Weakly noncollapsed RCD spaces with upper curvature bounds

We show that if a $CD(K,n)$ space $(X,d,f\mathcal{H}^n)$ with $n\geq 2$ has curvature bounded from above by $\kappa$ in the sense of Alexandrov then $f=const$.


Introduction
In [DPG18] Gigli and De Philippis introduced the following notion of a noncollapsed RCD(K, n) space. An RCD(K, n) space (X, d, m) is noncollapsed if n is a natural number and m = H n . A similar notion was considered by Kitabeppu in [Kit17].
Noncollapsed RCD(K, n) give a natural intrinsic generalization of noncollapsing limits of manifolds with lower Ricci curvature bounds which are noncollapsed in the above sense by work of Cheeger-Colding [CC97].
In [DPG18] Gigli and De Philippis also considered the following a-priori weaker notion. An RCD(K, n) space (X, d, m) is weakly noncollapsed if n is a natural number and m ≪ H n . Gigli and De Philippis gave several equivalent characterizations of weakly noncollapsed RCD(K, n) spaces and studied their properties. By work of Gigli-Pasqualetto [GP16], Mondino-Kell [KM18] and Brué-Semola [BS18] it follows that an RCD(K, n) space is weakly noncollapsed iff R n = ∅ where R n is the rectifiable set of n-regular points in X.
It is well-known that if (X, d, m) = (M n , g, e −f d vol g ) where (M n , g) is a smooth n-dimensional Riemannian manifold and f is a smooth function on M then (X, d, m) is RCD(K, n) iff f = const. More precisely, the classical Bakry-Emery condition BE(K, N ), K ∈ R and N ≥ n, for a (compact) smooth metric measure space (M n , g, e −f d vol g ), f ∈ C ∞ (M ), is 1 2 L|∇u| 2 g ≥ ∇Lu, ∇u g + 1 N (Lu) 2 + K|∇u| 2 g , ∀u ∈ C ∞ (M ), where L = ∆ − ∇f . In [Bak94, Proposition 6.2] Bakry shows that BE(K, N ) holds if and only if ∇f ⊗ ∇f ≤ (N − n) ric g +∇ 2 f − Kg .
In particular, if N = n, then f is locally constant.
On the other hand, in [EKS15,AGS15] it was proven that a metric measure space (X, d, m) satisfies RCD(K, N ) if and only if the corresponding Cheeger energy satifies a weak version of BE(K, N ) that is equivalent to the classical version for (M, g, e −f vol g ) from above.
In [DPG18] Gigli and De Philippis conjectured that a weakly noncollapsed RCD(K, n) space is already noncollapsed up to rescaling of the measure by a constant. Our main result is that this conjecture holds if a weakly noncollapsed space has curvature bounded above in the sense of Alexandrov.
Theorem 1.1. Let n ≥ 2 and let (X, d, f H n ) (where f is L 1 loc with respect to H n and supp(f H n ) = X) be a complete metric measure space which is CBA(κ) (has curvature bounded above by κ in the sense of Alexandrov) and satisfies CD(K, n). Then f = const 1 .
Since smooth Riemannian manifolds locally have curvature bounded above this immediately implies Corollary 1.2. Let (M n , g) be a smooth Riemannian manifold and suppose (M n , g, f H n ) is CD(K, n) where K is finite and f ≥ 0 is L 1 loc with respect to H n and supp(f H n ) = M . Then f = const.
As was mentioned above, this corollary was well-known in case of smooth f but was not known in case of general locally integrable f .
In [KK18] it was shown that if a (X, d, m) is CD(K, n) and has curvature bounded above then X is RCD(K, n) and if in addition m = H n then X is Alexandrov with two sided curvature bounds. Combined with Theorem 1.1 this implies that the same remains true if the assumption on the measure is weakened to m ≪ H n . Corollary 1.3. Let n ≥ 2 and let (X, d, f H n ) where f is L 1 loc with respect to H n and supp(f H n ) = X be a complete metric measure space which is CBA(κ) (has curvature bounded above by κ in the sense of Alexandrov) and satisfies CD(K, n). Then X is RCD(K, n), f = const, κ(n − 1) ≥ K, and (X, d) is an Alexandrov space of curvature bounded below by K − κ(n − 2).
Remark 1.4. Note that since a space (X, d, f H n ) satisfying the assumptions of Theorem 1.1 is automatically RCD(K, n), as was remarked in [DPG18] it follows from the results of [KM18] that n must be an integer.
Bakry's proof for smooth manifolds does not easily generalize to a non-smooth context. But let us describe a strategy that does generalize to RCD + CAT spaces.
Assume that (X, d) is induced by a smooth manifold (M n , g) and the density function f is smooth and positive such that (X, d, f m) satisfies RCD(K, n). Then, by integration by parts on (M, g) the induced Laplace operator L is given by where ∆u is the classical Laplace-Beltrami operator of (M, g) for smooth functions. By a recent result of Han one has for any RCD(K, n) space that the operator L is equal to the trace of Gigli's Hessian [Gig18] on the set of n-regular points R n . Hence, after one identifies the trace of Gigli's Hessian with the Laplace-Beltrami operator ∆ of M (what is true on (M n , g)), one obtains immediately that ∇ log f = 0. If M is connected, this yields the claim.
The advantage of this approach is that it does not involve the Ricci curvature tensor and in non-smooth context one might follow the same strategy. However, we have to overcome several difficulties that arise from the non-smoothness of the density function f and of the space (X, d, m).
In particular, we apply the recently developed DC-calculus by Lytchak-Nagano for spaces with upper curvature bounds to show that on the regular part of X the Laplace operator with respect to H n is equal to the trace of the Hessian. We also show that the combination of CD and CAT condition implies that f is locally semiconcave [KK18] and hence locally Lipschitz on the regular part of X. This allows us to generalize the above argument for smooth Riemannian manifolds to the general case.
In section 2 we provide necessary preliminaries. We present the setting of RCD spaces and the calculus for them. We state important results by Mondino-Cavalletti (Theorem 2.4), Han (Theorem 2.11) and Gigli (Theorem 2.7, Proposition 2.9). We also give a brief introduction to the calculus of BV and DC function for spaces with upper curvature bounds.
In section 3 we develop a structure theory for general RCD + CAT spaces where we adapt the DC-calculus of Lytchak-Nagano [LN18]. This might be of independent interest.
Finally, in section 4 we prove our main theorem following the above idea.
is a complete and separable metric space and m is a locally finite measure. P 2 (X) denotes the set of Borel probability measures µ on (X, d) such that X d(x 0 , x) 2 dµ(x) < ∞ for some x 0 ∈ X equipped with L 2 -Wasserstein distance W 2 . The sub-space of m-absolutely continuous probability measures in P 2 (X) is denoted by P 2 (X, m).
The N -Renyi entropy is S N is lower semi-continuous, and S N (µ) ≥ −m(supp µ) 1 N by Jensen's inequality. For κ ∈ R we define Let π κ be the diameter of a simply connected space form S 2 κ of constant curvature κ, i.e.
Remark 2.2. If (X, d, m) is complete and satisfies the condition CD(K, N ) for N < ∞, then (supp m, d) is a geodesic space and (supp m, d, m) is CD(K, N ). In the following we always assume that supp m = X.
Remark 2.3. For the variants CD * (K, N ) and CD e (K, N ) of the curvature-dimension condition we refer to [BS10,EKS15].
If f ∈ L 2 (m), a function g ∈ L 2 (m) is called relaxed gradient if there exists sequence of Lipschitz functions f n which L 2 -converges to f , and there exists h such that Lipf n weakly converges to h in L 2 (m) and h ≤ g m-a.e. . g ∈ L 2 (m) is called the minimal relaxed gradient of f and denoted by |∇f | if it is a relaxed gradient and minimal w.r.t. the L 2 -norm amongst all relaxed gradients. The space of L 2 -Sobolev functions is then W 1,2 (X) equipped with the norm f 2 W 1,2 (X) = f 2 L 2 + |∇f | 2 L 2 is a Banach space. If W 1,2 (X) is a Hilbert space, we say (X, d, m) is infinitesimally Hilbertian. In this case we can define (f, g) ∈ W 1,2 (X) 2 → ∇f, ∇g := 1 4 Assuming X is locally compact, if U is an open subset of X, we say f ∈ W 1,2 (X) is in the domain D(∆, U ) of the measure valued Laplace ∆ on U if there exists a signed Radon functional ∆f on the set of Lipschitz function g with bounded support in U such that Theorem 2.4 (Cavalletti-Mondino, [CM18]). Let (X, d, m) be an essentially non-branching CD(K, N ) space for some K ∈ R and N > 1. For p ∈ X consider d p = d(p, ·) and the asso- Then d p ∈ D(∆, X\ {p}) and ∆d p | X\{p} has the following representation formula: Moreover Remark 2.5. The sets X α in the previous disintegration are geodesic segments [a(X α ), p] with initial point a(X α ) and endpoint p. In particular, the set of points q ∈ X such that there exists a geodesic connecting p and q that is extendible beyond q, is a set of full measure.
In [Gig18] Gigli introduced a notion of Hess f in the context of RCD spaces. Hess f is tensorial and defined for f ∈ W 2,2 (X) that is the second order Sobolev space. An important property of W 2,2 (X) that we will need in the following is The next proposition [Gig18, Proposition 3.3.22 i)] allows to compute the Hess f explicitely.
Then, there exist n ∈ N and such that set of n-regular points R n has full measure. Corollary 2.12. Let (X, d, m) be a metric measure space as before. If f ∈ D L 2 (m) (∆), we have that ∆f = tr Hess f m-a.e. in the sense of the previous theorem.

2.3.
Spaces with upper curvature bounds. We will assume familiarity with the notion of CAT (κ) spaces. We refer to [BBI01,BH99] or [KK18] for the basics of the theory.
Definition 2.13. Given a point p in a CAT (κ) space X we say that two unit speed geodesics starting at p define the same direction if the angle between them is zero. This is an equivalence relation by the triangle inequality for angles and the angle induces a metric on the set S g p (X) of equivalence classes. The metric completion Σ g p X of S g p X is called the space of geodesic directions at p. The Euclidean cone C(Σ g p X) is called the geodesic tangent cone at p and will be denoted by T g p X.
The following theorem is due to Nikolaev [BH99, Theorem 3.19]: Theorem 2.14. T g p X is CAT (0) and Σ g p X is CAT (1).
Note that this theorem in particular implies that T g p X is a geodesic metric space which is not obvious from the definition. More precisely, it means that each path component of Σ g p X is CAT (1) (and hence geodesic) and the distance between points in different components is π. Note however, that Σ g p X itself need not be path connected. Remark 2.15. In [Per95] and [AB18] BV functions are called BV 0 if they are continuous away from an H n−1 -negligible set. However, for the purposes of the present paper it will be more convenient to work with the more restrictive definition above.
as signed Radon measures [Per95, Section 4, Lemma]. By taking the L n -absolutely continuous part of this equality it follows that (4) also holds a.e. in the sense of approximate derivatives. In fact, it holds at all points of approximate differentiability of f and g. This easily follows by a minor variation of the standard proof that d(f g) = f dg + gdf for differentiable functions.
Let (X, d) be a geodesic metric space. A function f : X → R is called a DC-function if it can be locally represented as the difference of two Lipschitz semi-convex functions. A map F : Z → Y between metric spaces Z and Y that is locally Lipschitz is called a DC-map if for each DC-function f that is defined on an open set U ⊂ Y the composition f • F is DC on F −1 (U ). In particular, a map F : Z → R l is DC if and only if its coordinates are DC. If F is a bi-Lipschitz homeomorphism and its inverse is DC, we say F is a DC-isomorphism.
2.5. DC-coordinates in CAT -spaces. The following was developed in [LN18] based on previous work by Perelman [Per95].
Assume (X, d) is a CAT -space, let p ∈ X such that there exists an open neighborhoodÛ of p that is homeomorphic to R n . It is well known (see e.g. [KK18, Lemma 3.1] ) that this implies that geodesics inÛ are locally extendible.
Suppose T g p X ∼ = R n . Then, there exist DC coordinates near p with respect to which the distance onÛ is induced by a BV Riemannian metric g.
More precisely, let a 1 , . . . , a n , b 1 , . . . , b n be points near p such that d(p, a i ) = d(p, b i ) = r, p is the midpoint of [a i , b i ] and ∠a i pa j = π/2 for all i = j and all comparison angles ∠a i pa j , ∠a i pb j , ∠b i pb j are sufficiently close to π/2 for all i = j. Let x :Û → R n be given by x = (x 1 , . . . , x n ) = (d (·, a 1 ), . . . , d(·, a n )). Then by [LN18,Corollary 11.12] for any sufficiently small 0 < ε < π k /4 the restriction x| B 2ε( p) is Bilipschitz onto an open subset of R n . Let U = B ε (p) and V = x(U ). By [LN18,Proposition 14.4 Further, the distance on U is induced by a BV Riemannian metric g which in x coordinates is given by a 2-tensor g ij (p) = cos α ij where α ij is the angle at p between geodesics connecting p and a i and a j respectively. By the first variation formula g ij is the derivative of d(a i , γ(t)) at 0 where γ is the geodesic with γ(0) = p and γ(1) = a j . Since d(a i , ·), i = 1, . . . n, are Lipschitz, g ij is in L ∞ . We denote v, w g (p) = g ij (p)v i w j the inner product of v, w ∈ R n at p. g ij induces a distance function d g on V such that x is a metric space isomorphism for ǫ > 0 sufficiently small.
If u is a Lipschitz function on U , u•x −1 is a Lipschitz function on V , and therefore differentiable L n -a.e. in V by Rademacher's theorem. Hence, we can define the gradient of u at points of differentiability of u in the usual way as the metric dual of its differential. Then the usual Riemannian formulas hold and ∇u = g ij ∂u ∂xi ∂ ∂xj and |∇u| 2 g = g ij ∂u ∂xi ∂u ∂xj a.e. .

Structure theory of RCD+CAT spaces
In this section we study metric measure spaces (X, d, m) satisfying (6) (X, d, m) is CAT (κ) and satisfies the conditions RCD(K, N ) for 1 ≤ N < ∞, K, κ < ∞.
Remark 3.2. It was shown in [KK18] that the above theorem also holds if the CD(K, N ) assumption in (6) is replaced by CD * (K, N ) or CD e (K, N ) conditions (see [KK18] for the definitions). Moreover, in a recent paper [MGPS18] Di Marino, Gigli, Pasqualetto and Soultanis show that a CAT (κ) space with any Radon measure is infinitesimally Hilbertian. For these reasons (6) is equivalent to assuming that X is CAT (κ) and satisfies one of the assumptions CD(K, N ), CD * (K, N ) or CD e (K, N ) with 1 ≤ N < ∞, K, κ < ∞.
In [KK18] we also established the following property of spaces satisfying (6): . Let X satisfy (6). Then X is non-branching.

Next we prove
Proposition 3.4. Let X satisfy (6). Then for almost all p ∈ X it holds that T g p X ∼ = R k for some k ≤ N .
Remark 3.5. Note that from the fact that X is an RCD space it follows that T p X is an Euclidean space for almost all p ∈ X [GMR15]. However, at this point in the proof we don't know if T p X ∼ = T g p X at all such points (we expect this to be true for all p). Proof. First, recall that by the CAT condition, geodesics of length less than π κ in X are unique. Moreover, since X is nonbranching and CD, for any p ∈ X the set E p of points q, such that the geodesic which connects p and q is not extendible, has measure zero (Remark 2.5). Let be a countable dense set of points in X, and let C = i∈N E pi . For any q ∈ X\C and any i with d(p i , q) < π κ the geodesic [p i q] can be extended slightly past q. Since A is dense this implies that for any q ∈ X\C there is a dense subset in T g q X consisting of directions v which have "opposites" (i.e. making angle π with v).
For every p ∈ X and every tangent cone T p X the geodesic tangent cone T g p X is naturally a closed convex subset of T p X. Since X is RCD this means that for almost all p the geodesic tangent cone T g p X is a convex subset of a Euclidean space. Thus, for almost all p ∈ X it holds that T g p X is a convex subset in R m for some m ≤ N , is a metric cone over Σ g p X and contains a dense subset of points with opposites also in T g p X. In particular, Σ g p X is a convex subset of S m . Since a closed convex subset of S m is either S k with k ≤ m or has boundary this means that for any such p T g p X is isometric to a Euclidean space of dimension k ≤ m.
i) Let p ∈ X satisfy T g p X ∼ = R m for some m ≤ N . Then an open neighbourhood W of p is homeomorphic to R m . ii) If an open neighborhood W of p is homeomorphic to R m then for any q ∈ W it holds that T g q X ∼ = T q X ∼ = R m . Moreover, for any compact set C ⊂ W there is ε = ε(C) > 0 such that every geodesic starting in C can be extended to length at least ε.
Proof. Let us first prove part i). Suppose T g p X ∼ = R m . By [Kra11, Theorem A] there is a small R > 0 such that B R (p)\{p} is homotopy equivalent to S m−1 . Since S m−1 is not contractible, by [LS07, TRheorem 1.5] there is 0 < ε < π κ /2 such that every geodesic starting at p extends to a geodesic of length ε. The natural "logarithm" map Φ :B ε (p) →B ε (0) ⊂ T g p X is Lipschitz since X is CAT (κ). By the above mentioned result of Lytchak and Schroeder [LS07, Theorem 1.5] Φ is onto.
We also claim that Φ is 1-1. If Φ is not 1-1 then there exist two distinct unit speed geodesics γ 1 , γ 2 of the same length ε ′ ≤ ε such that p = γ 1 (0) = γ 2 (0), γ ′ Then there is a geodesic γ 3 of length εstarting at p in the direction −v. Since X is CAT (κ) and 2ε < π k , the concatenation of γ 3 with γ 1 is a geodesic and the same is true for γ 2 . This contradicts the fact that X is nonbranching.
Thus, Φ is a continuous bijection and since bothB ε (p) andB ε (0) are compact and Hausdorff it's a homeomorphism. This proves part i).
Let us now prove part ii). Suppose an open neighborhood W of p is homeomorphic to R m . By [KK18, Lemma 3.1] or by the same argument as above using [Kra11] and [LS07], for any q ∈ W all geodesics starting at q can be extended to length at least ε(q) > 0. Therefore T g q X ∼ = T q X. By the splitting theorem T q X ∼ = R l where where l = l(q) ≤ N might a priori depend on q. However, using part i) we conclude that an open neighbourhood of q is homeomorphic to R l(q) . Since W is homeomorphic to R m this can only happen if l(q) = m.
The last part of ii) immediately follows from above and compactness of C.
3.1. DC-coordinates in RCD + CAT -spaces. Let X g reg be the set of points p in X with T p X ∼ = T g p X ∼ = R n . Then by Proposition 3.6 there is an open neighbourhoodÛ of p homeomorphic to R n such that every q ∈Û also lies in X g reg . In particular, X g reg is open. Further, geodesics inÛ are locally extendible by Proposition 3.6.
Thus the theory of Lytchak-Nagano from [LN18] applies, and let x : U → V with U = B 2ǫ (p) ⊂ U be DC-coordinates as in Subsection 2.5. The pushforward of the Hausdorff measure H n on U under x coordinates is given by |g|L where |g| is the determinant of g ij Consequently, the map With a slight abuse of notations we will identify these metric-measure spaces as well as functions on them, i.e we will identify any function u on U with u • x −1 on V .
Lemma 3.7. Angles between geodesics in U are continuous. That is if are converging sequences with q = s, q = t then ∠s i q i t i → ∠sqt.
Proof. Without loss of generality we can assume that q i ∈ U for all i. Let α i = ∠s i q i t i , α = ∠sqt. Let {α i k } be a converging subsequence and letᾱ = lim k→∞ α i k . Then by upper semicontinuity of angles in CAT (κ) spaces it holds that α ≥ᾱ. We claim that α =ᾱ.
By Proposition 3.6 we can extend [s i q i ] past q i as geodesics a definite amount δ to geodesics [s i z i ]. Let β i = ∠z i q i t i . By possibly passing to a subsequence of {i k } we can assume that [s i k z i k ] → [sz].
Let β = ∠zqt. Then since all spaces of directions T g qi X and T g q X are Euclidean by Proposition 3.6, we have that α i + β i = α + β = π for all i. Again using semicontinuity of angles we get that β ≥β.
Let A be the algebra of functions of the form ϕ(f 1 , . . . , f m ) where f i = d(·, q i ) for some q 1 , . . . , q m with |q i p| > ε and ϕ is smooth. Together with the first variation formula for distance functions Lemma 3.7 implies that for any u, h ∈ A it holds that ∇u, ∇h g is continuous on V . In particular, g ij = ∇x i , ∇x j g is continuous and hence g is BV 0 and not just BV.
Furthermore, since ∂ ∂xi = j g ij ∇x j where g ij is the pointwise inverse of g ij , Lemma 3.7 also implies that any u ∈ A is C 1 on V . Hence, any such u is DC 0 on V .
Recall that for a Lipschitz function u on V we have two a-priori different notions of the norm of the gradient defined m-a.e.: the "Riemannian" norm of the gradient |∇u| 2 g = g ij ∂u ∂x i ∂u ∂xj and the minimal weak upper gradient |∇u| when u is viewed as a Sobolev functions in W 1,2 (m). We observe that these two notions are equivalent.
In particular, g ij = ∇x i , ∇x j g = ∇x i , ∇x j m-a.e..
Proof. First note that since both ∇u, ∇h and ∇u, ∇h g satisfy the parallelogram rule, it's enough to prove that |∇u| = |∇u| g a.e.. Recall that g ij is continuous on U . Fix a point p where u is differentiable. Then = sup v, ∇u g(p) = |∇u| g(p) .
In the second equality we used that d is induced by g ij , and that g ij is continuous. Since (U, d, m) admits a local 1-1 Poincaré inequality and is doubling, the claim follows from [Che99] where it is proved that for such spaces Lip u = |∇u| a.e..
In view of the above Lemma from now on we will not distinguish between |∇u| and |∇u| g and between ∇u, ∇h and ∇u, ∇h g . Proposition 3.9. If u ∈ W 1,2 (m) ∩ BV (U ), then |∇u| 2 = g ij ∂ ap u ∂xi ∂ ap u ∂xj m-a.e. . Proof. We choose a set S ⊂ U of full measure such that u and |∇u| are defined pointwise on S and u is approximately differentiable at every x ∈ S. Since u is BV (U ), for η > 0 there exist u η ∈ C 1 (U ) such that for the set one has m(B η ) ≤ η [EG15, Theorem 6.13]. Note, since f is continuous, there exists a constant λ > 0 such that λ −1 m ≤ H n ≤ λm on U . Moreover, since g ij is continuous, one can check thatû η is Lipschitz w.r.t. d g , and hencef ∈ W 1,2 (m).
By [AGS14a, Proposition 4.8] we know that |∇u|| Aη = |∇û|| Aη m-a.e. for A η = S\B η . On the other hand, uniqueness of approximative derivatives also yields that g ij ∂ ap u ∂xi ∂ ap u ∂xj | Aη = g ij ∂ apû η ∂xi ∂ apû η ∂xj | Aη m-a.e. . Hence, sinceû is Lipschitz w.r.t. d, by Lemma 3.8. Now, we pick a sequence η k for k ∈ N such that ∞ k=1 η k < ∞. Then, by the Borel-Cantelli Lemma the set B = {x ∈ S : ∃ infinitely many k ∈ N s.t. x ∈ B η k } is of m-measure 0. Consequently, for x ∈ A = S\B we can pick a k ∈ N such that x ∈ A η k ⊂ S. It follows and hence m-a.e. . Following Gigli and De Philippis [DPG18] for any x ∈ X we consider the monotone quantity m(Br(x)) v k,n (r) which is non increasing in r by the Bishop-Gromov volume comparison. Let θ n,r (x) = m(Br(x)) ωnr n . Consider the density function θ n (x) = lim r→0 θ n,r (x) = lim r→0 m(Br (x)) ωnr n . Since n is fixed throughout the proof we will drop the subscripts n and from now on use the notations θ(x) and θ r (x) for θ n (x) and θ n,r (x) respectively.
By Propositions 3.4, 3.6 and [DPG18, Theorem 1.10] we have that for almost all p ∈ X it holds that T p X ∼ = T g p X ∼ = R n and θ(x) = f (x). Therefore we can and will assume from now on that f = θ everywhere. Corollary 4.4. θ = f is locally Lipschitz near any p ∈ X g reg . Proof. First observe that semiconcavity of θ, the fact that θ ≥ 0 and local extendability of geodesics on X g reg imply that θ must be locally bounded on X g reg . Now the corollary becomes an easy consequence of Lemma 4.3, the fact that geodesics are locally extendible a definite amount near p by Proposition 3.6 and the fact that a semiconcave function on (0, 1) is locally Lipschitz.
1. Since small balls in spaces with curvature bounded above are geodesically convex, we can assume that diam X < π κ . Let p ∈ X, x : U → R n and A be as in the previous subsection.
By the same argument as in [Per95, Section 4] (cf. [Pet11], [AB18]) it follows that any u ∈ A lies in D(∆, U, H n ) and the H n -absolutely continuous part of ∆ 0 u can be computed using standard Riemannian geometry formulas that is where |g| denotes the pointwise determinant of g ij . Here ∆ 0 denotes the measure valued Laplacian on (U, d, H n ). Note that g, |g| and ∂u ∂xi are BV 0 -functions, and the derivatives on the right are understood as approximate derivatives.
Indeed, w.l.o.g. let u ∈ DC 0 (U ), and let v be Lipschitz with compact support in U . As before we identify u and v with their representatives in x coordinates. First, we note that, since g, |g| and ∂u ∂xi are BV 0 , their product is also in BV 0 , as well as the product with v. Then, the Leibniz rule (4) for the approximate partial derivatives yieds that Again using (4) we also have that (8) and the absolutely continuous with respect to L n part of this equation is given by the previous identity.
The fundamental theorem of calculus for BV functions (see [EG15,Theorem 5.6]) yields that Moreover, by Lemma 3.8 ∇v, ∇u is given in x coordinates by g ij ∂v ∂xj ∂u ∂xi L n -a.e. . Combining the above formulas gives that where µ is some signed measure such that µ ⊥ L n . This implies (7).
2. Since (X, d, m) is RCD(K, n) for any q ∈ X, we have that d q lies in D(∆, U \{q}, m) and ∆d q is locally bounded above on U \{q} by const · m by Theorem 2.4. Furthermore, since by Proposition 3.6 all geodesics in U are locally extendible we have ∆d q = [∆d q ] reg · m on U \ {q} and [∆d q ] reg is locally bounded below on U \{q} again by Theorem 2.4. Therefore [∆d q ] reg is in L ∞ loc (U \{q}) with respect to m (and also H n ), and in particular, ∆d q is locally L 2 .
By the chain rule for ∆ [Gig15] the same holds for any u, h ∈ A on all of U as by construction u and h only involve distance functions to points outside U .
Remark 4.6. It is not clear that u itself is in the domain of Gigli's Hessian since u is not contained D L 2 (m) (∆) (integration by parts for u would involve boundary terms). Nevertheless, the equality and the RHS in (10) are well-defined on B δ (p). We denote the RHS in (10) with Hu(h 1 , h 2 ).
3. The aim of this paragraph is to compute Hu(x i , x j )g ij on B δ (p) in the DC 0 coordinate chart x. In the following we assume w.l.o.g. that B ε (p) = B δ (p) for δ like in the previous paragraph.
Hence, with the help of Proposition 3.9 the RHS of (10) can be computed pointwise in x coordinates at points of approximate differentiability of ∂u ∂xi , ∂h1 ∂xi and ∂h2 ∂xi , i = 1, . . . n, and (10) can be understood to hold a.e. in the sense of approximate derivatives. That is, we can write and do the same for the other two terms in the RHS of (10).
Using that g ij = ∇x i , ∇x j and ∂ ∂xi = j g ij ∇x j a standard computation shows that for any u ∈ A it holds that The easiest way to verify formula (12) is as follows. Let S be the set of points in V where ∇u, g ij have approximate derivatives and ∂ ap ∂xi ( ∂u ∂xj ) = ∂ ap ∂xj ( ∂u ∂xi ). Then by (5) S has full measure in V , and hence it's enough to verify (12) pointwise on S.
Let q ∈ S. Letĝ be a smooth metric on a neighborhood of q which such thatĝ(q) = g(q) and Dĝ(q) = D ap g(q). Likewise letû be a smooth function on a neighborhood of q such that u(q) = u(q), Dû(q) = Du(q) and D ∂û ∂xi (q) = D ap ∂u ∂xi (q) for all i. Suchû exists (we can take it to be quadratic in x) since ∂ ap ∂xi ( ∂u ∂xi )(q) = ∂ ap ∂xj ( ∂u ∂xi )(q). Then 1 |g| ∂ ap ∂x j g jk |g| ∂u ∂x k (q) = 1 |ĝ| where all the derivatives are approximate derivatives. Similarly Hu(x i , x j )(q)g ij (q) = Hû(x i , x j )(q)ĝ ij (q) where again all the derivatives in (10) and (11) are approximate derivatives.
for every u ∈ A where Hess is the Hessian in the sense of Gigli, and H(u) is denotes the RHS of (10). The first equality in (13) is the definition of Tr, the second equality is the L ∞ -homogeneity of the tensor Hess(χu), and the third equality is the identity (10).
Since f is locally Lipschitz and positive on B δ (p), we can perform the following integration by parts in DC 0 coordinates. Let u ∈ A and let g be Lipschitz with compact support in B δ (p). χu ∈ D L 2 (m) (∆) implies u| B δ (p) ∈ D(∆, B δ (p)). Then a.e. for any u ∈ A.
5. Therefore f ∇ log f | B δ (p) = ∇f | B δ (p) = 0. Indeed, since f is semiconcave, f • x −1 is DC by [LN18]. Hence ∇f = g ij ∂f ∂xi is continuous on a set of full measure Z in B δ (p) since this is true for convex functions on R n . Let q ∈ Z be a point of continuity of ∇f | Z and v = ∇f (q). Assume v = 0. Then due to extendability of geodesics there exists z / ∈ U such that ∇d z (q) = v |v| . Since ∇d z is continuous near q and ∇f is continuous on Z it follows ∇f, ∇d z = 0 on a set of positive measure. Hence ∇f | B δ (p) = 0 and f | B δ (p) = const.
6. We claim that this implies that f is constant on X g reg . (This is not immediate since we don't know yet that X g reg is connected.) Indeed, since X is essentially nonbranching, radial disintegration of m centered at p (Theorem 2.4) implies that for almost all q ∈ X the set [pq] ∩ X g reg has full measure in [p, q]. It is also open in [p, q] since X g reg is open. Suppose q ∈ X g reg is as above. Since θ is semiconcave on X and locally constant on X g reg it is locally Lipschitz (and hence Lipschitz) on the geodesic segment [p, q]. A Lipschitz function on [0, 1] which is locally constant on an open set of full L 1 measure is constant. Therefore θ is constant on [p, q] and hence θ is constant on X g reg which has full measure. Therefore f = θ = const a.e. globally.