Diameter bounds for metric measure spacesAbstract. Consider a metric measure space with non-negative...

INSCRIBED RADIUS BOUNDS FOR LOWER RICCI BOUNDED METRIC

MEASURE SPACES WITH MEAN CONVEX BOUNDARY∗

ANNEGRET BURTSCHER, CHRISTIAN KETTERER, ROBERT J MCCANN, AND ERIC WOOLGAR

Abstract. Consider an essentially nonbranching metric measure space with the measure con-

traction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott,Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose

boundary has a suitably signed lower bound on its generalized mean curvature. This providesa nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability state-

ment concerning such bounds and — in the Riemannian curvature-dimension (RCD) setting —

characterize the cases of equality.

1. Introduction

Kasue proved a sharp estimate for the inscribed radius (or inradius, denoted InRad) of a smooth,n-dimensional Riemannian manifold M with nonnegative Ricci curvature and smooth bound-ary ∂M whose mean curvature is bounded from below by n − 1. More precisely, he concludedInRadM ≤ 1 [Kas83]. This result was also rediscovered by Li [Li14] and extended for Bakry-Emery curvature bounds by Li-Wei [LW15b, LW15a] and Sakurai [Sak19]. Their result can beseen either as a manifold-with-boundary analog of Bonnet and Myers’ diameter bound, or as aRiemannian analog of the Hawking singularity theorem from general relativity [Haw66] (for theprecise statement see [Min19, Theorem 6.49]). There has been considerable interest in generalizingHawking’s result to a nonsmooth setting [KSSV15, LMO19, Gra20]. Motivated in part by thisgoal, we give a generalization of Kasue’s result which is interesting in itself and can serve as amodel for the Lorentzian case. Independently and simultaneously, Cavalletti and Mondino haveproposed a synthetic new framework for Lorentzian geometry (also under investigation by one ofus independently [McC]) in which they establish an analogue of the Hawking result [CM20].

In this note we generalize Kasue and Li’s estimate to subsets Ω of a (potentially nonsmooth)space X satisfying a curvature dimension condition CD(K,N) with K ∈ R and N > 1, providedthe topological boundary ∂Ω has a lower bound on its inner mean curvature in the sense of [Ket20].The notion of inner mean curvature in [Ket20] is defined by means of the 1D-localisation (needledecomposition) technique of Cavalletti and Mondino [CM17b] and coincides with the classicalmean curvature of a hypersurface in the smooth context. We also assume that the boundary ∂Ωsatisfies a measure theoretic regularity condition that is implied by an exterior ball condition.Hence, our result not only covers Kasue’s theorem but also holds for a large class of domainsin Alexandrov spaces or in Finsler manifolds. Kasue (and Li) were also able to prove a rigidityresult analogous to Cheng’s theorem [Che75] from the Bonnet-Myers context: namely that, among

∗A previous draft of the manuscript circulated under the title “Diameter bounds for metric measure spaces with

almost positive Ricci curvature and mean convex boundary”. The authors are grateful to Yohei Sakurai for directing

us to the work of Kasue, and to two anonymous referees for very constructive comments.AB is supported by the Dutch Research Council (NWO) – Project number VI.Veni.192.208.CK is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer

396662902, ”Synthetische Krummungsschranken durch Methoden des optimal Transports”.RM’s research is supported in part by NSERC Discovery Grants RGPIN–2015–04383 and 2020–04162.

EW’s research is supported in part by NSERC Discovery Grant RGPIN-2017-04896.

2010 Mathematics Subject classification. Primary 51K10; also 53C21 30L99 83C75. Keywords: curvature-dimension condition, synthetic mean curvature, optimal transport, comparison geometry, diameter bounds, trapped

surfaces, singularity theorems, inscribed radius, inradius bounds, measure contraction property. c©August 26, 2020.

1

2 ANNEGRET BURTSCHER, CHRISTIAN KETTERER, ROBERT J MCCANN, AND ERIC WOOLGAR

smooth manifolds, their inscribed radius bound is obtained precisely by the Euclidean unit ball.In the nonsmooth case there are also truncated cones that attain the maximal inradius; underan additional hypothesis known as RCD, we prove that these are the only nonsmooth optimizersprovided Ω is compact and its interior is connected.

To state our results first we recall the following definition. For κ ∈ R we define cosκ : [0,∞)→ Ras the solution of

v′′ + κv = 0, with v(0) = 1 and v′(0) = 0;(1)

The function sinκ : [0,∞)→ R is defined as solution of the same ODE with initial values v(0) = 0and v′(0) = 1. We define

πκ := supr > 0 : sinκ(r) > 0 =

π√κ

if κ > 0,

∞ otherwise,

and Iκ = [0, πκ). Let K,H ∈ R and N > 1. The Jacobian function is

r ∈ R 7→ JH,K,N (r) :=

(cosK/(N−1)(r) +

H

N − 1sinK/(N−1)(r)

)N−1

+

.(2)

where (a)+ := maxa, 0 for a ∈ R. We denote the upper limit of its domain of positivity by

rK,H,N := supr ∈ R : JH,K,N (r) > 0.(3)

For instance r0,κ(N−1),N = 1κ if κ > 0, rN−1,0,N = π

2 and r0,κ(N−1),N =∞ if κ ≤ 0.Our main theorem reads as follows:

Theorem 1.1 (Inscribed radius bounds for metric measure spaces). Let (X, d,m) be an essentiallynonbranching CD(K ′, N) space with K ′ ∈ R, N ∈ (1,∞) and spt m = X. Let Ω ⊂ X be closedwith Ω 6= X, m(Ω) > 0 and m(∂Ω) = 0 such that Ω satisfies the restricted curvature-dimensioncondition CDr(K,N) for K ∈ R (Definition 2.3) and ∂Ω = S has finite inner curvature (Definition2.19). Assume the inner mean curvature H−S satisfies H−S ≥ κ(N − 1) mS-a.e. for κ ∈ R wheremS denotes the surface measure (Definition 2.16). Then

InRad Ω ≤ rK,κ(N−1),N

where InRad Ω = supx∈Ω dΩc(x) is the inscribed radius of Ω.

We also show:

Theorem 1.2 (Stability). Consider (X, d,m) and Ω ⊂ X as in the previous theorem. Then, forevery ε > 0 there exists δ > 0 such that

InRad Ω ≤ rK,H,N + ε

provided K ≥ K − δ, H−S ≥ H − δ mS-a.e. and N ≤ N + δ for K, H ∈ R and N ∈ (1,∞).

Remark 1.3 (Definitions and improvements). (1) The curvature-dimension conditions CD(K,N)and the restricted curvature-dimension condition CDr(K,N) for an essentially nonbranch-ing metric measure space (X, d,m) are defined in Definition 2.3. If (X, d,m) satisfies thecondition CD(K,N) then Ω 6= ∅ trivially satisfies CDr(K,N) for the same K. For this wenote that for essentially nonbranching CD(K,N) spaces L2-Wasserstein geodesics betweenm-absolutely continuous probability measures are unique [CM17a].Appendix A extends the conclusions of Theorems 1.1 and 1.2 to the case where theCD(K,N) hypothesis is replaced by the measure contraction property MCP (K,N) pro-posed in [Stu06, Oht07], still under the essentially nonbranching hypothesis.

(2) The backward mean curvature bound introduced in Appendix B also suffices for the con-clusion of the above theorems, and replaces the finiteness assumed of the inner curvatureof ∂Ω = S by the requirement that the surface measure mS0 be Radon. These hypothesesalso suffice for the rigidity result of Theorem 1.4 below. They are related to but distinctfrom a notion presented in [CM20].

INRADIUS BOUNDS FOR METRIC MEASURE SPACES 3

(3) The property “having finite inner curvature” (Definition 2.19) is implied by an exteriorball condition for Ω (Lemma 2.23). The surface measure mS is defined in Definition 2.16.

(4) For S with finite inner curvature, the definition of generalized inner mean curvature H−Sis given in Definition 2.19. Let us briefly sketch the idea. Using a needle decompositionassociated to the signed distance function dS := dΩ−dΩ

c , one can disintegrate the referencemeasure m with conditional measures mα, α ∈ Q, (for a quotient space Q) that aresupported on curves γα of maximal slope with respect to dS , the so-called needles. Forq-almost every curve γα for a quotient measure q on Q there exists a conditional densityhα of mα with respect to the 1-dimensional Hausdorff measure H1. Then the inner mean

curvature for mS-a.e. p = γα(t0) ∈ S is defined as d−

dt log hα(t0) = H−S (p). We postponedetails to the Sections 2.3 and 2.4. In the case (X, d,m) = (M,dg, volg) for a Riemannianmanifold (M, g) and ∂Ω is a hypersurface the inner mean curvature coincides with theclassical mean curvature.

(5) Our assumptions cover the case of a Riemannian manifold with boundary: If (X, d,m) =(M,dg, volg) for a n-dimensional Riemannian manifold (M, g) with boundary and ricM ≥K, then one can always construct a geodesically convex, n-dimensional Riemannian mani-fold M with boundary such that M isometrically embeds into M , and such that ricM ≥ K ′[Won08]. In particular, one can consider M as a CDr(K,n) space that is a subset of the

CD(K ′, n) space (M, dM , volM ) (Remark 5.8 in [Ket20]).

1.1. Cones and spherical suspension. For smooth Riemannian manifolds with boundary rigid-ity is obtained precisely by the unit ball (see [Kas83, Li14, LW15b]). In the nonsmooth case alsotruncated cones attain the maximal inradius.

Let (X, d,m) be a metric measure space.

(1) The Euclidean N -cone over (X, d,m) is defined as the metric measure space([0,∞)×X/ ∼, dEucl,mN

Eucl

)=: [0,∞)×Nr X,

where the equivalence relation ∼ is defined by (0, x) ∼ (0, y) ∀x, y ∈ X, and (t, x) ∼ (t, y)for t > 0 iff x = y. o denotes the tip of the cone. The distance dEucl is defined by

d2Eucl((t, x), (s, y)) := t2 + s2 − 2ts cos [d(x, y) ∧ π] ,

where a ∧ b := mina, b, and the measure mNEucl is given by rNdr ⊗ dmX .

(2) The hyperbolic N -cone is defined similarly:([0,∞)×X/ ∼, dHyp,mN

Hyp

)=: [0,∞)×Nsinh X,

The distance dHyp is defined by

d2Hyp((t, x), (s, y)) := cosh t cosh s− sinh t sinh s cos [d(x, y) ∧ π] ,

and the measure mNHyp is given by sinhN rdr ⊗ dmX .

(3) Similar, the spherical N -suspension over (X, d,m) is defined as the metric space([0, π]×X/ ∼, dSusp,mN

Susp

)=: [0, π]×Nsin X,

where the equivalence relation ∼ is defined by (0, x) ∼ (0, y) ∀x, y ∈ X, (π, x) ∼ (π, y)∀x, y ∈ X and (t, x) ∼ (t, y) for t ∈ (0, π) iff x = y. The distance dSusp is defined by

cos dSusp((t, x), (s, y)) := cos t cos s+ sin t sin s cos[d(x, y) ∧ π],

and the measure mNSusp is given by sinN tdt⊗ dm.

In each case 0 denotes the equivalence class of the points (0, x).The next result shows that in the following RCD setting the cones and suspensions are indeed

all maximizers.The Riemannian curvature-dimension condition RCD(K,N) (Definition 2.5) is a strengthening

of the curvature-dimension condition that rules out Finsler manifolds and allows to prove isometricrigidity theorems for metric measure spaces. This condition is crucial in the proof of the next


theorem, since it permits us to exploit the volume cone rigidity theorem by DePhilippis and Gigli[DPG16]. For K > 0 one can view Theorem 1.4 as a version of the maximal diameter theorem[Ket15a] adapted to mean convex subsets of RCD spaces. For rigidity results pertaining to otherinequalities in nonsmooth or even discrete settings, see recent work of Ketterer [Ket15b], Nakajima,Shioya [NS19] and Cushing et al [CKK+20].

Theorem 1.4 (Rigidity). Let (X, d,m) be RCD(K,N) for K ∈ R and N ∈ (1,∞) and let Ω ⊂ Xbe compact with Ω 6= X, m(Ω) > 0, connected and non-empty interior Ω and m(∂Ω) = 0. Weassume that K ∈ N − 1, 0,−(N − 1), ∂Ω = S has finite inner curvature, S 6= pt and the innermean curvature satisfies H−S ≥ κ(N − 1) mS-a.e. . Then, there exists x ∈ X such that

dS(x) = InRad Ω = rK,κ(N−1),N

if and only if (K,κ) ∈ −(N−1), 0× (0,∞)∪N−1×R and there exists an RCD(N−2, N−1)

space Y such that (Ω, dΩ) is isometric to (BrK,κ(N−1),N(0), d) in I K

N−1×N−1

sinK/(N−1)Y , where dΩ

and d are the induced intrinsic distances of Ω and BrK,κ(N−1),N(0), respectively.

Remark 1.5 (Easy direction). In the above rigidity theorem one direction is obvious. Let us explorethis just for the case K = 0 and κ = 1.

Let Y be an RCD(N−2, N−1) space. Then the truncated Euclidean N -cone [0, 1]×Nr Y ≡ Ω isgeodesically convex and satisfies RCD(0, N) [Ket13]. The signed distance function dS for S = ∂Ωin (X, d,m) restricted to Ω is given by dΩc(t, x) = 1 − t. In particular, r0,N−1,N = dS((0, x)) =dS(o) = 1.

Moreover, S has (inner) mean curvature equal to N − 1 in the sense of Definition 2.19 in[0,∞) ×N−1

r Y . Indeed, we can see that points (s, x) and (t, y) in Ω lie on the same needle iffx = y or either x or y is 0. Hence, the needles in Ω for the corresponding 1D-localization are

t ∈ (0, 1) 7→ γ(t) = (1 − t, x), x ∈ Y . One can also easily check that h1/N−1γ (t) = t for all needles

γ in the corresponding disintegration of m |Ω. Hence H−∂Ω ≡ N − 1.

2. Preliminaries

2.1. Curvature-dimension condition. Let (X, d) be a complete and separable metric spaceand let m be a locally finite Borel measure. We call (X, d,m) a metric measure space. We alwaysassume spt m = X and X 6= pt.

The length of a continuous curve γ : [a, b] → X is L(γ) = sup∑ki=1 d(γ(ti), γ(ti+1)) ∈ [0,∞]

where the supremum is w.r.t. any subdivision of [a, b] given by a = t1 ≤ t2 ≤ · · · ≤ tk−1 ≤ tk = band k ∈ N. A geodesic is a length minimizing curve γ : [a, b]→ X. We denote the set of constantspeed geodesics γ : [0, 1]→ X with G(X); these are characterized by the identity

d(γs, γt) = (t− s)d(γ0, γ1)

for all 0 ≤ s ≤ t ≤ 1. For t ∈ [0, 1] let et : γ ∈ G(X) 7→ γ(t) be the evaluation map. A subset ofgeodesics F ⊂ G(X) is said to be nonbranching if for any two geodesics γ, γ ∈ F such that thereexists ε ∈ (0, 1) with γ|(0,ε) = γ|(0,ε), it follows γ = γ.

Example 2.1 (Euclidean geodesics). When X ⊂ Rn is convex and d(x, y) = |x − y| then G(X)consists of the affine maps γ : [0, 1]→ X.

The set of (Borel) probability measures on (X, d,m) is denoted with P(X), the subset of proba-bility measures with finite second moment is P2(X), the set of probability measures in P2(X) thatare m-absolutely continuous is denoted with P2(X,m) and the subset of measures in P2(X,m)with bounded support is denoted with P2

b (X,m).The space P2(X) is equipped with the L2-Wasserstein distance W2, e.g. [Vil09]. A dynamical

optimal coupling is a probability measure Π ∈ P(G(X)) such that t ∈ [0, 1] 7→ (et)#Π is a W2-geodesic in P2(X) where (et)#Π denotes the pushforward under the map γ 7→ et(γ) := γ(t).The set of dynamical optimal couplings Π ∈ P(G(X)) between µ0, µ1 ∈ P2(X) is denoted withOptGeo(µ0, µ1).


A metric measure space (X, d,m) is called essentially nonbranching if for any pair µ0, µ1 ∈P2(X,m) any Π ∈ OptGeo(µ0, µ1) is concentrated on a set of nonbranching geodesics.

Definition 2.2 (Distortion coefficients). For K ∈ R, N ∈ (0,∞) and θ ≥ 0 we define the distortioncoefficient as

t ∈ [0, 1] 7→ σ(t)K,N (θ) :=

sinK/N (tθ)

sinK/N (θ) if θ ∈ [0, πK/N ),

∞ otherwise,

where πκ := ∞ if κ ≤ 0 and πκ := π√κ

if κ > 0. Here sinK/N was defined after (1), and

σ(t)K,N (0) = t. Moreover, for K ∈ R, N ∈ [1,∞) and θ ≥ 0 the modified distortion coefficient is

defined as

t ∈ [0, 1] 7→ τ(t)K,N (θ) := t

1N

[σ

(t)K,N−1(θ)

]1− 1N

where our conventions are 0 · ∞ := 0 and ∞0 := 1.

Definition 2.3 (Curvature-dimension conditions [Stu06, LV09]). An essentially nonbranching met-ric measure space (X, d,m) satisfies the curvature-dimension condition CD(K,N) for K ∈ R andN ∈ [1,∞) if for every µ0, µ1 ∈ P2

b (X,m) there exists a dynamical optimal coupling Π between µ0

and µ1 such that for all t ∈ (0, 1)

ρt(γt)− 1N ≥ τ (1−t)

K,N (d(γ0, γ1))ρ0(γ0)−1N + τ

(t)K,N (d(γ0, γ1))ρ1(γ1)−

1N for Π-a.e. γ ∈ G(X),(4)

where (et)#Π = ρt m.We say that Ω ⊂ X with m(Ω) > 0 for an essentially nonbranching metric measure space

(X, d,m) satisfies the restricted curvature-dimension condition CDr(K,N) if for every dynamicaloptimal coupling Π between µ0, µ1 ∈ P2

b (X,m) with (et)#Π(Ω) = 1 for all t ∈ [0, 1], (4) holds forall t ∈ [0, 1].

Remark 2.4 (Consequences). A CD(K,N) space (X, d,m) for N ∈ [1,∞) is geodesic and locallycompact. Hence, by the metric Hopf-Rinow theorem the space is proper [BBI01, Theorem 2.5.28].

We recall briefly the Riemannian curvature-dimension condition that is a strengthening of theCD(K,N) condition and the result of the combined efforts by several authors [AGS14b, Gig15,EKS15, AGMR15, AMS19, CM16].

The Cheeger energy Ch : L2(m)→ [0,∞] of metric measure space (X, d,m) is defined as

2 Ch(f) := lim infLip(X)3un

L2→f

∫(Lipun)2dmX(5)

where Lip(X) is the space of Lipschitz functions on (X, d,m) and Lipu(x) := lim supy→x|u(x)−u(y)|d(x,y)

is the local slope of u ∈ Lip(X). The L2-Sobolev space is defined as W 1,2(X) = f ∈ L2(m) :Ch(f) <∞ and equipped with the norm ‖f‖2 := ‖f‖2

L2(m)+ 2 Ch(f) [AGS13, AGS14a].

Definition 2.5 (RCD(K,N)). A metric measure space (X, d,m) satisfies the Riemannian curvature-dimension condition RCD(K,N) if (X, d,m) satisfies the condition CD(K,N) and W 1,2(X) is aHilbert space, meaning the 2-homogeneous Cheeger energy (5) satisfies the parallelogram law:

Ch(f + g) + Ch(f − g) = 2 Ch(f) + 2 Ch(g).

2.2. Disintegration of measures. For further details about the content of this section we referto [Fre06, Section 452].

Let (R,R) be a measurable space, and let Q : R → Q be a map for a set Q. One can equipQ with the σ-algebra Q that is induced by Q where B ∈ Q if Q−1(B) ∈ R. Given a probabilitymeasure m on (R,R), one can define a probability measure q on Q via the pushforward Q# m =: q.

Definition 2.6 (Disintegration of measures). A disintegration of m that is consistent with Q is amap (B,α) ∈ R×Q 7→ mα(B) ∈ [0, 1] such that it follows


• mα is a probability measure on (R,R) for every α ∈ Q,• α 7→ mα(B) is q-measurable for every B ∈ R,

and for all B ∈ R and C ∈ Q the consistency condition

m(B ∩Q−1(C)) =

∫C

mα(B)q(dα)

holds. We use the notation mαα∈Q for such a disintegration. We call the measures mα conditionalprobability measures.

A disintegration mαα∈Q consistent with Q is called strongly consistent if for q-a.e. α we havethe normalization mα(Q−1(α)) = 1.

Theorem 2.7 (Existence of unique disintegrations). Assume that (R,R,m) is a countably gener-ated probability space and R =

⋃α∈QRα is a partition of R. Let Q : R → Q be the quotient map

associated to this partition, that is α = Q(x) if and only if x ∈ Rα and assume the correspondingquotient space (Q,Q) is a Polish space.

Then, there exists a strongly consistent disintegration mαα∈Q of m with respect to Q : R→ Qthat is unique in the following sense: if m′αα∈Q is another consistent disintegration of m withrespect to Q then mα = m′α for q-a.e. α ∈ Q.

2.3. 1D-localization. In this section we will recall the basics of the localization technique intro-duced by Cavalletti and Mondino for 1-Lipschitz functions as a nonsmooth analogue of Klartag’sneedle decomposition; here needle refers to any geodesic along which the Lipschitz function attainsits maximum slope that Klartag and others also call transport rays [EG99, FM02, Kla17]. Thepresentation follows Section 3 and 4 in [CM17b]. We assume familiarity with basic concepts inoptimal transport (for instance [Vil09]).

Let (X, d,m) be a proper metric measure space. We assume that spt m = X.Let u : X → R be a 1-Lipschitz function. Then

Γu := (x, y) ∈ X ×X : u(y)− u(x) = d(x, y)

is a d-cyclically monotone set, and one defines Γ−1u = (x, y) ∈ X ×X : (y, x) ∈ Γu.

Note that we switch orientation in comparison to [CM17b] where Cavalletti and Mondino defineΓu as Γ−1

u .The union Γu ∪ Γ−1

u defines a relation Ru on X × X, and Ru induces the transport set withendpoint and branching points

Tu,e := P1(Ru\(x, y) : x = y ∈ X) ⊂ X

where P1(x, y) = x. For x ∈ Tu,e one defines Γu(x) := y ∈ X : (x, y) ∈ Γu, and similarly Γ−1u (x)

and Ru(x). Since u is 1-Lipschitz, Γu,Γ−1u and Ru are closed as well as Γu(x),Γ−1

u (x) and Ru(x).The forward and backward branching points are defined respectively as

A+ :=x ∈ Tu,e : ∃z, w ∈ Γu(x) & (z, w) /∈ Ru, A− :=x ∈ Tu,e : ∃z, w ∈ Γ−1u (x) & (z, w) /∈ Ru.

Then one considers the (nonbranched) transport set as Tu := Tu,e\(A+∪A−) and the (nonbranched)transport relation as the restriction of Ru to Tu × Tu.

The sets Tu,e, A+ and A− are σ-compact ([CM17b, Remark 3.3] and [Cav14, Lemma 4.3]respectively), and Tu is a Borel set. In [Cav14, Theorem 4.6] Cavalletti shows that the restrictionof Ru to Tu × Tu is an equivalence relation. Hence, from Ru one obtains a partition of Tu intoa disjoint family of equivalence classes Xαα∈Q. There exists a measurable section s : Tu → Tu[Cav14, Proposition 5.2], that is if (x, s(x)) ∈ Ru and (y, x) ∈ Ru then s(x) = s(y), and Q can beidentified with the image of Tu under s. Every Xα is isometric to an interva Iα ⊂ R (c.f. [CM17b,Lemma 3.1] and the comment after Proposition 3.7 in [CM17b]) via a distance preserving mapγα : Iα → Xα where γα is parametrized such that d(γα(t), s(γα(t))) = sign(γα(t))t, t ∈ Iα, andwhere signx is the sign of u(x)−u(s(x)). The map γα : Iα → X extends to a geodesic also denotedγα and defined on the closure Iα of Iα. We set Iα = [a(Xα), b(Xα)].


Then, the quotient map Q : Tu → Q given by the section s above is measurable, and we setq := Q# m |Tu . Hence, we can and will consider Q as a subset of X, namely the image of s, equippedwith the induced measurable structure, and q as a Borel measure on X. By inner regularity wereplace Q with a Borel set Q′ such that q(Q\Q′) = 0 and in the following we denote Q′ by Q(compare with [CM17b, Proposition 3.5] and the following remarks).

The following theorem is [CM18, Theorem 3.3].

Theorem 2.8 (Disintegration into needles/transport rays). Let (X, d,m) be a geodesic metricmeasure space with spt m = X and m σ-finite. Let u : X → R be a 1-Lipschitz function, letXαα∈Q be the induced partition of Tu via Ru, and let Q : Tu → Q be the induced quotient mapas above. Then, there exists a unique strongly consistent disintegration mαα∈Q of m |Tu withrespect to Q.

Now, we assume that (X, d,m) is an essentially nonbranching CD(K,N) space for K ∈ R andN > 1. The following is [CM18, Lemma 3.4].

Lemma 2.9 (Negligibility of branching points). Let (X, d,m) be an essentially nonbranchingCD(K,N) space for K ∈ R and N ∈ (1,∞) with spt m = X. Then, for any 1-Lipschitz functionu : X → R, it follows m(Tu,e\Tu) = 0.

The initial and final points are defined by

a :=x ∈ Tu,e : Γ−1

u (x) = x, b :=x ∈ Tu,e : Γu(x) = x .

In [CM16, Theorem 7.10] it was proved that under the assumption of the previous lemma there

exists Q ⊂ Q with q(Q\Q) = 0 such that for α ∈ Q one has Xα\Tu ⊂ a ∪ b. In particular, for

α ∈ Q we have

Ru(x) = Xα ⊃ Xα ⊃ (Ru(x)) ∀x ∈ Q−1(α) ⊂ Tu.(6)

where (Ru(x)) denotes the relative interior of the closed set Ru(x).The following is [CM18, Theorem 3.5].

Theorem 2.10 (Factor measures inherit curvature-dimension bounds). Let (X, d,m) be an essen-tially nonbranching CD(K,N) space with spt m = X, K ∈ R and N ∈ (1,∞).

Then, for any 1-Lipschitz function u : X → R there exists a disintegration mαα∈Q of m |Tuthat is strongly consistent with Ru.

Moreover, there exists Q such that q(Q\Q) = 0 and ∀α ∈ Q, mα is a Radon measure withmα = hαH1|Xα and (Xα, d,mα) verifies the condition CD(K,N).

More precisely, for all α ∈ Q it follows that

hα(γt)1

N−1 ≥ σ(1−t)K/N−1(|γ|)hα(γ0)

1N−1 + σ

(t)K/N−1(|γ|)hα(γ1)

1N−1(7)

for every affine map γ : [0, 1]→ (a(Xα), b(Xα)).

Remark 2.11 (Consequences). The property (7) yields that hα is locally Lipschitz continuous on(a(Xα), b(Xα)) [CM17b, Section 4], and that hα : (a(Xα), b(Xα))→ (0,∞) satisfies

d2

dr2h

1N−1α +

K

N − 1h

1N−1α ≤ 0 on (a(Xα), b(Xα)) in distributional sense;(8)

in particular, h1

N−1α is semiconcave on (a(Xα), b(Xα)), hence admits left and right derivatives at

each point.It is also well-known that (8) is equivalent to (7) provided hα is continuous.

Remark 2.12. We observe the following from the proof of [CM17b, Theorem 4.2]: when Ω ⊂ Xsatisfies the restricted condition CDr(K

′, N) and the 1-Lipschitz function u is the distance functionto Ωc, that is dΩc . then hγ satisfies (7) with K ′ replacing K.

Let us be a little bit more precise here. For the proof of Theorem 4.2 in [CM17b] the authorsconstruct L2-Wasserstein geodesics between m-absolutely continuous probability measures such


that the corresponding optimal dynamical plans are supported on transport geodesics of the 1-Lipschitz function φ that appears in the statement of [CM17b, Theorem 4.2].

In our situation, when φ is actually dΩc , all transport geodesics of positive length are inside ofΩ. Hence, the L2-Wasserstein geodesics constructed by Cavalletti and Mondino are concentratedin Ω and the restricted condition CDr(K

′, N) applies. Then we can follow verbatim the proof ofTheorem 4.2 in [CM17b].

Remark 2.13 (Extended densities). The Bishop-Gromov volume monotonicity implies that hα canalways be extended to continuous function on [a(Xα), b(Xα)] [CM18, Remark 2.14]. Then (7) holdsfor every affine map γ : [0, 1] → [a(Xα), b(Xα)]. We set (hα γα(r)) · 1[a(Xα),b(Xα)] = hα(r) andconsider hα as function that is defined everywhere on R. We also consider h′α : Xα → R defineda.e. via h′α(γα(r)) = h′α(r).

It is standard knowledge that the derivatives from the left and from the right

d+

drhα(r) = lim

t↓0

hα(r + t)− hα(r)

t,d−

drhα(r) = lim

t↑0

hα(r + t)− hα(r)

t

exist for r ∈ [a(Xα), b(Xα)) and r ∈ (a(Xα), b(Xα)] respectively. Moreover, we set d+

dr hα = −∞ in

b(Xα) and d−

dr hα =∞ in a(Xα).

Remark 2.14 (Generic geodesics). In the following we set Q† := Q ∩ Q, where Q and Q index thetransport rays identified between Lemma 2.9 and Theorem 2.10. Then, q(Q\Q†) = 0 and for everyα ∈ Q† the inequality (7) and (6) hold. We also set Q−1(Q†) =: T †u ⊂ Tu and

⋃x∈T †u Ru(x) =:

T †u,e ⊂ Tu,e.

2.4. Generalized mean curvature. Let (X, d,m) be a metric measure space as in Theorem 2.10.Let Ω ⊂ X be a closed subset, and let S = ∂Ω such that m(S) = 0. The function dΩ : X → R isgiven by

infy∈Ω

d(x, y) =: dΩ(x).

Let us also define d∗Ω := dΩc . The signed distance function dS for S is given by

dS := dΩ − d∗Ω : X → R.

It follows that dS(x) = 0 if and only if x ∈ S, and dS ≤ 0 if x ∈ Ω and dS ≥ 0 if x ∈ Ωc. It is clearthat dS |Ω = −d∗Ω and dS |Ωc = dΩ. Setting v = dS we can also write

dS(x) = sign(v(x))d(v = 0, x),∀x ∈ X.

Since X is proper, dS is 1-Lipschitz [CM18, Remark 8.4, Remark 8.5]. Let Ω denote the topologicalinterior of Ω.

Let TdS ,e be the transport set of dS with end- and branching points. We have TdS ,e ⊃ X\S. Inparticular, we have m(X\TdS ) = 0 by Lemma 2.9 and m(S) = 0.

Therefore, by Theorem 2.10 the 1-Lipschitz function dS induces a partition Xαα∈Q of (X, d,m)

up to a set of measure zero for a measurable quotient space Q, and a disintegration mαα∈Q thatis strongly consistent with the partition. The subset Xα, α ∈ Q, is the image of a geodesicγα : Iα → X.

We consider Q† ⊂ Q as in Remark 2.14. One has the representation

m(B) =

∫Q

mα(B)dq(α) =

∫Q†

∫γ−1α (B)

hα(r)drdq(α)

for all Borel B ⊂ X.For any transport ray Xα, α ∈ Q†, it follows that dS(γα(b(Xα))) ≥ 0 and dS(γα(a(Xα))) ≤ 0

(for instance compare with [CM18, Remark 4.12]).


Remark 2.15 (Measurability and zero-level selection). It is easy to see that A := Q−1(Q(S ∩TdS )) ⊂ TdS is a measurable subset. The set A ⊂ TdS is defined such that ∀α ∈ Q(A) we haveXα ∩ S = γ(tα) 6= ∅ for a unique tα ∈ Iα. Then, the map s : γ(t) ∈ A 7→ γ(tα) ∈ S ∩ TdS isa measurable section (i.e. selection) on A ⊂ TdS , one can identify the measurable set Q(A) ⊂ Qwith A ∩ S and one can parameterize γα such that tα = 0.

This measurable section s on A is fixed for the rest of the paper. The set A is the union of alldisjoint needles that intersect ∂Ω. We shall also define Bin as the union of all needles disjoint fromΩc and Bout as the union of all needles disjoint from Ω. The superscript † will be used indicate

intersection with T †dS . Thus

A ∩ T †dS =: A† and⋃x∈A† RdS (x) =: A†e.

The sets A† and A†e are measurable, and also

B†in := Ω ∩ T †dS\A† ⊂ T †dS and B†out := Ωc ∩ T †dS\A

† ⊂ TdS(9)

as well as⋃x∈B†out

RdS (x) =: B†out,e and⋃x∈B†in

RdS (x) =: B†in,e are measurable.

The map α ∈ Q(A†) 7→ hα(0) ∈ R is measurable (see [CM16, Proposition 10.4]).

Definition 2.16 (Surface measures). Taking S = ∂Ω as above, we use the disintegration of Remark2.15 to define the surface measure mS via∫

φ(x)dmS(x) :=

∫Q(A†)

φ(γα(0))hα(0)dq(α)

for any bounded and continuous function φ : X → R.

Remark 2.17. That is, mS is the pushforward of the measure hα(0)q(dα)|Q(A†) under the map

S : γ ∈ Q(A†) 7→ γ(0).

Remark 2.18 (Recast using ray maps). Let us briefly explain the previous definition from theviewpoint of the ray map [CM17b, Definition 3.6] or its precursor from the smooth setting [FM02].For the definition we fix measurable section s0 : TdS → TdS such that s0|A† = s as in Remark2.15. As was explained in Subsection 2.3 such a section allows us to identify the quotient space Qwith a Borel subset in X up to a q-set of measure 0. Following [CM17b, Definition3.6] we definethe ray map inside Ω defined as

g : V ⊂ Q(A ∪Bin)× R→ X

via its graph

graph(g) = (α, t, x) ∈ Q(A)× R× Ω : x ∈ Xα,−d(x, α) = t∪ (α, t, x) ∈ Q(Bin)× R× Ω : x ∈ Xα,−d(x, γα(b(Xα))) = t.

This is exactly the ray map as in [CM17b] up to a reparametrisation for α ∈ Q(Bin). Notethat g(α, 0) = γα(0) = α and g(α, t) = γα(t) if α ∈ Q(A) but γα(t + d(b(Xα), α)) = g(α, t) forα ∈ Q(Bin). Then the disintegration for a non-negative φ ∈ Cb(X) takes the form∫

X

φdm =

∫Q

∫Vαφ g(α, t)hα g(α, t)dL1(t)dq(α)

where Vα = P2(V ∩ α × R) ⊂ R and P2(α, t) = t. With Fubini’s theorem the right hand side is∫Vφ g(α, t)hα g(α, t)d(q⊗ L1)(α, t) =

∫ ∫Vtφ g(α, t)hα g(α, t)dq(α)dt

where Vt = P1(V ∩ Q × t) ⊂ Q and P1(α, t) = α. In particular, for L1-a.e. t ∈ R the mapα 7→ hα g(α, t) is measurable. Hence, for L1-a.e. t ∈ R we define pt = hα g(α, t)q|Vt on Q. Thendisintegration takes the form

m |Ω = m |Ω∩TdS =

∫g(·, t)#ptdt.


Now, we can consider the pushforward mS0= g(·, 0)#p0. We see that mS0

is concentrated onlarger set than mS but by construction one recognizes that mS = mS0 |A† .

Definition 2.19 (Inner mean curvature). Set S = ∂Ω and let Xαα∈Q be the disintegrationinduced by u := dS . Recalling (9), we say that S has finite inner (respectively outer) curvature if

m(B†in) = 0 (respectively m(B†out) = 0), and S has finite curvature if m(B†out ∪B†in) = 0. If S has

finite inner curvature we define the inner mean curvature of S mS-almost everywhere as

p ∈ S 7→ H−S (p) :=

d−

dr |r=0 log hα γα(r) if p = γα(0) ∈ S ∩A†,∞ if p ∈ B†out,e ∩ S

where we set d−

dr log hα(γα(0)) = −∞ if hα(γα(0)) = 0.

Remark 2.20 ((Sign) conventions). We point out two differences in comparison to [Ket20]: For thedefinition of A† we do not remove points that lie in a and b, and we switched signs in the definitionof inner mean curvature. The latter allows us to work with mean curvature bounded below insteadof bounded above.

Remark 2.21 (Smooth case). Let us briefly address the case of a Riemannian manifold (M, g)equipped with a measure of the form m = Ψ volg for Ψ ∈ C∞(M) and Ω with a boundary S whichis a smooth compact submanifold. For every x ∈ S there exist ax < 0 and bx > 0 such thatγx(r) = expx(r∇dS(x)) is a minimal geodesic on (ax, bx) ⊂ R, and we define

U = (x, r) ∈ S × R : r ∈ (αx, bx) ⊂ S × R

and the map T : U → M via T (x, r) = γx(r). The map T is a diffeomorphism on U , withvolg(M\T (U)) = 0 and the integrals can be computed effectively by the following formula:∫

gdm =

∫S

∫ bx

ax

g T (x, r) detDT(x,r)|TxSΨ T (x, r)drdvolS(x)

where volS is the induced Riemannian surface measure on S. By comparison with the needletechnique disintegration it is not difficult to see that dmS = Ψd volS . Moreover, the open needlesfor dS are the geodesics γx : (ax, bx)→M and the densities hx(r) are given by c(x) detDT(x,r)Ψ T (x, r) for some normalization constant c(x), x ∈ S.

A direct computation then yields

d

drlog hx(0) = HS(x) + 〈∇dS(x),∇ log Ψ〉(x), ∀x ∈ S.

where HS is the standard mean curvature, i.e. the trace of the second fundamental form of S.

Definition 2.22 (Exterior ball condition). Let Ω ⊂ X and ∂Ω = S. Then S satisfies the exteriorball condition if for all x ∈ S there exists r > 0 and px ∈ Ωc such that d(x, px) = r and Br(px) ⊂ Ωc.

Lemma 2.23 (Exterior ball criterion for finite inner curvature). Let Ω ⊂ X. If S = ∂Ω satisfiesthe exterior ball condition, then S has finite inner curvature.

Proof. Let S satisfy the exterior ball condition. Then for every x ∈ S there exists a point px ∈ Ωc

and a geodesic γx : [0, rx] → Ωc from x to px such that L(γx) = d(x, px) = rx and d(px, y) > rxfor any y ∈ S\x. Hence, dS(px) = rx and the image of γx is contained in RdS (x).

Recall the definition of Q† ⊂ Q (Remark 2.14). Since Q† has full q-measure, it is enough to show

that for all α ∈ Q† the endpoint b(Xα) > 0. Then also B†in = ∅. Assume the contrary. Let α′ ∈ Q†and let γ′ := γα′ be the corresponding geodesic such that b(Xα′) = 0, that is Im(γ′|(a(Xα′ ),0)) ⊂ Ω.The concatenation γ′′ : (a(Xα′), rx)→ X of γ′ with γx for γ′(0) = x satisfies γ′′(0) = x and

d(γ′′(s), γ′′(t)) ≤ d(γ′′(s), x) + d(x, γ′′(t)) = dS(γ′′(t))− dS(γ′′(s)).(10)

for s ∈ (a(Xα), 0] and t ∈ [0, rx).The first inequality in (10) is actually an equality. To see this let γ be the geodesic between γ′′(s)

and γ′′(t). There exists t0 ∈ [0, L(γ)] such that γ(t0) ∈ S. Since −d(γ′′(s), x) = dS(γ′′(s)) and


d(γ′′(t), x) = dS(γ′′(t)), it follows that d(γ′′(s), x) ≤ d(γ′′(s), γ(t0)) and d(γ′′(t), x) ≤ d(γ′′(t), γ(t0)).Hence the right hand side in (10) is smaller than d(γ′′(t), γ′′(s)).

Hence, Im(γ′′) ⊂ RdS (γ′′(s)), the points that are RdS -related to γ′′(s). These are exactly thepoints y that satisfy (10) with γ′′(t) replaced with y. But this contradicts Xγ′ = R(γ′′(s)) that isrequired by definition of Q†.

Remark 2.24 (Partial converse). As was pointed out to us by one of the referees the converseimplication in Lemma 2.23 holds in the following sense. If x ∈ S ∩A there exists p ∈ X that eitherbelongs to Ωc or Ω such that Br(p) is either fully contained in Ωc or Ω with r = d(p, x).

Remark 2.25 (Related literature). We note that the previous notion of mean curvature under theassumption that S has finite inner curvature, allows to assign to any point p ∈ S ∩ A† a numberthat is the mean curvature of S in p. This was useful for proving the Heintze-Karcher inequalityin [Ket20].

If one is just interested in lower bounds for the mean curvature, one can adapt a definition ofCavalletti-Mondino [CM20]. They define achronal FTC Borel subsets in a Lorentz length spacehaving forward mean curvature bounded below. We will not recall their definition for Lorentz lengthspaces but we give a corresponding definition for CD(K,N) metric measure spaces in the appendixof this article and outline how our result also follows for this notion of lower mean curvature bounds.

3. Proof of inradius bounds and stability (Theorems 1.1 and 1.2)

Recall the definitions (1)–(3) of the Jacobian JH,K,N (r) and its domain 0 < r < rK,H,N ofpositivity. Observe JH,K,N is pointwise monotone non-decreasing in H and K, and monotonenon-increasing in N . To prove our main theorems requires one more fact from [Ket20] :

Lemma 3.1 (Comparison inequality). Let h : [a, b] → [0,∞) be continuous such that a ≤ 0 < band every affine map γ : [0, 1]→ [a, b] satisfies

h(γt)1

N−1 ≥ σ(1−t)K/(N−1)(|γ|)h(γ0)

1N−1 + σ

(t)K/(N−1)(|γ|)h(γ1)

1N−1 .(11)

Then

(h(r))1

N−1 ≤ (h(0))1

N−1 cos KN−1

(r) +d+

dr

∣∣∣r=0

(h(r))1

N−1 sin KN−1

(r).

In particular, b ≤ rK,H,N , and if h(0) > 0, then h(r)h(0)−1 ≤ JK,H,N (r) for r ∈ (0, b) where

H = d+

dr log h(0).

Proof. If a < 0, the lemma is exactly the statement of Corollary 4.3 in [Ket20].

For a = 0 we pick rn ↓ 0. Then, the statement follows since d+

dr h(r) is continuous from the leftfor a semiconcave function h.

Proof of the Theorems 1.1 and 1.2. Let (X, d,m) be CD(K ′, N) and consider Ω ⊂ X satisfyingCDr(K,N) as assumed in Theorem 1.1. Let u = dS be corresponding signed distance function.Let Xαα∈Q be the decomposition of TdS and

∫mα dq(α) be the disintegration of m given by

Theorem 2.10 and Remark 2.15. In Remark 2.14 we define Q† ⊂ Q. Recall that Q† is a subset ofQ with full q-measure and for all α ∈ Q† one has dmα = hαdH1, Xα,e = Xα and hα satisfies

(h1

N−1α )′′ +

K

N − 1h

1N−1α ≤ 0 on (a(Xα), 0)(12)

in the distributional sense. For hα(r) := hα(−r), (12) still holds. 1. Assume K ∈ R andH−S ≥ κ(N − 1) mS-a.e. . Recall that

H−S (γα(0)) =d−

drlog hα(0) = −d

+

drlog hα(0).

In particular, hα(0) > 0 for q-a.e. α ∈ Q†, since hα(0) = 0 yields d−

dr log hα(r) = −∞ < κ(N − 1)by Definition 2.19.


By Lemma 3.1 it follows that r ∈ [0,−a(Xα)] 7→ hα(r)hα(0)−1 is bounded from above by

JN−1K,κ(N−1),N . Hence −a(Xα) ≤ rK,κ(N−1),N for any α ∈ Q†.

We show dΩc |Ω = −dS |Ωc ≤ rK,κ(N−1),N . Assume there exists x ∈ Ω such that dΩc(x) >rK,κ(N−1),N . If rK,κ(N−1),N =∞, this is already impossible.

Otherwise there exists ε > 0 such that Bε(x) ∩ BrK,κ(N−1),N(Ωc) = ∅ and m(Bε(x)) > 0. Since

0 < m(Bε(x)) =∫Q†

∫γ−1(Bε(x))

hα(r)drdq(α), there exists at least one α ∈ Q† such that∫γ−1α (Bε(x))

hα(r)dr > 0

and every r ∈ γ−1α (Bε(x)) satisfies r > rK,κ(N−1),N because Bε(x) ⊂ (BrK,κ(N−1),N

(Ωc))c. Hence

r = −dS(γα(r)) > rK,κ(N−1),N for such r, and therefore dΩc |Bε(x) > rK,κ(N−1),N . This is a con-tradiction. In particular, the inscribed radius satisfies InRad Ω ≤ rK,κ(N−1),N .

2. Assume K ≥ K − δ, H−S ≥ H − δ mS-a.e. and N ≤ N + δ. Arguing as before yields

r ∈ [0,−a(Xα)] 7→ hα(r)hα(0)−1 is bounded from above by JK−δ,H−δ,N+δ for every α ∈ Q† andconsequently

InRad Ω ≤ rK−δ,H−δ,N+δ.

It follows that for any ε > 0 there exists δ > 0 such that rK−δ,H−δ,N+δ ≤ rK,H,N + ε since

f(r) = cos KN−1

(r) + HN−1 sin K

N−1(r) that appears in the definition of JK,H,N depends continuously

on K, H and N .

4. Rigidity

4.1. Volume cone implies metric cone. The following theorem by Gigli and De Philippis willbe crucial in the proof the main result.

Theorem 4.1 (Volume cone implies metric cone [DPG16]). Let K ∈ 0, N − 1, N ∈ (1,∞) andX an RCD(K,N) space with spt mX = X. Assume there exists o ∈ X and R > r > 0 such that

mX(BR(o)) =

∫ R0

(sin K

N−1u)N−1

du∫ r0

(sin K

N−1v)N−1

dv

mX(Br(o)).(13)

Then the following holds:

(1) If ∂BR/2(o) contains only one point, then X is isometric to [0,diamX ] or [0,∞) in the caseK ≤ 0 with an isometry that sends o to 0, or X is isometric to [0, πK/(N−1)] in the case

K > 0 with an isometry that sends o to 0. The measure mX |BR(o) is given by c sinN−1KN−1

dx

for some positive constant c.(2) If BR/2(o) contains more than one point then N ≥ 2 and there exists an RCD(N−2, N−1)

space Z with diamZ ≤ π and a local isometry U : BR(o)→ [0, R)×Nsin KN−1

Z that is also a

measure preserving bijection.

Remark 4.2. In the proof of Theorem 4.1 Gigli and De Philippis show that the map U has aninverse V : BR(0)→ BR(o) that is also a local isometry.

There is also a third case in the conclusion of the theorem when BR(o) contains exactly twopoints. However, in this case, one has necessarily N = 1 which is excluded by our assumptions.

4.2. Distributional Laplacian and strong maximum principle. We recall the notion of thedistributional Laplacian for RCD spaces (cf. [Gig15, CM18]).

Let X be an RCD space, and Lipc(Ω) denote the set of Lipschitz function compactly supportedin an open subset Ω ⊂ X. A Radon functional over Ω is a linear functional T : Lipc(Ω)→ R such


that for every compact subset W in Ω there exists a constant CW ≥ 0 such that

|T (f)| ≤ CW maxW|f | ∀f ∈ Lipc(Ω) with spt f ⊂W.(14)

One says T is non-negative if for all f ∈ Lipc(Ω) with f ≥ 0 it follows that T (f) ≥ 0.The classical Riesz-Markov-Kakutani representation theorem says that for every non-negative

Radon functional T from (14) there exists a non-negative Radon measure µT such that T (f) =∫fdµT for all f ∈ Lipc(Ω).When (X, d,m) is an RCD space, its Cheeger energy (5) is a quadratic form and one can

introduce a symmetric bilinear form 〈·, ·〉 on the Sobolev space W 1,2(X) with values in L1(m) via

(f, g) ∈W 1,2(X)×W 1,2(X) 7→ 〈∇f,∇g〉 :=1

4|∇(f + g)|2 − 1

4|∇(f − g)|2 ∈ L1(m).

Definition 4.3 (Nonsmooth Laplacian). Let Ω ⊂ X be an open subset and let u ∈ Lip(X). Onesays u is in domain of the distributional Laplacian on Ω provided there exists a Radon functionalT over Ω such that

T (f) =

∫〈∇u,∇f〉dmX ∀f ∈ Lipc(Ω).

In this case we write u ∈ D(∆,Ω). If T is represented as a measure µT , one writes µT ∈ ∆u|Ω,and if there is only one such measure µT by abuse of notation we will identify µT with T and writeµT = ∆u|Ω.

We also recall that u ∈W 1,2loc (Ω) for an open set Ω ⊂ X if and only if for any Lipschitz function φ

with compact support in Ω we have φ·u ∈W 1,2(X). In particular, if u ∈ Lip(X) then u ∈W 1,2loc (Ω).

Remark 4.4 (Locality and linearity). (i) If u ∈ D(∆,Ω) and Ω′ is open in X with Ω′ ⊂ Ω,then u ∈ D(∆,Ω′) and for µ ∈∆u|Ω it follows that µ|Ω′ ∈∆u|Ω′ .

(ii) If u, v ∈ D(∆,Ω), then u+ v ∈ D(∆,Ω) and for µu ∈∆u|Ω and µv ∈∆vΩ it follows thatµu + µv ∈∆(u+ v)|Ω.

Recall that u ∈W 1,2(Ω) is sub-harmonic if∫Ω

|∇u|2dmX ≤∫

Ω

|∇(u+ g)|2dmX ∀g ∈W 1,2(Ω) with g ≤ 0.

One says u is super-harmonic if −u is sub-harmonic, and u is harmonic if it is both sub- andsuper-harmonic. The following is well known.

Theorem 4.5 (Characterizing sub-harmonicity). Let X be RCD, let Ω ⊂ X be open and u ∈W 1,2loc (Ω). Then u is sub-harmonic if and only if u ∈ D(∆,Ω) and there exists µ ∈∆u|Ω such that

µ ≥ 0.

The following is [BB11, Theorem 9.13]:

Theorem 4.6 (Strong Maximum Principle). Let X be an RCD(K,N) space with K ∈ R and

N ∈ [1,∞), let Ω ⊂ X be open with compact closure and connected and let u ∈W 1,2loc (Ω)∩C(Ω) be

sub-harmonic. If there exists x0 ∈ Ω such that u(x0) = maxΩ u then u is constant.

Let us recall another result of Cavalletti-Mondino:

Theorem 4.7 (Laplacian of a signed distance [CM18]). Let (X, d,m) be a CD(K,N) space, andΩ and S = ∂Ω as above. Then dS ∈ D(∆, X\S), and one element of ∆dS |X\S that we also denotewith ∆dS |X\S is the Radon functional on X\S given by the representation formula

∆dS |X\S = (log hα)′m |X\S +

∫Q

(hαδa(Xα)∩dS>0 − hαδb(Xα)∩dS<0)dq(α).

We note that the Radon functional ∆dS |X\S can be represented as the difference of two measures

[∆dS ]+ and [∆dS |X\S ]− such that

[∆dS |X\S ]+reg − [∆dS |X\S ]−reg = (log hα)′ m -a.e.


where [∆dS |X\S ]±reg denotes the m-absolutely continuous part in the Lebesgue decomposition of

[∆dS |X\S ]±. In particular, −(log hα)′ coincides with a measurable function m-a.e. .

To prove the rigidity asserted in Theorem 1.4, we need one more lemma:

Lemma 4.8 (Riccati comparison). Let u : [0, b] → R be continuous such that u′′ + κu ≤ 0 in thedistributional sense, u(0) = 1 and u′(0) ≤ d. Let v : [0, b]→ R be the maximal positive solution of

v′′ + κu = 0 with v(0) = 1 and v′(0) = d. Then b ≥ b and d+

dt log u ≤ (log v)′ on [0, b).

Proof. Note that v(r) = cosκ(r) + d sinκ(r) and b = rκ(N−1),d(N−1),N . Then Lemma 3.1 already

yields that b ≤ b and u ≤ v on [0, b]. Therefore, without loss of generality we restrict v to [0, b].We pick ϕ ∈ C2

c ((−1, 1)) with∫ϕL1 = 1 and define ϕε(x) = εϕ(xε ). Let ε > 0 and ε ∈ (0, ε),

and let uε =∫ϕε(t)u(s− t)dt be the mollification of u by ϕε. One can check that uε is well-defined

on [ε, b− ε] and uε ∈ C2([ε, b− ε]) satisfies

u′′ε + (κ+ δ)uε ≤ 0

in the classical sense with δ = δ(ε) → 0 for ε → 0. Since u is continuous, uε(t) → u(t) for allt ∈ [ε, b− ε]. Moreover, u′ε(t)→ u′(t) for every t ∈ [ε, b− ε] where u is differentiable.

Let vε : [0, bε]→ [0,∞) be the maximal positive solution of v′′ε + (κ+ δ(ε))vε = 0 with vε(0) = 1and v′ε(0) = d. By the previous comparison principle of Lemma 3.1 it follows that b ≤ bε, and sinceδ(ε)→ 0 for ε→ 0 we have bε → b, vε → v and v′ε → v pointwise on [0, b] if ε→ 0.

We pick ε ∈ (0, b) and t ∈ [ε, b− ε] where u is differentiable. Then

0 ≥∫ t

ε

[vε(u′′ε + (κ+ δ)uε)− uε(v′′ε + (κ+ δ)vε)] dL1 =

∫ t

ε

[vεu

′ε]′ − [uεv

′ε]′dL1

= vε(t)u′ε(t)− uε(t)v′ε(t) + uε(ε)v

′ε(ε)− vε(ε)u′ε(ε)→ v(t)u′(t)− u(t)v′(t) + u(ε)v′(ε)− v(ε)u′(ε)

Since u is semiconcave and continuous on [0, b], the right derivative d+

dt u : [0, b] → R ∪ ∞ iscontinuous from the right. Hence, for ε ↓ 0 and any t ∈ (0, b) it follows

0 ≥ v(t)d+

dtu(t)− u(t)v′(t) + u(0)v′(0)− v(0)

d+

dtu(0) ≥ v(t)

d+

dtu(t)− u(t)v′(t)

Hence d+

dt log u =d+

dt u

u ≤ v′

v = (log v)′ as desired.

4.3. Proof of rigidity (Theorem 1.4). Let (X, d,m) be RCD(K,N) and consider Ω ⊂ X asin the main theorem. Let u = dS be the corresponding signed distance function. Let (Xα)α∈Qbe the decomposition of Tu and

∫mα dq(α) be the disintegration of m given by Theorem 2.8 and

Remark 2.15. We consider again the set Q† of full q-measure in Q. For every α ∈ Q† we have thatmα = hαH1, Xα,e = Xα and hα satisfies

(h1

N−1α )′′ +

K

N − 1h

1N−1α ≤ 0 on (a(Xα), 0) ∀α ∈ Q†(15)

in the distributional sense. As usual we write hα = hα γα. Note the properties of hα discussedin Remark 2.11. For hα(r) := hα(−r), (15) is still true.

Also recall that by the lower mean curvature bound the set of αs in Q with hα(0) = 0 has q-measure 0. Hence hα(r) > 0 for q-a.e. α.

We only consider the cases K ≥ 0. A straightforward modification of the K > 0 proof alsoyields the analogous result for K < 0.

1. Assume K = 0 and H−S ≥ κ(N − 1) mS-a.e. with κ ∈ R. Recall that

H−S (γα(0)) =d−

drlog hα(0) = −d

+

drlog hα(0).


By Theorem 4.7, dS |Ω ∈ D(∆,Ω) and

∆dS |Ω = (log hα)′m |Ω +

∫Q

(hαδa(Xα)∩dS>0 − hαδb(Xα)∩dS<0)dq(α).

Any γα for α ∈ Q that starts inside of Ω satisfies γα(b(Xα)) ∈ Ωc. Hence∫Qhαδb(Xα)∩Ωdq(α) = 0.

It follows that

∆dΩc |Ω = ∆(−dS |Ω)|Ω ≤ (log hα)′m |Ω .

Recall that −dS |Ω = dΩc |Ω and by locality of the distributional Laplacian (∆dS |X\S)|Ω =∆dS |Ω .

Recall that−(log hα)′(r) = (log hα)′(−r). By the mean curvature bound and Riccati comparisonlemma (Lemma 4.8) it follows that

∆dΩc |Ω ≤ (N − 1)−κ

1− κdΩcm |Ω .

Now we assume that there exists p ∈ Ω such that dΩc(p) = r0,κ(N−1),N from (3). Note that

r0,κ(N−1),N = 1κ if κ > 0 and ∞ otherwise, and hence κ > 0 necessarily. In particular, there exists

a geodesic γ∗ : [a, b] → Ω such that γ∗(a) = p, γ∗(b) ∈ S and L(γ∗) = d(γ∗(a), γ∗(b)) = 1/κ.Moreover B1(p) ⊂ Ω. By Laplace comparison [Gig15] the corresponding distance dp satisfies

∆dp ≤ (N − 1)1

dpm .

We can add the previous two inequalities on Ω and obtain

∆(dΩc + dp)|Ω ≤ (N − 1)1− κdΩc − κdp(1− κdΩc)dp

m |Ω ≤ 0

Since dΩc + dp ≥ 1κ , dΩc + dp is a super-harmonic function on Ω. Moreover, (dp + dΩc)|Ω ≥ 1

κand attains its minimum on the geodesic γ∗ in Ω .

Hence, by the strong maximum principle (Theorem 4.6) it follows that dΩc + dp = 1κ and

∆dp = −∆dΩc = (N − 1)κ

1− κdΩcon Ω.(16)

We observe that ∆dp = −∆dΩc is m-absolutely continuous on Ω, Ω = B 1κ

(p) and a(Xα) = − 1κ

for q-a.e. α ∈ Q. Moreover, by Theorem 4.7

(∆dp) γα(r) = −(∆dΩc) γα(r) = (log hα)′ γα(r) for r ∈ [− 1κ , 0) and q-a.e. α ∈ Q.

Recall at this point that (log hα)′ γα(r) is in fact given by (log hα)′(r) for the density hα of mα

with respect to L1 on [a(Xα), b(Xα)].It follows for q-a.e. α ∈ Q and r ∈ [− 1

κ , 0) that

(N − 1)1

1κ + r

= (N − 1)1

1κ − dΩc

γα(r) = (log hα)′ γα(r).(17)

Solving this ODE we obtain that

hα(r) = hα(0)(1 + κr)N−1 = hα(0)J0,κ(N−1),N (−r) for r ∈ [− 1κ , 0) and q-a.e. α ∈ Q.

Then it follows that

m(BR(p)) =

∫ ∫ R− 1κ

− 1κ

hα(r)drdq(α) =

∫ ∫ R

0

hα(0)κN−1rN−1drdq = λ

∫ R

0

rN−1dr ∀R ∈ [0,1

κ],

where λ = κN−1∫hα(0)dq(α). Hence B 1

κ(p) is a volume cone in the sense of (13) in Theorem 4.1.

2. Assume that K > 0 and H−S ≥ κ(N − 1) mS-a.e. for κ ∈ R. By rescaling, we may thenassume without loss of generality that K = N − 1.


The proof is similar to the case K = 0. We deduce from Theorem 4.7, the curvature bounds andthe Riccati comparison lemma (Lemma 4.8) that dS |Ω = −dΩc |Ω ∈ D(∆,Ω) and

∆dΩc |Ω ≤ (N − 1)− sin dΩc + κ cos dΩc

cos dΩc + κ sin dΩcm |Ω .(18)

Assume that there is a p ∈ Ω such that dΩc(p) = rK,κ(N−1),N . Then BrK,κ(N−1),N(p) ⊂ Ω and

∆dp|Ω ≤ (N − 1)cos dpsin dp

m |Ω .(19)

Adding the previous two inequalities yields

∆(dp + dΩc)|Ω ≤ (N − 1)cos dp cos dΩc + κ cos dp sin dΩc − sin dp sin dΩc + κ sin dp cos dΩc

sin dp(cos dΩc + κ sin dΩc)m |Ω

= (N − 1)cos(dp + dΩc) + κ sin(dp + dΩc)

sin dp(cos dΩc + κ sin dΩc)m |Ω ≤ 0.

The last inequality follows since rK,κ(N−1),N ≤ dp + dΩc by the triangle inequality and sincecos(r) + κ sin(r) ≤ 0 for r ≥ rK,κ(N−1),N by definition of rK,κ(N−1),N . For this we also note that

sin dp(cos dΩc + κ sin dΩc) = sin dp(JK,κ(N−1),N dΩc)1

N−1 > 0

on Ω because dp ≤ π for any p ∈ X where X is CD(N − 1, N) by the Bonnet-Myers diameterestimate (e.g. [Stu06]) and because JK,κ(N−1),N (dΩc) > 0 on Ω by Theorem 1.1.

Hence dp + dΩc on Ω is a super-harmonic function that attains its minimum inside Ω andis therefore constant equal to dp + dΩc = rK,κ(N−1),N on Ω. In particular, it follows Ω =BrK,κ(N−1),N

(p). Moreover, the inequality in (18) must be an equality on Ω:

∆dΩc |Ω = (N − 1)− sin dΩc + κ cos dΩc

cos dΩc + κ sin dΩcm |Ω .

Solving the ODE analogous to (17) yields

hα(r) = hα(0)JN−1,κ(N−1),N (−r) for r ∈ [−rN−1,κ(N−1),N , 0) and q-a.e. α ∈ Q

and therefore

m(BR(p)) = λ

∫ R

0

sinN−1 rdr ∀R ∈ [0, rK,κ(N−1),N ]

for some constant λ > 0. Hence BrK,κ(N−1),N(p) is a volume cone in the sense of (13).

3. Now we can finish the proof of the main theorem for both cases by application of Theorem 4.1.First, we note that by assumption S = ∂BrK,κ(N−1),N

(p) is not just a point. Hence, one can

conclude that ∂BrK,κ(N−1),N/2(p) is also not just a point. Otherwise, one can easily construct an

L2-Wasserstein geodesic between δp and an m-absolutely continuous measure µ in P2(X) that issupported on set of branching geodesics. But this contradicts the fact that the space is essentiallynonbranching because of the RCD condition. For instance

µ := c

∫mα|Xα∩[−η+rK,κ(N−1),N ,rK,κ(N−1),N ]dq(α)

with η < R/2 is such a measure (c is a normalisation constant).Second, S must contain more than 2 points. Otherwise, we can apply Remark 4.2 and it follows

that N = 1 , which is also excluded.Hence, only the last case in Theorem 4.1 remains relevant to us. By Remark 4.2 there exist

local isometries U and V between Ω = BrK,κ(N−1),N(p) and BrK,κ(N−1),N

(0) in the correspondingcone that are also measure preserving bijections.

Now, it is standard knowledge that U is an isometry with respect to the induced intrinsicdistances.


Let us be more precise. Set rK,κ(N−1),N = R and let dε be the induced intrinsic distance

on BR−ε(p). We denote by d∗ the cone (or suspension distance) and by d∗ and d∗ε the inducedintrinsic distances of BR(0) and BR−ε(0), respectively. Then U is an isometry between BR−ε(p)and BR−ε(0) with respect to the induced intrinsic distances.

To prove this let γ : [0, 1]→ BR−ε(p) be a geodesic with respect to dε between x, y ∈ BR−ε(p).We can divide γ into k ∈ N small pieces γ|[ti−1,ti] with i = 1, . . . , k and t0 = 0, tk = 1 such thateach piece stays inside a small ball that is mapped isometrically with respect to d via U to a smallball in BR(0). We obtain

k∑i=1

d∗(U(γ(ti−1)), U(γ(ti))) =

k∑i=1

d(γ(ti−1), γ(ti)) ≤k∑i=1

dε(γ(ti−1), γ(ti)) = dε(x, y).

The first equality holds because U is an isometry with respect to d and d∗ on the small ballsthat contain γ|[ti−1,ti]. The last equality holds because γ is geodesic with respect to dε and theinequality holds because the intrinsic distance is always equal or larger than d itself.

On the left hand side we can take the supremum with respect to all such subdivisions (ti)i=0,...,k−1.

This yields d∗ε (U(x), U(y)) ≤ L(U γ) ≤ dε(x, y) where L(U γ) is the length of the continuouscurve U γ. In particular U γ is a rectifiable curve(that means has finite length) in BR−ε(0).

We can argue in the same way for the inverse map V and obtain that U : BR−ε(p)→ BR−ε(0)

is an isometry with respect to the induced intrinsic distances dε and d∗ε .

Finally, we let ε → 0 and observe that dε → d on BR−ε(p) and the same for d∗ε and d∗. Thisfinishes the proof.

Appendix A. Substituting measure contraction for lower Ricci bounds

In this appendix we will sketch why the results of Theorem 1.1 also hold when one replacesthe condition CD(K,N) with the weaker measure contraction property MCP (K,N) that wasintroduced in [Stu06, Oht07]. We will not repeat the technical details but focus on necessarymodifications for this setup.

For a proper metric measure space (X, d,m) that is essentially nonbranching there are severalequivalent ways to define the MCP (K,N). The following one can be found in [CM16, Section 9].

Definition A.1 (Measure contraction property). Let (X, d,m) be proper and essentially non-branching. The measure contraction property MCP (K,N), K ∈ R and N ∈ (1,∞) holds if forevery pair µ0, µ1 ∈ P2(X) such that µ0 is m-absolutely continuous there exists a dynamical optimalplan Π such that (et)#Π = µt ∈ P2(m) and

ρt(γt)− 1N ≥ τ (1−t)

K,N (d(γ0, γ1))ρ0(γ0)−1N for Π-a.e. geodesic γ.

where µt = ρt m.For Ω ⊂ X with m(Ω) > 0 the restricted measure contraction property MCPr is defined similarly

as the condition CDr (compare with Definition 2.3).

All the technical results in Section 2.3 still hold when we replace the condition CD(K,N) withMCP (K,N) (c.f. [CM18]). Only in Theorem 2.10 the density hα of the conditional measuremα will not satisfy (7) and therefore need not be semiconcave, although it does remain Lipschitz.Instead one has only

hα(γt)1

N−1 ≥ σ(1−t)K,N (d(γ0, γ1))hα(γ0)

1N−1(20)

for every affine function γ : [0, 1]→ [a(Xα), b(Xα)].Considering Ω ⊂ X with ∂Ω = S and m(S) = 0 then the same definition for mS as in Definition

2.16 makes sense. We also can define the notion of finite inner curvature of Ω. However, sincehα is not semiconcave in general, the right and the left derivative might not exist for every t ∈


[a(Xα), b(Xα)]. Therefore, for a Lipschitz function f : [a, b]→ R we set

d−

dtf(t) = lim sup

t↑0

1

h[f(t+ h)− f(t)] for t ∈ (a, b].

We can set up a definition of mean curvature for subsets in MCP spaces in the following way.

Definition A.2 (Inner mean curvature revisited in the MCP setting). Set S = ∂Ω and letXαα∈Q be the disintegration induced by u := dS . Recalling (9), we say that S has finite

inner (respectively outer) curvature if m(B†in) = 0 (respectively m(B†out) = 0). If S has finite innercurvature we define the inner mean curvature of S mS-almost everywhere as

p ∈ S 7→ H−S (p) :=

d−

dr |r=0 log hα γα if p = γα(0) ∈ S ∩A†,∞ if p ∈ B†out,e ∩ S

where we set d−

dr log hα(γα(0)) = −∞ if hα(γα(0)) = 0.

Theorem A.3 (Inscribed radius bounds under MCP). Let (X, d,m) be an essentially nonbranchingMCP (K ′, N) space with K ′ ∈ R, N ∈ (1,∞) and spt m = X. Let Ω ⊂ X be closed with Ω 6= X,m(Ω) > 0 and m(∂Ω) = 0 such that Ω satisfies the restricted curvature-dimension conditionMCPr(K,N) for K ∈ R and ∂Ω = S has finite inner curvature. Assume the inner mean curvatureH−S satisfies H−S ≥ κ(N − 1) mS-a.e. for κ ∈ R where mS denotes the surface measure. Then

InRad Ω ≤ rK,κ(N−1),N

where InRad Ω = supx∈Ω dΩc(x) is the inscribed radius of Ω.

Proof. From

hα(γα(t))1

N−1 ≥ σ(1− ta(Xα) )

K/N−1 (a(Xα))hα(γα(0))1

N−1

for t ∈ (a(Xα), 0) and any α ∈ Q† follows

d−

dthα(γα(0)) ≤ d−

dt

∣∣∣t=0

σ( a(Xα)−ta(Xα) )

K/N−1 (a(Xα))N−1hα(γα(0)).

If the inner mean curvature is bounded below by H, then

H ≤ d−

dt

∣∣∣t=0


K/N−1 (a(Xα))N−1 = (N − 1)cosK/(N−1)(a(Xα))

sinK/(N−1)(a(Xα)).

Hence HN−1 ≤

cosK/(N−1)(a(Xα))

sinK/(N−1)(a(Xα)) for q-a.e. α ∈ Q†. The symmetry properties of sines and cosines

now imply

0 ≤ cosK/(N−1)(−a(Xα)) +H

N − 1sinK/(N−1)(−a(Xα)).

Thus −a(Xα) ≤ rK,H,N . Otherwise −a(Xα) > rK,H,N implies that the right hand side in the lastinequality is negative by the definition of rK,H,N .

Now, one can finish the argument exactly as for CD(K,N) spaces.

Appendix B. A weaker form of mean curvature bound also suffices

Inspired by [CM20], in this appendix we introduce another new notion of mean curvaturebounded from below which yields Theorems 1.1 and Theorem 1.4 without requiring finite innercurvature of Ω but assuming that the measure p0 in Remark 2.18 is a Radon measure on Q.


Definition B.1 (Backward mean curvature bounded below). Let (X, d,m) be an essentiallynonbranching metric measure space that satisfies MCP or CD. Recall the family of measurespt = hα g(α, t)q|Vt , t ∈ [0,∞) on Q, that we introduced in Remark 2.18. Recall that Q isconstructed as a Borel subset of X and (α, t) 7→ g(α, t) is the ray map constructed in Remark 2.18.

Then S = ∂Ω has backward mean curvature bounded from below by H ∈ R if the measure p0 isa Radon measure and

lim supt↑0

1

t

(∫φdpt −

∫φdp0

)≥ H

∫φdp0

for all bounded and continuous functions φ : X → [0,∞).

Remark B.2. Since it is not assumed that pt for t < 0 is a Radon measure,∫φdpt can be infinite.

Lemma B.3 (Backward versus inner mean curvature). Let (X, d,m) be an essentially nonbranch-ing MCP space. Assume that S = ∂Ω for some Borel set Ω that has finite inner curvature inthe sense of Definition A.2 and that p0 is a Radon measure. Then mS-almost everywhere, thepointwise inner mean curvature of S is bounded from below by H if and only if S has backwardmean curvature bounded from below by H.

Proof. 1. Assume backward mean curvature bounded below by K ∈ R. We can compute for t < 0:∫φdpt −

∫φdp0 =

∫φ(α) (1Vt(α)hα g(α, t)− 1V0(α)hα g(α, 0)) dq(α).

Fatou’s lemma yields

H

∫V0φ(α)hα g(α, 0)dq(α) = H

∫V0φ(α)dp0(α)

≤∫φ(α) lim sup

t↑0

1

t(1Vt(α)hα g(α, t)− 1V0(α)hα g(α, 0)) dq(α)

≤∫φ(α) lim sup

t↑0

1

t(1Vt∩V0(α)hα g(α, t)− 1V0(α)hα g(α, 0)) dq(α)

=

∫V0φ(α)

d−

dt|t=0hα g(α, t)dq(α)

for any bounded and continuous function φ : X → [0,∞). It follows that

Hhα g(α, 0) ≤ d−

dt|t=0hα g(α, t) for q-a.e. α ∈ V0.(21)

In particular, for q-a.e. α ∈ V0 with hα g(α, 0) = 0, it follows that d−

dt |t=0hα g(α, t) ≥ 0. On theother hand for α ∈ V0 with hα g(α, 0) = 0 one has

d−

dr|r=0hα g(α, r) = lim

t↑0

1

thα g(α, t) ≤ 0.

Hence for q-almost every α ∈ V0 with hα g(α, 0) = 0 one has d−

dr |r=0hα g(α, r) = 0. Then, itfollows that

d−

dr|r=0[hα g(α, r)]

1N−1 = lim

t↑0

hα g(α, t)

t[hα g(α, t)]

1N−1−1 = 0 for q-a.e. α s.t. hα g(α, 0) = 0

In the case of a CD space directly by Lemma 3.1 and in the case of an MCP space by (20) itfollows that hα g(α, t) ≡ 0 for t < 0 for such α, and by the disintegration formula the set of suchα has q-measure 0.


In particular, we can assume hα g(α, 0) > 0 for q-almost every α ∈ V0. Therefore, it followsthat

H

∫V0φ(α)hα g(α, 0)dq(α) ≤

∫V0φ(α)

d−

dt|t=0hα g(α, t)dq(α)

=

∫V0φ(α)

d−

dt|t=0 log(hα g(α, t))hα g(α, 0)dq(α)

Now, we recall that V0 ⊂ Q(A†)∪B†in) with q(Q(A† ∪B†in)\V0) = 0 and m(B†in) = q(Q(B†in)) = 0(because we assume finite inner curvature). Moreover g(α, t) = γα(t), hα g(α, t) = hα γα(t) =hα(t) and α = γα(0) if α ∈ Q(A). Hence

H

∫φdmS = H

∫Q(A†)

φ(γα(0))hα(0)dq(α) ≤∫Q(A†)

φ(γα(0))H−S γα(0)hα(0)dq(α) =

∫φH−S dmS .

and consequently H ≤ H−S mS-almost everywhere.

2. Assuming inner mean curvature bounded from below by H we have already observed in theprove of Theorem 1.1 that for q-a.e. α ∈ Q† one has hα(0) > 0. Using Fatou’s lemma again, itfollows that

H

∫φdp0 =

∫V0φ(α)Hhα g(α, 0)dq(α) =

∫Q(A†)

φ(α)Hhα(0)dq(α)

≤∫V0φ(α)

d−

dtlog hα(0)hα(0)dq(α)

≤ lim inft↑0

∫φ(α)

1

t(1Vt(α)hα(t)− 1V0(α)hα(0)) dq(α)

≤ lim supt↑0

∫φ(α)

1

t(1Vt(α)hα(t)− 1V0(α)hα(0)) dq(α)

= lim supt↑0

1

t

(∫φdpt −

∫φdp0

).

Hence, the backward mean curvature is bounded from below H.

We state a theorem under MCP . The corresponding statement for CD then follows since CDimplies MCP for essentially nonbranching proper metric measure spaces.

Theorem B.4 (Inradius bounds under backward mean curvature bounded below). Let (X, d,m)be an essentially nonbranching MCP (K,N) space with K ∈ R, N ∈ (1,∞) and spt m = X. LetΩ ⊂ X be closed with Ω 6= X, m(Ω) > 0 and m(∂Ω) = 0. Assume ∂Ω = S has backward meancurvature bounded from below by H ∈ R. Then

InRad Ω ≤ rK,H,N .

Proof. As in the previous appendix we have

hα(t)1

N−1 ≥ σ(1− ta(Xα) )

K/N−1 (a(Xα))hα(0)1

N−1

for an t ∈ (a(Xα, 0) and any α ∈ Q†. Therefore, it follows that

d−

dt|t=0hα g(α, t) = lim sup

t↑0

1

t(hα(g(α, t))− hα(g(α, 0))) ≤ d−

dt

∣∣∣t=0


K/N−1 (a(Xα))N−1hα(g(α, 0))

Since the backward mean curvature is bounded below by H, it follows that

H


∫V0φ(α)

d−

dt|t=0hα g(α, t)dq(α).(22)


We obtain again the inequality (21) exactly as in the beginning of step 1 of the proof of LemmaB.3 (where we did not use finite inner curvature), and this yields again hα(0) > 0 for q-almostevery α. Hence, it follows that

H


∫V0φ(α)

d−

dt

∣∣∣t=0

log(hα g(α, t))hα g(α, 0)dq(α)(23)

Hence HN−1 ≤

d−

dt

∣∣∣t=0

log hα g(α, t) ≤ cosK/(N−1)(a(Xα))

sinK/(N−1)(a(Xα)) for q-a.e. α ∈ V0 = Q(A ∪Bin).

At this point it is clear that we can finish the proof as before.

Theorem B.5 (Rigidity under backward mean curvature bounded from below). Let (X, d,m) beRCD(K,N) for K ∈ R and N ∈ (1,∞) and let Ω ⊂ X be compact with Ω 6= X, m(Ω) > 0,connected and non-empty interior Ω and m(∂Ω) = 0. We assume that K ∈ N − 1, 0,−(N − 1),∂Ω = S 6= pt and S has backward mean curvature bounded below by κ(N − 1) ∈ R. Then, thereexists x ∈ X such that

dS(x) = InRad Ω = rK,κ(N−1),N

if and only if (K,κ) ∈ −(N−1), 0× (0,∞)∪N−1×R and there exists an RCD(N−2, N−1)

space Y such that (Ω, dΩ) is isometric to (BrK,κ(N−1),N(0), d) in I K

N−1×N−1

sinK/(N−1)Y , where dΩ

and d are the induced intrinsic distances of Ω and BrK,κ(N−1),N(0), respectively.

Proof. We observe that the inequality (23) implies H ≤ d−

dt

∣∣∣t=0

log hα g(α, t) for q-a.e. α ∈ V0,

so in particular for α ∈ Q(Bin). Then using the Riccatti comparison and the maximum principlewe can follow verbatim the same proof as in Section 4.

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Department of Mathematics, Radboud University, PO Box 9010, Postvak 59, 6500 GL Nijmegen, The

Netherlands

E-mail address: [email protected]

Department of Mathematics, University of Toronto, 40 St George St, Toronto Ontario, Canada

M5S 2E4E-mail address: [email protected]

Department of Mathematics, University of Toronto, 40 St George St, Toronto Ontario, Canada

M5S 2E4E-mail address: [email protected]

Department of Mathematical and Statistical Sciences and Theoretical Physics Institute, Universityof Alberta, Edmonton AB, Canada T6G 2G1

E-mail address: [email protected]

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