consider a point x ∈ X \ ∂X. Assume that x is not contained in X δ . By Lemma 3.1, we find some r > 0 such that B 2r (x) \ X δ contains only the point x. By Proposition 6.1, B 2r (x) \ {x} has curvature ≥ κ in the sense of Alexandrov.
Due to Lemma 2.4, for any y, z ∈ B r (x) \ {x}, any geodesic connecting y and z in X is contained in B 2r (x)\{x}. Toponogov's globalization theorem in the version of [Pet16], now shows that B r (x) \ {x} satisfies the (1 + 3)-point comparison. By continuity, B r (x) satisfies the (1 + 3)-point comparison as well.
We have just verified: Corollary 7.3. The subspace X \ ∂X of X has curvature ≥ κ.
7.3. Setting. We now fix a point z ∈ ∂X with b(z) = 1 2 . We consider a sequence of points x i ∈ X converging to z and a sequence r i of positive numbers converging to 0. After choosing a subsequence we may and will assume that the blow-up (Y, y) = lim( 1 r i X, x i ) exists. Furthermore, we consider the sequence of non-negative numbers s i = d(x i , ∂X). By choosing a further subsequence, we may and will assume that the following limit exists: As we have seen in Section 2, any ball of any radius t in the blow-up Y has H 2 -measure at least π 2 t 2 . From the volume-cone rigidity, [DPG16], and Lemma 2.3 we deduce: Lemma 7.4. The RCD(0, 2) space Y can have non-empty boundary only if Y is isometric to the flat half-plane H.
Our next aim is to show that (non) boundary points of X converge to (non) boundary points in the blow-up Y . After that we will show that the blow-up Y is isometric to a plane or a half-plane. 7.4. Stability of the boundary. In the previous notation we are going to show: Lemma 7.5. If A > 0 then the point y is not contained in ∂Y .
Proof. Assume the contrary. By Lemma 7.4, Y is isometric to the half-plane H and y ∈ ∂H.
By rescaling, we may assume that A > 9. Thus, for all i large enough, the compact ballB i in X i = 1 r i X of radius 6 around x i does not contain boundary points. Due to Corollary 7.3, the open ball B i of radius 3 around x i in X i satisfies the (1 + 3)-point comparison, see Section 6 and [Pet16].
A contradiction to the fact that the surfaces without boundary B i converge to a surface with boundary (the ball in H around the limit point y) can now be deduced in several ways. We choose a way relying on (the simplest case of) Perelman's topological stability theorem, [Per91], [Kap07].
From [Pet07, Theorem 7.1.3] we deduce the existence of some δ > 0 and closed compact convex subsets C i ⊂ B i containing the ball B δ (x i ) ⊂ B i . The spaces C i are compact 2-dimensional Alexandrov spaces converging (after choosing a subsequence) to an Alexandrov space C ⊂ H which contains the ball B δ (y) ⊂ H. By Perelman's stability theorem [Per91], [Kap07], for all large i, there exists a homeomorphism Φ i : C i → C close to the identity. Since B δ (x i ) is a 2-manifold without boundary, we deduce that y must have in Y a neighborhood homeomorphic to a 2-manifold without boundary which is impossible.
This contradiction finishes the proof.
Lemma 7.6. If A = 0 then Y is isometric to the half-plane H and y is on the boundary ∂H.
The assumption A = 0 implies that the points z i ∈ X i = 1 r i X converge to the same point y ∈ Y .
Since z i ∈ ∂X, we see that the density b(y) at Y is at most 1 2 . The volume-cone rigidity argument implies that b(y) = 1 2 and Y is isometric to the Euclidean cone T y Y . It remains to show that T y Y cannot be the Euclidean cone over the circle of length π.
We assume that Y is a Euclidean cone over a circle and are going to derive a contradiction.
For all i large enough, there exist no points p i in ∂X with r i < d(p i , z i ) < 2r i , since Y \ {y} is locally Euclidean. Denote by K i the set ∂X ∩B r i (z i ) and byK i the complement ∂X \ K i . Then, for all i large enough, K i is compact andK i is closed in ∂X. Moreover, d(z, z i ) ≥ t i := d(K i ,K i ) ≥ r i .
Consider points k i ∈ K i andk i ∈K i realizing the distance between K i andK i and take the blow-up (choosing a subsequence) (Ŷ , k) = lim( 1 t i X, k i ). As before, the volume-cone rigidity implies thatŶ is either a halfplane or a cone over a circle. However, the pointsk i converge (after taking a subsequence) to a non-Euclidean pointk ∈Ŷ with distance 1 to k. Therefore,Ŷ must be isometric to H and the geodesic between k andk must lie on the boundary ∂H.
Hence, the midpoints m i of any geodesic between k i andk i in X converge to a point on the boundary ofŶ .
But, by construction, the point m i in X has distance t i 2 from ∂X. We obtain a contradiction with Lemma 7.5 and finish the proof.
Lemma 7.7. If A = ∞ then Y is isometric to R 2 .
Proof. Due to Lemma 7.6 and Lemma 7.5, the sequence ( 1 s i X, x i ) converges to a half-plane (H,ŷ) whereŷ has distance 1 from ∂H. Therefore, H 2 (B 1 (ŷ)) = π. Stability of the Hausdorff measures implies, Applying the Bishop-Gromov inequality, and the assumption A = ∞, we see that for any fixed t > 0, Thus, the ball B t (y) in Y has H 2 -measure π · t 2 . Since t is arbitrary, the volume-cone rigidity implies that Y is isometric to T y Y = R 2 .
Combining the last three statements we easily arrive at Corollary 7.8. The space Y is either R 2 or the half-plane H. The case Y = H happens if and only if A < ∞. Moreover, in this case ∂H coincides with the limit of the boundary ∂X.
Proof. The only statement not directly contained in Lemma 7.5, Lemma 7.6 and Lemma 7.7 is that for 0 < A < ∞ the space Y is isometric to H. However, if A < ∞ we can replace the base points x i by closest points z i ∈ ∂X and apply Lemma 7.6 to deduce the statement.
7.5. Reifenberg's lemma twice. We can now finish the Proof of Proposition 7.1. Due to Corollary 7.8, for any sequence z i ∈ ∂X converging to z and any sequence r i → 0 the sequence ( 1 r i ∂X, z i ) converges to the line ∂H = R. Applying the Cheeger-Colding-Reifenberg Lemma, [CC97, Theorem A1], we deduce that a neighborhood of z in ∂X is homeomorphic to an interval.
Consider now the doubling W = X ∪ ∂X X of X with its natural length metric, compare [Per91, Section 5].
We claim that for any sequence x i ∈ W converging to z and any sequence of positive numbers r i converging to 0 the blow-up lim( 1 r i W, x i ) is isometric to R 2 .
Choosing a subsequence and using the symmetry we may assume that x i ∈ X ⊂ W and that for s i := d(x i , ∂X) the quotients s i /r i converge to a number A ∈ [0, ∞]. Applying Corollary 7.8, we see that in the case A = ∞, the blow-up coincides with lim( 1 r i X, x i ) = R 2 . On the other hand, if A < ∞, we can change the base points x i and assume x i ∈ ∂X, without changing the isometry class of the blow-up. Then the limit lim( 1 r i W, x i ) is the doubling of H = lim( 1 r i W, x i ) along the boundary ∂H = lim( 1 r i ∂X, z i ). Having proved the claim, we can now apply the Cheeger-Colding-Reifenberg Lemma a second time and deduce that a neighborhood V of z in W is homeomorphic to an open disc. Passing to a smaller subdisc if necessary, we obtain a homeomorphism between V and the plane which takes V ∩ ∂X to a line.
By connectedness reasons we see that X ∩ V must be homeomorphic to a half-plane.