aa r X i v : . [ m a t h . DG ] A ug Needle decompositions in Riemannian geometry
Bo’az Klartag ∗ Abstract
The localization technique from convex geometry is generalized to the setting of Rieman-nian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method isbased on the following observation: When the Ricci curvature is non-negative, log-concavemeasures are obtained when conditioning the Riemannian volume measure with respect toan integrable geodesic foliation. The Monge mass transfer problem plays an important rolein our analysis.
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12 Regularity of geodesic foliations . . . . . . . . . . . . . . . . . . . . . . . . C , . . . . . . . . . . . . . . . . . . . . 112.3 Riemann normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Proof of the regularity theorem . . . . . . . . . . . . . . . . . . . . . . . 21 . . C , -hypersurface . . . . . . . . . . . . . . . 273.2 Decomposition into ray clusters . . . . . . . . . . . . . . . . . . . . . . . 373.3 Needles and Ricci curvature . . . . . . . . . . . . . . . . . . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . . . . .
535 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]
Introduction
The localization technique in convex geometry is a method for reducing n -dimensionalproblems to one-dimensional problems, that was developed by Gromov and Milman [21],Lov´asz and Simonovits [30] and Kannan, Lov´asz and Simonovits [26]. Its earliest appear-ance seems to be found in the work of Payne and Weinberger [34], where the followinginequality is stated: For any bounded, open, convex set K ⊂ R n and an integrable, C -function f : K → R , Z K f = 0 = ⇒ Z K f ≤ Diam ( K ) π Z K |∇ f | , (1)where Diam ( K ) = sup x,y ∈ K | x − y | is the diameter of K , and | · | is the standard Euclideannorm in R n . The localization proof of (1) goes roughly as follows: Given f with R K f = 0 ,one finds a hyperplane H ⊂ R n such that R K ∩ H + f = R K ∩ H − f = 0 , where H − , H + ⊂ R n are the two half-spaces determined by the hyperplane H . The problem of proving (1) isreduced to proving the two inequalities: Z K ∩ H ± f ≤ Diam ( K ∩ H ± ) π Z K ∩ H ± |∇ f | . The next step is to again bisect each of the two half-spaces separately, retaining the require-ment that the integral of f is zero. Thus one recursively obtains finer and finer partitions of R n into convex cells. At the k th step, the proof of (1) is reduced to k “smaller” problemsof a similar nature. At the limit, the original problem is reduced to a lower-dimensionalproblem, and eventually even to a one-dimensional problem. This one-dimensional problemhas turned out to be relatively simple to solve.This bisection technique has no clear analog in the context of an abstract Riemannianmanifold. The purpose of this manuscript is to try and bridge this gap between convexgeometry and Riemannian geometry.There are only two parameters of a given Riemannian manifold that play a role in ouranalysis: the dimension of the manifold, and a uniform lower bound κ for its Ricci curvature.We say that an n -dimensional Riemannian manifold M satisfies the curvature-dimensioncondition CD ( κ, N ) for κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] if Ric M ( v, v ) ≥ κ · g ( v, v ) for p ∈ M , v ∈ T p M , (2)where g is the Riemannian metric tensor and Ric M is the Ricci tensor of M . The con-tribution of Bakry and ´Emery [2] has made it clear that weighted Riemannian manifoldsare convenient for the study of curvature-dimension conditions. A weighted Riemannianmanifold is a triplet ( M , d, µ ) , where M is an n -dimensional Riemannian manifold withRiemannian distance function d , and where the measure µ has a smooth, positive density e − ρ with respect to the Riemannian volume measure on M . The generalized Ricci tensor ofthe weighted Riemannian manifold ( M , d, µ ) is defined via Ric µ ( v, v ) := Ric M ( v, v ) + Hess ρ ( v, v ) for p ∈ M , v ∈ T p M , (3) here Hess ρ is the Hessian form associated with the smooth function ρ : M → R . For N ∈ ( −∞ , ∪ [ n, + ∞ ] , p ∈ M and v ∈ T p M we define the generalized Ricci tensor withparameter N as follows: Ric µ,N ( v, v ) := Ric µ ( v, v ) − ( ∂ v ρ ) N − n N = n, + ∞ Ric µ ( v, v ) N = + ∞ Ric M ( v, v ) N = n, ρ ≡ Const (4)The standard agreement is that
Ric µ,n ( v, v ) is undefined unless ρ is a constant function. For κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] we say that ( M , d, µ ) satisfies the curvature-dimensioncondition CD ( κ, N ) when Ric µ,N ( v, v ) ≥ κ · g ( v, v ) for p ∈ M , v ∈ T p M . For instance, the CD (0 , ∞ ) -condition is equivalent to the requirement that the generalizedRicci tensor be non-negative. We refer the reader to Bakry, Gentil and Ledoux [4] for back-ground on weighted Riemannian manifolds of class CD ( κ, N ) . In this manuscript, a mini-mizing geodesic is a curve γ : A → M , where A ⊆ R is a connected set, such that d ( γ ( s ) , γ ( t )) = | s − t | for all s, t ∈ A. Definition 1.1.
Let κ ∈ R , = N ∈ R ∪ {∞} and let ν be a measure on the Riemannianmanifold M . We say that ν is a “ CD ( κ, N ) -needle” if there exist a non-empty, connectedopen set A ⊆ R , a smooth function Ψ : A → R and a minimizing geodesic γ : A → M such that:(i) Denote by θ the measure on A ⊆ R whose density with respect to the Lebesgue mea-sure is e − Ψ . Then ν is the push-forward of θ under the map γ .(ii) The following inequality holds in the entire set A : Ψ ′′ ≥ κ + (Ψ ′ ) N − , (5) where in the case N = ∞ , we interpret the term (Ψ ′ ) / ( N − as zero. Condition (5) is equivalent to condition CD ( κ, N ) for the weighted Riemannian mani-fold ( A, d, θ ) with d ( x, y ) = | x − y | . Examples of needles include:1. Log-concave needles which are defined to be CD (0 , ∞ ) -needles. In this case, Ψ is aconvex function. Log-concave needles are valuable when studying the uniform mea-sure on convex sets in R n for large n .2. A sin n -concave needle is a CD ( n − , n ) -needle. These are relevant to the sphere S n ,since the n -dimensional unit sphere is of class CD ( n − , n ) .3. The N -concave needles are CD (0 , N + 1) -needles with N > . Here, f /N is aconcave function, where f = e − Ψ is the density of the measure θ . For N < , the CD (0 , N + 1) -condition is equivalent to the convexity of f − / | N | .4. A κ -log-concave needle is a CD ( κ, ∞ ) -needle. hese examples are discussed by Gromov [24, Section 4]. We say that the Riemannianmanifold M is geodesically-convex if any two points in M may be connected by a minimiz-ing geodesic. By the Hopf-Rinow theorem, any complete, connected Riemannian manifoldis geodesically-convex. A partition of M is a collection of non-empty disjoint subsets of M whose union equals M . Theorem 1.2 (“Localization theorem”) . Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . As-sume that ( M , d, µ ) is an n -dimensional weighted Riemannian manifold of class CD ( κ, N ) which is geodesically-convex. Let f : M → R be a µ -integrable function with R M f dµ = 0 .Assume that there exists a point x ∈ M with R M | f ( x ) | · d ( x , x ) dµ ( x ) < ∞ .Then there exist a partition Ω of M , a measure ν on Ω and a family { µ I } I∈ Ω of measureson M such that:(i) For any Lebesgue-measurable set A ⊆ M , µ ( A ) = Z Ω µ I ( A ) dν ( I ) (In particular, the map I 7→ µ I ( A ) is well-defined ν -almost everywhere and it is a ν -measurable map). In other words, we have a “disintegration of the measure µ ”.(ii) For ν -almost any I ∈ Ω , the set I ⊆ M is the image of a minimizing geodesic, themeasure µ I is supported on I , and either I is a singleton or else µ I is a CD ( κ, N ) -needle.(iii) For ν -almost any I ∈ Ω we have R I f dµ I = 0 . We demonstrate in Section 5 that Theorem 1.2 may be used in order to obtain alter-native proofs of some familiar inequalities from convex and Riemannian geometry. Theseinclude the isoperimetric inequality, the Poincar´e and log-Sobolev inequalities, the Payne-Weiberger/Yang-Zhong inequality, the inequality of Cordero-Erausquin, McCann and Schmuck-enschlaeger, among others. Some of these inequalities are consequences of the followingRiemannian analog of the four functions theorem of Kannan, Lov´asz and Simonovits [26]:
Theorem 1.3 (“The four functions theorem”) . Let n ≥ , α, β > , κ ∈ R , N ∈ ( −∞ , ∪ [ n, + ∞ ] . Let ( M , d, µ ) be an n -dimensional weighted Riemannian manifold of class CD ( κ, N ) which is geodesically-convex. Let f , f , f , f : M → [0 , + ∞ ) be measurable functionssuch that there exists x ∈ M with Z M ( | f ( x ) | + | f ( x ) | + | f ( x ) | + | f ( x ) | ) · (1 + d ( x , x )) dµ ( x ) < ∞ . Assume that f α f β ≤ f α f β almost-everywhere in M and that for any probability measure η on M which is a CD ( κ, N ) -needle, (cid:18)Z M f dη (cid:19) α (cid:18)Z M f dη (cid:19) β ≤ (cid:18)Z M f dη (cid:19) α (cid:18)Z M f dη (cid:19) β (6) whenever f , f , f , f are η -integrable. Then, (cid:18)Z M f dµ (cid:19) α (cid:18)Z M f dµ (cid:19) β ≤ (cid:18)Z M f dµ (cid:19) α (cid:18)Z M f dµ (cid:19) β . (7) heorem 1.2 was certainly known in the case where M = R n or M = S n − . How-ever, even in these symmetric spaces, our proof of Theorem 1.2 is very different from thetraditional bisection proofs given in Gromov and Milman [21] or Lov´asz and Simonovits[30]. The geodesic foliations that we construct in Theorem 1.2 are integrable , meaning thatthere is a function u : M → R such that the geodesics appearing in the partition are integralcurves of ∇ u . This integrability property makes the construction of the partition somewhatmore “canonical”. In contrast, there are many arbitrary choices that one makes during thebisection process, as there could be many hyperplanes that bisect a domain in R n into twosubsets of equal volumes. For a function u : M → R we define its Lipschitz seminorm by k u k Lip = sup x = y ∈M | u ( x ) − u ( y ) | d ( x, y ) . Given a -Lipschitz function u : M → R and a point y ∈ M , we say that y is a strain point of u if there exist x, z ∈ M for which u ( y ) − u ( x ) = d ( x, y ) > , u ( z ) − u ( y ) = d ( y, z ) > , d ( x, z ) = d ( x, y ) + d ( y, z ) . Write
Strain [ u ] ⊆ M for the collection of all strain points of u . The set Strain [ u ] resemblesthe transport set defined at the beginning of Section 3 in Evans and Gangbo [17]. It isexplained below that Strain [ u ] is a measurable subset of M . It is also proven below that therelation x ∼ y ⇐⇒ | u ( x ) − u ( y ) | = d ( x, y ) is an equivalence relation on Strain [ u ] , and that each equivalence class is the image of aminimizing geodesic. Write T ◦ [ u ] for the collection of all equivalence classes. It followsthat for any I ∈ T ◦ [ u ] there exists a minimizing geodesic γ : A → M with γ ( A ) = I and u ( γ ( t )) = t for all t ∈ A. (8)Let π : Strain [ u ] → T ◦ [ u ] be the partition map, i.e., x ∈ π ( x ) ∈ T ◦ [ u ] for all x ∈ Strain [ u ] .The conditioning of µ with respect to the geodesic foliation T ◦ [ u ] is described in the follow-ing theorem: Theorem 1.4.
Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) isan n -dimensional weighted Riemannian manifold of class CD ( κ, N ) which is geodesically-convex. Let u : M → R satisfy k u k Lip ≤ . Then there exist a measure ν on the set T ◦ [ u ] and a family { µ I } I∈ T ◦ [ u ] of measures on M such that:(i) For any Lebesgue-measurable set A ⊆ M , the map I 7→ µ I ( A ) is well-defined ν -almost everywhere and is a ν -measurable map. If a subset S ⊆ T ◦ [ u ] is ν -measurablethen π − ( S ) ⊆ Strain [ u ] is a measurable subset of M .(ii) For any Lebesgue-measurable set A ⊆ M , µ ( A ∩ Strain [ u ]) = Z T ◦ [ u ] µ I ( A ) dν ( I ) . (iii) For ν -almost any I ∈ T ◦ [ u ] , the measure µ I is a CD ( κ, N ) -needle supported on I ⊆ M . Furthermore, the set A ⊆ R and the minimizing geodesic γ : A → M fromDefinition 1.1 may be selected so that I = γ ( A ) and so that (8) holds true. e call the -Lipschitz function u from Theorem 1.4 the guiding function of the needle-decomposition. In the case where the function u from Theorem 1.4 is the distance functionfrom a smooth hypersurface, the conclusion of Theorem 1.4 is essentially a classical com-putation in Riemannian geometry which may be found in Gromov [22, 23], Heintze andKarcher [25] and Morgan [33]. That computation is related to Paul Levy’s proof of theisoperimetric inequality. It is beneficial to analyze arbitrary Lipschitz functions in Theo-rem 1.4, because of the relation to the dual Monge-Kantorovich problem presented in thefollowing: Theorem 1.5 (“Localization theorem with a guiding function”) . Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) is an n -dimensional weighted Rieman-nian manifold of class CD ( κ, N ) which is geodesically-convex. Let f : M → R be a µ -integrable function with R M f dµ = 0 . Assume that there exists a point x ∈ M with R M | f ( x ) | · d ( x , x ) dµ ( x ) < ∞ . Then,(A) There exists a -Lipschitz function u : M → R such that Z M uf dµ = sup k v k Lip ≤ Z M vf dµ. (9) (B) For any such function u , the function f vanishes µ -almost everywhere in M\ Strain [ u ] .Furthermore, let ν and { µ I } I∈ T ◦ [ u ] be measures on T ◦ [ u ] and M , respectively, satis-fying conclusions (i), (ii) and (iii) of Theorem 1.4. Then for ν -almost any I ∈ T ◦ [ u ] , Z I f dµ I = 0 . (10) (C) For any such function u , there exist Ω , ν, { µ I } I∈ Ω satisfying the conclusions of Theo-rem 1.2, which also satisfy the following property: For ν -almost any I ∈ Ω , there exista connected set A ⊆ R and a minimizing geodesic γ : A → M with γ ( A ) = I and u ( γ ( t )) = t for all t ∈ A. Our manuscript owes much to previous investigations of the Monge-Kantorovich prob-lem. An integrable foliation by straight lines satisfying an analog of (10) was mentionedalready by Monge in 1781, albeit on a heuristic level (see, e.g., Cayley’s review of Monge’swork [10]). The optimization problem (9) entered the arena with the work of Kantorovich[27, Section VIII.4].An analytic resolution of the Monge-Kantorovich problem which is satisfactory for ourneeds is provided by Evans and Gangbo [17], with subsequent developments by Ambrosio[1], Caffarelli, Feldman and McCann [9], Feldman and McCann [18] and Trudinger andWang [36]. Ideas from these papers have helped us in dealing with the following difficulty:We are obliged to work with the second fundamental form of the level set { u = t } inorder to use the Ricci curvature and conclude that µ I is a CD ( κ, N ) -needle. However, thefunction u is an arbitrary Lipschitz function, and it is not entirely clear how to interpret itsHessian. Section 2 is devoted to overcoming this difficulty, by showing that inside the set Strain [ u ] the function u behaves as if it were a C , -function. The conditioning of µ with espect to the partition T ◦ [ u ] is discussed in Section 3, in which we prove Theorem 1.4.Section 4 is dedicated to the proofs of Theorem 1.2 and Theorem 1.5.Throughout this note, by a smooth function or manifold we always mean C ∞ -smooth.All differentiable manifolds are assumed smooth and all of our Riemannian manifolds havesmooth metric tensors. We do not consider Riemannian manifolds with a boundary. Whenwe mention a measure ν on a set X we implicitly consider a σ -algebra of ν -measurablesubsets of X . All of our measures in this paper are complete , meaning that if ν ( A ) = 0 and B ⊆ A , then B is ν -measurable. When we push-forward the measure ν , we implicitlyalso push-forward its σ -algebra. Note that the concept of a Lebesgue-measurable subsetof a differentiable manifold is well-defined (e.g., Section 3.1 below). When we write “ameasurable set”, without any reference to a specific measure, we simply mean Lebesgue-measurable. We write log for the natural logarithm. Acknowledgements.
I would like to thank Emanuel Milman for introducing me to thesubject of Riemannian manifolds with lower bounds on their Ricci curvature. Supported bya grant from the European Research Council.
Let M be an n -dimensional Riemannian manifold which is geodesically-convex and let d be the Riemannian distance function on M . As before, a curve γ : I → M is a minimizinggeodesic if I ⊆ R is a connected subset and d ( γ ( s ) , γ ( t )) = d ( s, t ) for all s, t ∈ I. A curve γ : J → M is a geodesic if J ⊆ R is connected, and for any x ∈ J there existsa relatively-open subset I ⊆ J containing x such that γ | I is a minimizing geodesic. Thus,we only discuss geodesics of speed one, and not of arbitrary speed as is customary. For thebasic concepts in Riemannian geometry that we use here we refer the reader, e.g., to the firstten pages of Cheeger and Ebin [12]. In particular, it is well-known that all geodesic curvesare smooth, and that for p ∈ M and a unit vector v ∈ T p M there is a unique geodesic curve γ p,v with γ p,v (0) = p and ˙ γ p,v (0) = v . Let I p,v ⊆ R be the maximal set on which γ p,v iswell-defined, which is an open, connected set containing zero. Denote exp p ( tv ) = γ p,v ( t ) for t ∈ I p,v . The exponential map exp p : T p M → M is a partially-defined function, which is well-defined and smooth on an open subset of T p M containing the origin. Lemma 2.1.
Let A ⊆ R be an arbitrary subset, and let γ : A → M satisfy d ( γ ( s ) , γ ( t )) = | s − t | for all s, t ∈ A. (1) Denote conv ( A ) = { λt + (1 − λ ) s ; s, t ∈ A, ≤ λ ≤ } . Then there exists a minimizinggeodesic ˜ γ : conv ( A ) → M with ˜ γ | A = γ . roof. We may assume that A ) ≥ , because if A contains only two points then we mayconnect them by a minimizing geodesic. Fix s ∈ A with inf A < s < sup A . According to(1), for any r, t ∈ A with r < s < t , d ( γ ( r ) , γ ( s )) + d ( γ ( s ) , γ ( t )) = d ( γ ( r ) , γ ( t )) . (2)Denote a = γ ( r ) , b = γ ( s ) , c = γ ( t ) . Select any minimizing geodesic γ from a to b ,and any minimizing geodesic γ from b to c . We claim that γ and γ make a zero angleat the point b . Indeed by (2), the concatenation of the curves γ and γ forms a minimizinggeodesic from a to c , which is necessarily smooth, hence the curves γ and γ must fittogether at the point b . We conclude that there exists a unit vector v ∈ T γ ( s ) M , such that forany x ∈ A \ { s } , the vector sgn( x − s ) v is tangent to any minimizing geodesic from γ ( s ) to γ ( x ) . Here, sgn( x ) is the sign of x ∈ R \ { } . Denote ˜ γ ( x ) = exp γ ( s ) (( x − s ) v ) . Then ˜ γ is the geodesic emanating from γ ( s ) in the direction of v , and it satisfies ˜ γ ( x ) = γ ( x ) for any x ∈ A . The geodesic curve ˜ γ is thus well-defined on the interval conv ( A ) , with ˜ γ | A = γ . Furthermore, it follows from (1) that the geodesic ˜ γ : conv ( A ) → M is aminimizing geodesic, and the lemma is proven.The following definition was proposed by Evans and Gangbo [16] who worked under theassumption that M is a Euclidean space, see Feldman and McCann [18] for the generaliza-tion to complete Riemannian manifolds. Definition 2.2.
Let u : M → R be a function with k u k Lip ≤ . A subset I ⊆ M is a“transport ray” associated with u if | u ( x ) − u ( y ) | = d ( x, y ) for all x, y ∈ I (3) and if for any J ) I there exist x, y ∈ J with | u ( x ) − u ( y ) | 6 = d ( x, y ) . In other words, I is a maximal set that satisfies condition (3). We write T [ u ] for the collection of all transportrays associated with u . By continuity, the closure of a transport ray is also a transport ray, and by maximalityany transport ray is a closed set. By Zorn’s lemma, any subset
I ⊆ M satisfying (3)is contained in a certain transport ray. For the rest of this subsection, we fix a function u : M → R with k u k Lip ≤ . The following lemma shows that transport rays are geodesicarcs in M on which u grows at speed one. For a map F defined on a set A we write F ( A ) = { F ( x ) ; x ∈ A } . Lemma 2.3.
Any
J ∈ T [ u ] is the image of a minimizing geodesic γ : A → M , where A = u ( J ) is a connected set in R , and we have u ( γ ( t )) = t for t ∈ A. (4) roof. Denote A = u ( J ) ⊆ R . From (3) the map u : J → A is invertible. By defining γ ( u ( x )) = x for x ∈ J , we see from (3) that d ( γ ( s ) , γ ( t )) = | s − t | for any s, t ∈ A. (5)We may apply Lemma 2.1 in view of (5), and conclude that γ may be extended to a curve ˜ γ : conv ( A ) → M which is a minimizing geodesic. Furthermore, since k u k Lip ≤ with u ( γ ( t )) = t for t ∈ A , then necessarily u (˜ γ ( t )) = t for t ∈ conv ( A ) . (6)The curve ˜ γ is a minimizing geodesic, and its image I = ˜ γ ( conv ( A )) satisfies (3), thanksto (6). The maximality property of J entails that I = J and A = conv ( A ) . Consequently J is the image of the minimizing geodesic γ ≡ ˜ γ , and (4) follows from (6).Lemma 2.3 states that we may identify between a transport ray I ⊆ M and the imageof a certain minimizing geodesic γ : A → M . When we write that a unit vector v ∈ T M istangent to I we mean that v = ˙ γ ( t ) for some t ∈ A . We say that { γ ( t ) ; t ∈ int ( A ) } is the relative interior of the transport ray I , where int ( A ) ⊆ R is the interior of the set A ⊆ R . Note that a transport ray I could be a singleton, and then its relative interior turnsout to be empty. The set { γ ( t ) ; t ∈ A \ int ( A ) } is defined to be the relative boundary of the transport ray I . Since A ⊆ R is connected, thenthe relative boundary of any transport ray contains at most two points. The short proof of thefollowing lemma appears in Feldman and McCann [18, Lemma 10]: Lemma 2.4.
For any transport ray
I ∈ T [ u ] and a point x in the relative interior of I , thefunction u is differentiable at x , and ∇ u ( x ) is a unit vector tangent to I . In this subsection we define the set
Strain [ u ] ⊆ M to be the union of all relative interiorsof transport rays associated with u . Very soon we will show that this definition, in fact,coincides with the definition of Strain [ u ] provided in Section 1. Lemma 2.5.
For any x ∈ Strain [ u ] there exists a unique I ∈ T [ u ] such that x ∈ I .Furthermore, x belongs to the relative interior of I .Proof. From Lemma 2.4 we know that u is differentiable at x and that ∇ u ( x ) is a unit vector.Consider the geodesic ˜ γ ( t ) = exp x ( t ∇ u ( x )) (7)which is well-defined in a maximal subset ( a, b ) ⊆ R containing zero. Define A = { t ∈ ( a, b ) ; u (˜ γ ( t )) = u ( x ) + t } . (8)Note that ∈ A . Since ˜ γ is a geodesic and k u k Lip ≤ , then A is necessarily connected and ˜ γ : A → M is a minimizing geodesic. In fact, by (8) the set ˜ γ ( A ) is contained in a certaintransport ray. e will show that ˜ γ ( A ) is the unique transport ray containing x . Indeed, x ∈ Strain [ u ] and hence there exists I ∈ T [ u ] with x ∈ I . Since x is contained in the relative interiorof a certain transport ray, then I is not a singleton by the maximality property of transportrays. Note that ∇ u ( x ) is necessarily tangent to I : this follows from equation (4) of Lemma2.3 and from the fact that ∇ u ( x ) is a unit vector. We conclude from (7), (8) and Lemma2.3 that I ⊆ ˜ γ ( A ) . However, we said earlier that ˜ γ ( A ) is contained in a transport ray, andby maximality I = ˜ γ ( A ) . Therefore ˜ γ ( A ) is the unique transport ray containing x . Since x ∈ Strain [ u ] then the point x necessarily belongs to the relative interior of the transport ray ˜ γ ( A ) .For a point y ∈ Strain [ u ] define α u ( y ) = u ( y ) − inf z ∈J u ( z ) , β u ( y ) = (cid:20) sup z ∈J u ( z ) (cid:21) − u ( y ) , where J ∈ T [ u ] is the unique transport ray containing y . For y Strain [ u ] we set α u ( y ) = β u ( y ) = −∞ . Thus, the functions α u , β u are positive on Strain [ u ] , and equal to −∞ outside Strain [ u ] . Lemma 2.3 and Lemma 2.4 admit the following immediate corollary: Corollary 2.6.
Let y ∈ Strain [ u ] . Set A = ( − α u ( y ) , β u ( y )) ⊆ R . Then there exists aminimizing geodesic γ : A → M whose image is the relative interior of a transport ray,such that γ (0) = y and for all t ∈ A , u ( γ ( t )) = u ( y ) + t, ˙ γ ( t ) = ∇ u ( γ ( t )) . Recall that the set
Strain [ u ] = { x ∈ M ; α u ( x ) > } = { x ∈ M ; β u ( x ) > } wasdefined a bit differently in Section 1. The equivalence of the two definitions follows fromour next little lemma: Lemma 2.7.
Let y ∈ M . Then α u ( y ) equals the supremum over all ε > for which thereexist x, z ∈ M with d ( x, y ) = u ( y ) − u ( x ) ≥ ε, d ( y, z ) = u ( z ) − u ( y ) > , d ( x, y ) + d ( y, z ) = d ( x, z ) . (9) The supremum over an empty set is defined to be −∞ .Proof. Write ˜ α u ( y ) for the supremum over all ε > for which there exist x, z ∈ M suchthat (9) holds. We need to show that α u ( y ) = ˜ α u ( y ) for all y ∈ M . (10)Corollary 2.6 implies that α u ( y ) ≤ ˜ α u ( y ) for any y ∈ Strain [ u ] . Clearly α u ( y ) ≤ ˜ α u ( y ) for any y Strain [ u ] , since α u ( y ) = −∞ for such y . It thus remains to prove the “ ≥ ”inequality between the terms in (10). To this end, we fix y ∈ M for which ˜ α u ( y ) > −∞ .Then there exist x, z ∈ M satisfying (9) with some ε > . The triplet I = { x, y, z } satisfies I is contained in a transport ray J , and the point y must belong tothe relative interior of J as u ( x ) < u ( y ) < u ( z ) . By Lemma 2.5, the point y does not belong to any transport ray other than J . Additionally,any points x, z ∈ M satisfying (9) must belong to the transport ray J . It follows fromCorollary 2.6 that ˜ α u ( y ) ≤ α u ( y ) , and (10) is proven.A transport ray which is a singleton is called a degenerate transport ray. According toLemma 2.3, a transport ray I ∈ T [ u ] is non-degenerate if and only if its relative interior isnon-empty. Lemma 2.8.
The following relation is an equivalence relation on Strain [ u ] : x ∼ y ⇐⇒ | u ( x ) − u ( y ) | = d ( x, y ) . (11) As in Section 1, we write T ◦ [ u ] for the collection of all equivalence classes. Then T ◦ [ u ] isthe collection of all relative interiors of non-degenerate transport rays.Proof. According to Lemma 2.5, The collection of all relative interiors of non-degeneratetransport rays is a partition of
Strain [ u ] . Let x, y ∈ Strain [ u ] . We need to show that x ∼ y ifand only if x and y belong to the relative interior of the same transport ray.Assume first that x ∼ y . Then I = { x, y } satisfies (3), and hence there exists a transportray J ∈ T [ u ] such that x, y ∈ J . However, x, y ∈ Strain [ u ] and J is a transport raycontaining x and y . From Lemma 2.5 we conclude that x and y belong to the relative interiorof J . Conversely, suppose that x, y ∈ Strain [ u ] belong to the relative interior of a certaintransport ray J ∈ T [ u ] . By (11) and Definition 2.2, we have x ∼ y . The proof is complete.A σ -compact set is a countable union of compact sets. A topological space is second-countable if its topology has a countable basis of open sets. Note that any geodesically-convex, Riemannian manifold M is second-countable: Indeed, since M is a metric space, itsuffices to find a countable, dense subset. Fix a ∈ M and a countable, dense subset of T a M .Since M is geodesically-convex, the image of the latter subset under exp a is a countable,dense subset of M . Therefore M is second-countable, and any open cover of any subset S ⊆ M has a countable subcover. Since M is locally-compact and second-countable, it is σ -compact.Define ℓ u ( y ) = min { α u ( y ) , β u ( y ) } for y ∈ M . Then ℓ u is positive on Strain [ u ] , and itequals −∞ outside Strain [ u ] . Lemma 2.9.
The functions α u , β u , ℓ u : M → R ∪ {±∞} are Borel-measurable.Proof. We will only prove that α u is Borel-measurable. The argument for β u is similar,while ℓ u is Borel-measurable as ℓ u = min { α u , β u } . For ε, δ > we define A ε,δ to be thecollection of all triplets ( x, y, z ) ∈ M with d ( x, y ) = u ( y ) − u ( x ) ≥ ε, d ( y, z ) = u ( z ) − u ( y ) ≥ δ, d ( x, y ) + d ( y, z ) = d ( x, z ) . hen A ε,δ is a closed set, by the continuity of u and of the distance function. The Riemannianmanifold M is σ -compact, hence there exist compacts K ⊆ K ⊆ . . . such that M = ∪ i K i . Define A i,ε,δ = A ε,δ ∩ ( K i × K i × K i ) ( i ≥ , ε > , δ > . Note that A i,ε,δ is compact and hence π ( A i,ε,δ ) is also compact, where π ( x, y, z ) = y .Clearly, A ε,δ = ∪ i A i,ε,δ . Let α i,ε,δ : M → R ∪ {−∞} be the function that equals ε on thecompact set π ( A i,ε,δ ) and equals −∞ otherwise. Then α i,ε,δ is a Borel-measurable functionand by Lemma 2.7, for any y ∈ M , α u ( y ) = sup { ε > ∃ δ > , y ∈ π ( A ε,δ ) } = sup { α i,ε,δ ( y ) ; ε, δ ∈ Q ∩ (0 , ∞ ) , i ≥ } . Hence α u is the supremum of countably many Borel-measurable functions, and is thus nec-essarily Borel-measurable.For ε > denote Strain ε [ u ] = { x ∈ M ; ℓ u ( x ) > ε } . Thus, Strain [ u ] = [ ε> Strain ε [ u ] = { x ∈ M ; ℓ u ( x ) > } . The function u is basically an arbitrary Lipschitz function, yet the following theorem assertshigher regularity of u inside the set Strain [ u ] . Denote B M ( p, δ ) = { x ∈ M ; d ( x, p ) < δ } . Theorem 2.10.
Let M be a geodesically-convex Riemannian manifold. Let u : M → R bea function with k u k Lip ≤ . Let p ∈ M , ε > . Then there exist δ > and a C , -function ˜ u : B M ( p, δ ) → R such that for any x ∈ M , x ∈ B M ( p, δ ) ∩ Strain ε [ u ] = ⇒ ˜ u ( x ) = u ( x ) , ∇ ˜ u ( x ) = ∇ u ( x ) . (12)Section 2.2 contains the standard background on C , -functions. In Section 2.3 we dis-cuss the Riemann normal coordinates, and in Section 2.4 we complete the proof of Theorem2.10. Our proof of Theorem 2.10 is related to the arguments of Evans and Gangbo [17] andto the contributions by Ambrosio [1], Caffarelli, Feldman and McCann [9], Feldman andMcCann [18] and Trudinger and Wang [36]. The new ingredient in our analysis is the use ofWhitney’s extension theorem. C , Given a function f : R n → R we write ∂ i f = ∂f /∂x i for its i th partial derivative, so that ∇ f = ( ∂ f, . . . , ∂ n f ) . Denote by | · | the standard Euclidean norm in R n , and x · y is theusual scalar product of x, y ∈ R n . For an open, convex set K ⊆ R n and a C -function ϕ = ( ϕ , . . . , ϕ m ) : K → R m we set k ϕ k C , = sup x ∈ K (cid:0) | ϕ ( x ) | + k ϕ ′ ( x ) k op (cid:1) + sup x = y ∈ K k ϕ ′ ( x ) − ϕ ′ ( y ) k op | x − y | , (1) here the derivative ϕ ′ ( x ) is an m × n matrix whose ( i, j ) -entry is ∂ j ϕ i ( x ) , and k A k op = sup = v ∈ R n | Av | / | v | is the operator norm. Similarly, we may define the C , -norm of a function ϕ : K → Y ,where X and Y are finite-dimensional linear spaces with inner products and where K ⊆ X is an open, convex set. In fact, formula (1) remains valid in the latter scenario, yet in thiscase we need to interpret ϕ ′ ( x ) as a linear map from X to Y and not as a matrix. For anopen set U ⊆ R n , we say that f : U → R m is a C , -function if for any x ∈ U there exists δ > such that (cid:13)(cid:13)(cid:13) f | B ( x,δ ) (cid:13)(cid:13)(cid:13) C , < ∞ where f | B ( x,δ ) is the restriction of f to the open ball B ( x, δ ) = { y ∈ R n ; | y − x | < δ } . Inother words, a C -function f : U → R m is a C , -function if and only if the derivative f ′ isa locally-Lipschitz map into the space of m × n matrices. Any C -function f : U → R m is automatically a C , -function. A map ϕ : U → V is a C , -diffeomorphism, for opensets U, V ⊆ R n , if ϕ is an invertible C , -map and the inverse map ϕ − : V → U is also C , . The C -version of the following lemma may be found in any textbook on multivariatecalculus. Lemma 2.11. (i) Let U ⊆ R n and U ⊆ R m be open sets. Let f : U → R k and f : U → U be C , -functions. Then f ◦ f is also a C , -function.(ii) Let U ⊆ R n be an open set and let f : U → R n be a C , -function. Assume that x ∈ U is such that det f ′ ( x ) = 0 . Then there exists δ > such that f | B ( x ,δ ) is a C , -diffeomorphism onto some open set V ⊆ R n .(iii) Let U ⊆ R n be an open set and let f : U → R be a C , -function. Assume that x ∈ U is such that ∇ f ( x ) = 0 . Then there exists an open set V ⊆ U containing thepoint x , an open set Ω ⊆ R n − × R of the form Ω = Ω × ( a, b ) ⊆ R n − × R and a C , -diffeomorphism G : Ω → V such that for any ( y, t ) ∈ Ω , f ( G ( y, t )) = t. Proof. (i) We know that h = f ◦ f is a C -function. The map x f ′ ( f ( x )) islocally-Lipschitz, since it is the composition of two locally-Lipschitz maps. Since f ′ is locally-Lipschitz, the product h ′ ( x ) = f ′ ( f ( x )) · f ′ ( x ) is also locally-Lipschitz.Hence h is a C , -function.(ii) The usual inverse function theorem for C guarantees the existence of δ > and anopen set V ⊆ R n such that f : B ( x , δ ) → V is a C -diffeomorphism. Let g : V → B ( x , δ ) be the inverse map. The map g ′ ( x ) = ( f ′ ( g ( x ))) − is the composition ofthree locally-Lipschitz maps, hence it is locally-Lipschitz and g is C , .(iii) This follows from (ii) in exactly the same way that the implicit function theorem fol-lows from the inverse function theorem in the C case, see e.g. Edwards [15, ChapterIII.3]. emma 2.11(i) shows that the concept of a C , -function on a differentiable manifold iswell-defined: Definition 2.12.
Let M and N be differentiable manifolds. A function f : M → N isa C , -function if f is C , in any local chart. A C , -function f : M → N is a C , -diffeomorphism if it is invertible and the inverse function f − : N → M is also C , . Let K ⊆ R n be an open, convex set and let f : K → R satisfy M := k f k C , < ∞ . Itfollows from the definition (1) that for x, y ∈ K , |∇ f ( x ) − ∇ f ( y ) | ≤ M | x − y | . (2)For x, y ∈ K we also have, denoting x t = (1 − t ) x + ty , | f ( x ) + ∇ f ( x ) · ( y − x ) − f ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z [ ∇ f ( x ) − ∇ f ( x t )] · ( y − x ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ M | x − y | . (3)Conditions (2) and (3), which are basically Taylor’s theorem for C , -functions, capture theessence of the concept of a C , -function, as is demonstrated in Theorem 2.13 below. Forpoints x, y ∈ R n and for f : { x, y } → R and V : { x, y } → R n we define k ( f, V ) k x,y to bethe infimum over all M ≥ for which the following three conditions hold:(i) | f ( x ) | ≤ M, | V ( x ) | ≤ M ,(ii) | V ( y ) − V ( x ) | ≤ M | y − x | ,(iii) | f ( x ) + V ( x ) · ( y − x ) − f ( y ) | ≤ M | y − x | .This infimum is in fact a minimum. Note that k ( f, V ) k x,y is not necessarily the same as k ( f, V ) k y,x . Theorem 2.13 (Whitney’s extension theorem for C , ) . Let A ⊆ R n be an arbitrary set, let f : A → R and V : A → R n . Assume that sup x,y ∈ A k ( f, V ) k x,y < ∞ . (4) Then there exists a C , -function ˜ f : R n → R such that for any x ∈ A , ˜ f ( x ) = f ( x ) , ∇ ˜ f ( x ) = V ( x ) . For a proof of Theorem 2.13 see Stein [35, Chapter VI.2.3] or the original paper byWhitney [37]. Whitney’s theorem is usually stated under the additional assumption that A ⊆ R n is a closed set, but it is straightforward to extend f and V from A to the closure A by continuity, preserving the validity of assumption (4).Given a differentiable manifold M and a subset A ⊆ M , a -form on A is a map ω : A → T ∗ M with ω ( x ) ∈ T ∗ x M for x ∈ A . Let M , N be differentiable manifolds andlet ϕ : M → N be a C -map. For a -form ω on A ⊆ N we write ϕ ∗ ω for the pull-back of under the map ϕ . Thus ϕ ∗ ω is a -form on ϕ − ( A ) . Write R n ∗ for the space of all linearfunctionals from R n to R . With any ℓ ∈ R n ∗ we associate the vector V ℓ ∈ R n which satisfies ℓ ( x ) = x · V ℓ for any x ∈ R n . Since T ∗ x ( R n ) is canonically isomorphic to R n ∗ , any -form ω on a subset A ⊆ R n may beidentified with a map ω : A → R n ∗ . Defining V ω ( x ) := V ω ( x ) ∈ R n we recall the formula V ϕ ∗ ω ( x ) = ϕ ′ ( x ) ∗ · V ω ( ϕ ( x )) , (5)where B ∗ is the transpose of the matrix B . Here, ω is a -form on a subset A ⊆ R m , thefunction ϕ is a C -map from an open set U ⊆ R n to R m , and the formula (5) is valid forany x ∈ ϕ − ( A ) . For x, y ∈ R n and for f : { x, y } → R , ω : { x, y } → R n ∗ we define k ( f, ω ) k x,y = k ( f, V ω ) k x,y . Lemma 2.14.
Let K , K ⊆ R n be open, convex sets. Let R ≥ and let ϕ : K → K bea C -diffeomorphism with k ϕ − k C , ≤ R. (6) Let x, y ∈ K , denote A = { x, y } , let f : A → R , and let ω : A → R n ∗ be a -form on A .Denote ˜ A = ϕ − ( A ) , ˜ ω = ϕ ∗ ω, ˜ f = f ◦ ϕ , and ˜ x = ϕ − ( x ) , ˜ y = ϕ − ( y ) . Then, k ( f, ω ) k x,y ≤ C n,R (cid:13)(cid:13)(cid:13) ( ˜ f , ˜ ω ) (cid:13)(cid:13)(cid:13) ˜ x, ˜ y , where C n,R > is a constant depending solely on n and R .Proof. It follows from (1), (6) and the convexity of K that the map ψ := ϕ − is R -Lipschitz. Thus, | ˜ y − ˜ x | = | ψ ( y ) − ψ ( x ) | ≤ R | y − x | . (7)Set V = V ω : A → R n and ˜ V = V ˜ ω : ˜ A → R n . Since ˜ ω = ϕ ∗ ω then ω = ψ ∗ ˜ ω and from(5), V ( x ) = ψ ′ ( x ) ∗ · ˜ V (˜ x ) . Denote M = k ( ˜ f , ˜ ω ) k ˜ x, ˜ y = k ( ˜ f , ˜ V ) k ˜ x, ˜ y . It suffices to show that f and V satisfy conditions(i), (ii) and (iii) from the definition of k ( f, V ) k x,y with M replaced by M ( R + nR + 1) .To that end, observe that | f ( x ) | = | ˜ f (˜ x ) | ≤ M, | V ( x ) | = | ψ ′ ( x ) ∗ · ˜ V (˜ x ) | ≤ M R. (8)Thus condition (i) is satisfied. To prove condition (ii), we compute that | V ( y ) − V ( x ) | = (cid:12)(cid:12)(cid:12) ψ ′ ( y ) ∗ ˜ V (˜ y ) − ψ ′ ( x ) ∗ ˜ V (˜ x ) (cid:12)(cid:12)(cid:12) (9) ≤ (cid:12)(cid:12)(cid:12) ψ ′ ( y ) ∗ ( ˜ V (˜ y ) − ˜ V (˜ x )) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( ψ ′ ( y ) ∗ − ψ ′ ( x ) ∗ ) ˜ V (˜ x ) (cid:12)(cid:12)(cid:12) ≤ RM ( | ˜ y − ˜ x | + | y − x | ) . Condition (ii) holds in view of (7) and (9). Denote ψ = ( ψ , . . . , ψ n ) . From (3) and (7), | f ( x ) + V ( x ) · ( y − x ) − f ( y ) | = | ˜ f (˜ x ) + ψ ′ ( x ) ∗ ˜ V (˜ x ) · ( y − x ) − ˜ f (˜ y ) |≤ | ˜ f (˜ x ) + ˜ V (˜ x ) · (˜ y − ˜ x ) − ˜ f (˜ y ) | + | ˜ V (˜ x ) | · (cid:12)(cid:12) ψ ′ ( x )( y − x ) − ( ψ ( y ) − ψ ( x )) (cid:12)(cid:12) ≤ M | ˜ x − ˜ y | + M n X i =1 |∇ ψ i ( x ) · ( y − x ) − ( ψ i ( y ) − ψ i ( x )) | ≤ ( M R + nM R ) | y − x | . ondition (iii) is thus satisfied and the lemma is proven. Corollary 2.15.
Let M be an n -dimensional differentiable manifold, let R ≥ and let U ⊆ M be an open set. Assume that for any a ∈ U we are given a convex, open set U a ⊆ R n and a C , -diffeomorphism ϕ a : U a → U . Suppose that for any a, b ∈ U , k ϕ − b ◦ ϕ a k C , ≤ R. (10) Let A ⊆ U . Let f : A → R and let ω be a -form on A . For a ∈ U set f a = f ◦ ϕ a and w a = ϕ ∗ a w . Suppose that for any x, y ∈ A there exists a ∈ U for which k ( f a , ω a ) k ϕ − a ( x ) ,ϕ − a ( y ) ≤ R. (11) Then there exists a C , -function ˜ f : U → R with ˜ f | A = f, d ˜ f | A = ω, (12) where d ˜ f is the differential of the function ˜ f .Proof. Fix b ∈ U and denote A b = ϕ − b ( A ) ⊆ U b ⊆ R n . Abbreviate ϕ b,a = ϕ − a ◦ ϕ b . Let x, y ∈ A b ⊆ R n . According to (11) there exists a ∈ U for which k ( f a , ω a ) k ϕ b,a ( x ) ,ϕ b,a ( y ) ≤ R. (13)We may apply Lemma 2.14, thanks to (10) and (13), and conclude that for any x, y ∈ A b , k ( f b , ω b ) k x,y ≤ C n,R , (14)for some C n,R > depending only on n and R . Recall that for any linear functional ℓ ∈ R n ∗ there corresponds a vector V ℓ ∈ R n defined via ℓ ( z ) = V ℓ · z ( z ∈ R n ) . In particular, for x ∈ A b we have ω b ( x ) ∈ R n ∗ and let us set V b ( x ) := V ω b ( x ) ∈ R n .According to (14), the function f b : A b → R and the vector field V b : A b → R n satisfy sup x,y ∈ A b k ( f b , V b ) k x,y ≤ C n,R < ∞ . Theorem 2.13 thus produces a C , -function ˜ f b : U b → R with ˜ f b ( x ) = f b ( x ) , ∇ ˜ f b ( x ) = V b ( x ) ( x ∈ A b ) . In particular d ˜ f b | A b = ω b . Setting ˜ f ( x ) = ˜ f b ( ϕ − b ( x )) for x ∈ U , we obtain a function ˜ f : U → R satisfying (12). The function ˜ f is a C , -function since it is the composition oftwo C , -functions. Remark 2.16.
Corollary 2.15 admits the following formal generalization: Rather than stipu-lating that U a is a subset of R n for any a ∈ U , we may assume that U a ⊆ X a , where X a is an n -dimensional linear space with an inner product. This generalization is completely straight-forward, and it does not involve any substantial modifications to neither the formulation northe proof of Corollary 2.15. .3 Riemann normal coordinates Let M be an n -dimensional Riemannian manifold with Riemannian distance function d . For a ∈ M we write h· , ·i for the Riemannian scalar product in T a M , and |·| is the norm inducedby this scalar product. Given a C -function g : T a M → R and a point X ∈ T a M we mayspeak of the gradient ∇ g ( X ) ∈ T a M and of the Hessian operator ∇ g ( X ) : T a M → T a M ,which is a symmetric operator such that g ( Y ) = g ( X ) + h∇ g ( X ) , Y − X i + 12 (cid:10) ∇ g ( X )( Y − X ) , Y − X (cid:11) + o ( | Y − X | ) . (1)On a very formal level, since T a M is a linear space, we canonically identify T X ( T a M ) ∼ = T a M for any X ∈ T a M . Therefore the gradient ∇ g ( X ) belongs to T a M ∼ = T X ( T a M ) .A subset U ⊆ M is strongly convex if for any two points x, y ∈ U there exists a unique minimizing geodesic in M that connects x and y , and furthermore this minimizing geodesicis contained in U , while there are no other geodesic curves contained in U that join x and y . See, e.g., Chavel [11, Section IX.6] for more information. The following standard lemmaexpresses the fact that a Riemannian manifold is “locally-Euclidean”. Lemma 2.17.
Let M be a Riemannian manifold and let p ∈ M . Then there exists δ = δ ( p ) > such that the following hold:(i) For any x ∈ B M ( p, δ ) and < δ ≤ δ , the ball B M ( x, δ ) is strongly convex and itsclosure is compact.(ii) Denote U = B M ( p, δ / and for a ∈ U set U a = exp − a ( U ) . Then U a ⊆ T a M is abounded, open set and exp a is a smooth diffeomorphism between U a and U .(iii) Define f a,X ( Y ) = · d (exp a X, exp a Y ) for a ∈ U, X, Y ∈ U a . Then f a,X : U a → R is a smooth function, and its Hessian operator ∇ f a,X satisfies · Id ≤ ∇ f a,X ( Y ) ≤ · Id ( a ∈ U, X, Y ∈ U a ) , (2) in the sense of symmetric operators, where Id is the identity operator.(iv) For any a, x ∈ U and < δ ≤ δ , the set exp − a ( B M ( x, δ )) is a convex subset of T a M . In particular, U a is convex.(v) For any a ∈ U, X, Y ∈ U a , · | X − Y | ≤ d (exp a X, exp a Y ) ≤ · | X − Y | . (vi) For a, b ∈ U consider the transition map ϕ a,b : U a → U b defined by ϕ a,b = exp − b ◦ exp a .Then, sup a,b ∈ U k ϕ a,b k C , < ∞ . (3) Proof.
We will see that the conclusions of the lemma hold for any sufficiently small δ , i.e.,there exists ˜ δ > such that the conclusions of the lemma hold for any < δ < ˜ δ . For a ∈ M , X ∈ T a M and δ > we define B T a M ( X, δ ) = { Y ∈ T a M ; | X − Y | < δ } . tem (i) is the content of Whitehead’s theorem, see [12, Theorem 5.14] or [11, TheoremIX.6.1]. Regarding (ii), the openness of U a and the fact that exp a : U a → U is a smoothdiffeomorphism are standard, see [12, Chapter I]. Furthermore, U a ⊆ B T a M (0 , δ ) , andhence U a is bounded and (ii) holds true.We move to item (iii). The function f a,X ( Y ) := d (exp a X, exp a Y ) / is a smoothfunction, which depends smoothly also on a ∈ U and X ∈ U a . The Hessian operator of f p, at the point ∈ T p M is precisely the identity, as follows from (1) and [12, Corollary1.9]. By smoothness, the Hessian operator of f a,X at the point Y ∈ T a M is at least · Id and at most Id , whenever a is sufficiently close to p and X, Y are sufficiently close to zero.In other words, assuming that δ is at most a certain positive constant determined by p , weknow that for a ∈ B M ( p, δ ) and X, Y ∈ B T a M (0 , δ ) , · Id ≤ ∇ f a,X ( Y ) ≤ · Id. (4)Thus (iii) is proven. It follows from (4) that the function f a,X is convex in the Euclidean ball B T a M (0 , δ ) . Let a, x ∈ U and < δ ≤ δ . Then B M ( x, δ ) ⊆ B M ( a, δ ) . Denoting X = exp − a ( x ) we observe that { Y ∈ T a M ; f a,X ( Y ) ≤ δ / } = exp − a ( B M ( x, δ )) ⊆ B T a M (0 , δ ) . (5)Since f a,X is convex in B T a M (0 , δ ) , then (5) implies that the set exp − a ( B M ( x, δ )) is con-vex. Therefore (iv) is proven. Thanks to the convexity of U a we may use Taylor’s theorem,and conclude from (2) that for a ∈ U, X, Y ∈ U a , · | X − Y | ≤ | f a,X ( Y ) − ( f a,X ( X ) + ∇ f a,X ( X ) · ( Y − X )) | ≤ | X − Y | . (6)However f a,X ( X ) = 0 , and also ∇ f a,X ( X ) = 0 since Y f a,X ( Y ) attains its minimum atthe point X . Therefore (v) follows from (6). Finally, the smooth map ϕ a,b = exp − b ◦ exp a : U a → U b smoothly depends also on a, b ∈ U . Since the closure of U is compact, thecontinuous function k ϕ a,b k C , is bounded over a, b ∈ U , and (3) follows.For the rest of this subsection, we fix a point p ∈ M , and let δ > be the radius whoseexistence is guaranteed by Lemma 2.17. Set U = B M ( p, δ / and U a = exp − a ( U ) for a ∈ U . When we say that a constant C depends on p , we implicitly allow this constant todepend on the choice of δ , on the Riemannian structure of M and on the dimension n .Since T X ( T a M ) ∼ = T a M for any a ∈ M and X ∈ T a M , we may view the differentialof the map exp a at the point X ∈ T a M as a map dexp X : T a M → T x M , where x = exp a ( X ) . We define Π x,a : T x M → T a M to be the adjoint map, where weidentify T x M ∼ = T ∗ x M and T a M ∼ = T ∗ a M by using the Riemannian scalar products. Inother words, for V ∈ T x M we define Π x,a ( V ) ∈ T a M via h Π x,a ( V ) , W i a = h V, dexp X ( W ) i x for all W ∈ T a M . (7) ere, h· , ·i a is the Riemannian scalar product in T a M , and h· , ·i x is the Riemannian scalarproduct in T x M . Following Feldman and McCann [18], for a ∈ U and X, Y ∈ U a wedenote x = exp a ( X ) , y = exp a ( Y ) and define F a ( X, Y ) := exp − x y. It follows from Lemma 2.17 that the vector F a ( X, Y ) ∈ U x is well-defined, as x, y ∈ U and exp x : U x → U is a diffeomorphism. Equivalently, F a ( X, Y ) is the unique vector V ∈ U x ⊆ T x M for which exp x ( V ) = y . Given a ∈ U and X, Y ∈ U a we define −−→ XY = Π x,a ( F a ( X, Y )) ∈ T a M . (8)Intuitively, we think of −−→ XY as a vector in T a M which represents “how exp a ( Y ) is viewedfrom exp a ( X ) ”. Lemma 2.18.
Let f : U → R , t ∈ R , a ∈ U and X, Y ∈ U a . Denote x = exp a ( X ) , y =exp a ( Y ) . Assume that f is differentiable at x with ∇ f ( x ) = t · F a ( X, Y ) and set f a = f ◦ exp a . Then ∇ f a ( X ) = t · −−→ XY .Proof. Let us pass to -forms. Then df a = exp ∗ a ( df ) , and for any W ∈ T a M , h∇ f a ( X ) , W i a = ( df a ) X ( W ) = ( df ) x (dexp X ( W )) (9) = h∇ f ( x ) , dexp X ( W ) i x = h tF a ( X, Y ) , dexp X ( W ) i x . From (7) and (9) we obtain that ∇ f a ( X ) = Π x,a ( tF a ( X )) = t Π x,a ( F a ( X )) . The lemmathus follows from (8). Lemma 2.19.
Let a ∈ U, X, Y ∈ U a . Assume that there exists α ∈ R such that X = αY .Then, −−→ XY = Y − X, (10) and |−−→ XY | = d (exp a X, exp a Y ) . (11) Proof.
Let Z ∈ T a M be a unit vector such that X and Y are proportional to Z . Write γ ( t ) = exp a ( tZ ) for the geodesic leaving a in direction Z . Then exp a ( X ) and exp a ( Y ) lieon this geodesic and by the strong convexity of U , d (exp a ( X ) , exp a ( Y )) = | X − Y | . Therefore (11) would follow once we prove (10). In order to prove (10) we denote x =exp a ( X ) and claim that h Y − X, Z i a = h F a ( X, Y ) , dexp X ( Z ) i x . (12)Indeed, F a ( X, Y ) ∈ T x M is a vector of length d (exp a X, exp a Y ) = | Y − X | which istangential to the curve γ . The vector dexp X ( Z ) ∈ T x M is a unit tangent to γ . Therefore a ( X, Y ) is proportional to the unit vector dexp X ( Z ) , in exactly the same way that Y − X is proportional to the unit vector Z . Thus (12) follows. The Gauss lemma [12, Lemma 1.8]states that for any W ∈ T a M , h Z, W i a = 0 = ⇒ h dexp X ( Z ) , dexp X ( W ) i x = 0 . (13)Recall that −−→ XY = Π x,a ( F a ( X, Y )) and that F a ( X, Y ) is proportional to the unit vector dexp X ( Z ) . From (7) and (13) we learn that −−→ XY = βZ for some β ∈ R . From (7) and (12), h Y − X, Z i a = h F a ( X, Y ) , dexp X ( Z ) i x = h−−→ XY , Z i a = h βZ, Z i a = β. (14)Since X and Y are proportional to the unit vector Z , then −−→ XY = h Y − X, Z i a · Z = Y − X according to (14). Thus (10) is proven. Lemma 2.20.
Let a ∈ U and t ∈ R . Assume that V, Z ∈ U a are such that t V ∈ U a .Then, in the notation of Lemma 2.17(iii), f a,t V ( Z ) ≤ f a,t V ( V ) + h (1 − t ) V, Z − V i + | Z − V | . (15) Proof.
Fix X , Y ∈ U a and define x = exp a ( X ) ∈ U, y = exp a ( Y ) ∈ U . Considerthe function g x ( y ) = · d ( x , y ) , defined for y ∈ U . Then ∇ g x ( y ) equals the vector V ∈ U y ⊆ T y M for which x = exp y ( − V ) . Consequently, ∇ g x ( y ) = − exp − y ( x ) = − F a ( Y , X ) . (16)Since f a,X = g x ◦ exp a , then from (16) and Lemma 2.18, ∇ f a,X ( Y ) = −−−−→ Y X . (17)According to (17) and Lemma 2.19, if X, Y ∈ U a lie on the same line through the origin,then ∇ f a,X ( Y ) = −−−→ Y X = − ( X − Y ) = Y − X. In particular, ∇ f a,t V ( V ) = V − t V = (1 − t ) V. (18)We may use Taylor’s theorem in the convex set U a ⊆ T a M , and deduce from the bound (2)in Lemma 2.17(iii) that | f a,t V ( Z ) − ( f a,t V ( V ) + h∇ f a,t V ( V ) , Z − V i ) | ≤ · · | Z − V | . (19)Now (15) follows from (18) and (19). emma 2.21. Let a ∈ U and X, X , X , Y, Y , Y ∈ U a . Then, (cid:12)(cid:12)(cid:12) −−→ XY − −−→ XY − ( Y − Y ) (cid:12)(cid:12)(cid:12) ≤ C p · | X | · | Y − Y | , (20) and (cid:12)(cid:12)(cid:12) −−→ X Y − −−→ X Y − ( X − X ) (cid:12)(cid:12)(cid:12) ≤ C p · | Y | · | X − X | . (21) Here, C p > is a constant depending on p .Proof. For a ∈ U, X, Y ∈ U a denote H a,X ( Y ) = −−→ XY − Y. (22)Then H a,X : U a → T a M is a smooth function. Since T a M is a linear space, then at thepoint Y ∈ U a the derivative H ′ a,X ( Y ) is a linear operator from the space T a M to itself. Weclaim that there exists a constant C p > depending on p such that (cid:13)(cid:13) H ′ a,X ( Y ) − H ′ a,X ( Y ) (cid:13)(cid:13) op ≤ C p · | X − X | for a ∈ U, X , X , Y ∈ U a , (23)where k S k op = sup = V | S ( V ) | / | V | is the operator norm. Write L ( T a M ) for the space oflinear operators on T a M , equipped with the operator norm. For a ∈ U, Y ∈ U a the map U a ∋ X H ′ a,X ( Y ) ∈ L ( T a M ) (24)is a smooth map. In fact, the map in (24) may be extended smoothly to the larger domain a ∈ B M ( p, δ ) , X, Y ∈ exp − a ( B M ( p, δ )) . Since U a is convex with a compact closure,the smooth map in (24) is necessarily a Lipschitz map, and the Lipschitz constant of thismap depends continuously on a ∈ U and Y ∈ U a . Since the closure of U is compact, theLipschitz constant of the map in (24) is bounded over a ∈ U and Y ∈ U a . This completesthe proof of (23). From (22) and Lemma 2.19, H a, ( Y ) = 0 for any Y ∈ U a . (25)From (25) we have H ′ a, ( Y ) = 0 for any Y ∈ U a . The set U a is convex, and by applying(23) with X = X and X = 0 we obtain sup Y ,Y ∈ UaY = Y | H a,X ( Y ) − H a,X ( Y ) || Y − Y | = sup Y ∈ U a (cid:13)(cid:13) H ′ a,X ( Y ) (cid:13)(cid:13) op ≤ C p · | X | for all X ∈ U a , and (20) is proven. In order to prove (21), one needs to analyze ˜ H a,Y ( X ) = −−→ XY + X .According to Lemma 2.19 we know that ˜ H a, ( X ) = 0 for any X ∈ U a . The latter equalityreplaces (25), and the rest of the proof of (21) is entirely parallel to the analysis of H a,X presented above. .4 Proof of the regularity theorem In this subsection we prove Theorem 2.10. We begin with a geometric lemma:
Lemma 2.22 (Feldman and Mccann [18]) . Let M be a Riemannian manifold with distancefunction d , and let p ∈ M . Then there exists δ = δ ( p ) > with the following property:Let x , x , x , y , y , y ∈ B M ( p, δ ) . Assume that there exists σ > such that d ( x i , x j ) = d ( y i , y j ) = σ | i − j | ≤ d ( x i , y j ) for i, j ∈ { , , } . (1) Then, max { d ( x , y ) , d ( x , y ) } ≤ · d ( x , y ) . (2)Together with Whitney’s extension theorem, Lemma 2.22 is the central ingredient in ourproof of Theorem 2.10. The proof of Lemma 2.22 provided by Feldman and McCann in [18,Lemma 16] is very clear and detailed, yet the notation is a bit different from ours. For theconvenience of the reader, their proof is reproduced in the Appendix below.Let us recall the assumptions of Theorem 2.10. The Riemannian manifold M is geodesically-convex and the function u : M → R satisfies k u k Lip ≤ . We are given a point p ∈ M anda number ε > . Set: δ = min (cid:26) C p , δ , δ (cid:27) > (3)where C p is the constant from Lemma 2.21, the constant δ = δ ( p ) is provided by Lemma2.17, and δ = δ ( p ) is the constant from Lemma 2.22. As before, we denote for a ∈ U , U = B M ( p, δ / , U a = exp − a ( U ) ⊆ T a M . Recall from the previous subsection that U ⊆ M is strongly convex, and that for a ∈ U and X, Y ∈ U a we defined a certain vector −−→ XY ∈ T a M . Lemma 2.23.
Let ε, σ > . Let x, x , x , x , y , y , y ∈ B M ( p, δ ) ⊆ U . Assume that d ( x, y ) = ε , that x lies on the geodesic arc between x and x , and that for i, j ∈ { , , } , d ( x i , x j ) = d ( y i , y j ) = σ | i − j | ≤ d ( x i , y j ) . (4) Denote a = x and let X, X , X , X , Y , Y , Y ∈ U a = exp − a ( U ) be such that x =exp a ( X ) and x i = exp a ( X i ) , y i = exp a ( Y i ) for i = 0 , , . Then, (cid:12)(cid:12)(cid:12) −−→ Y Y − −−−→ X X (cid:12)(cid:12)(cid:12) ≤ · ε, (5) and |h X , Y − X i| ≤ · ε . (6) roof. From (4), the point x is the midpoint of the geodesic arc between x and x . Thepoint x also lies on the geodesic between x and x . Let K ∈ { , } be such that x lies onthe geodesic from x to x K . According to (4), d ( x , x ) + d ( x, x K ) = d ( x , x K ) = σ. (7)From (4) and (7), σ ≤ d ( x K , y ) ≤ d ( x K , x ) + d ( x, y ) = ( σ − d ( x, x )) + d ( x, y ) . (8)By using (8) and our assumption that d ( x, y ) = ε we obtain d ( x , y ) ≤ d ( x , x ) + d ( x, y ) ≤ d ( x, y ) = 2 ε. (9)We would like to apply Lemma 2.22. Recall from (3) that δ ≤ δ , where δ = δ ( p ) isthe constant from Lemma 2.22. Therefore x , x , x , y , y , y ∈ B M ( p, δ ) . Moreover,assumption (1) holds in view of (4). We may therefore apply Lemma 2.22, and according toits conclusion, d ( x i , y i ) ≤ · d ( x , y ) ≤ ε ( i = 0 , , , (10)where we used (9) in the last passage. By Lemma 2.17(v), the inequality (10) yields | X i − Y i | ≤ ε ( i = 0 , , . (11)Since a = x and exp a ( X ) = x , then X = 0 . According to Lemma 2.19, for i = 0 , , , | Y i | = |−−−→ X Y i | = d ( x , y i ) ≤ δ , | X i | = |−−−→ X X i | = d ( x , x i ) ≤ δ , (12)as x , x , x , y , y , y ∈ B M ( p, δ ) . From Lemma 2.21 combined with (11) and (12), (cid:12)(cid:12)(cid:12) −−→ Y Y − −−−→ X Y − ( X − Y ) (cid:12)(cid:12)(cid:12) ≤ C p · | Y | · | Y − X | ≤ C p · δ · ε ≤ ε, (13)where we used the fact that δ C p ≤ / in the last passage, as follows from (3). Similarly,according to Lemma 2.21 and the inequalities (11) and (12), (cid:12)(cid:12)(cid:12) −−−→ X Y − −−−→ X X − ( Y − X ) (cid:12)(cid:12)(cid:12) ≤ C p · | X | · | Y − X | ≤ C p · δ · ε ≤ ε. (14)Finally, by using (11), (13) and (14), |−−→ Y Y − −−−→ X X | = | ( −−→ Y Y − −−−→ X Y ) + ( −−−→ X Y − −−−→ X X ) |≤ ε + | ( X − Y ) + ( Y − X ) | ≤ ε + | X − Y | + | Y − X | ≤ ε, and (5) is proven. We move on to the proof of (6). For a ∈ U and W, Z ∈ U a define d a ( W, Z ) := d (exp a W, exp a Z ) . (15)Then d a ( W, Z ) = 2 f a,W ( Z ) , in the notation of Lemma 2.17(iii). Using Lemma 2.20 with V = X , t = 0 and Z = Y , d a ( X , Y ) ≤ d a ( X , X ) + h X , Y − X i + 2 | Y − X | . (16) rom (4) and (15), d a ( X , Y ) = d ( x , y ) ≥ d ( x , x ) = d a ( X , X ) . Therefore (16) entails h X , Y − X i ≥ −| Y − X | . (17)Since x is the midpoint of the geodesic between a = x and x , then x = exp a ( X ) =exp a (2 X ) . Hence X = 2 X . By using Lemma 2.20 with V = X , t = 2 and Z = Y we obtain d a ( X , Y ) ≤ d a ( X , X ) + h− X , Y − X i + 2 | Y − X | . (18)As before, from (4) and (15) we deduce that d a ( X , Y ) ≥ d a ( X , X ) . Therefore (18) leadsto h X , Y − X i ≤ | Y − X | . (19)The desired conclusion (6) follows from (11), (17) and (19). Proof of Theorem 2.10.
Denote σ = min { ε / , δ / } . (20)We will prove the theorem with δ = min { σ/ , } . (21)We would like to apply Whitney’s extension theorem, in the form of Corollary 2.15 andRemark 2.16. Denote ϕ a = exp a : U a → U for any a ∈ U . Then ϕ a is a smoothdiffeomorphism between the convex, open set U a ⊆ T a M and the open set U ⊆ M . Thanksto Lemma 2.17(vi), there exists a constant R = R p > depending on p with the followingproperty: For any a, b ∈ U , condition (10) from Corollary 2.15 holds true. Furthermore,since u is a Lipschitz function, R := 1 + sup x ∈ B M ( p,δ ) | u ( x ) | < ∞ . (22)Denote A = { x ∈ B M ( p, δ ) ; ℓ u ( x ) > ε } = B M ( p, δ ) ∩ Strain ε [ u ] . (23)Then A ⊆ U = B M ( p, δ / according to (3), (20) and (21). The function u is differentiableon the entire set A , according to Lemma 2.4. Define a -form ω on A by setting ω = du | A .We will verify that the scalar function u : A → R and the -form ω on the set A satisfycondition (11) from Corollary 2.15. In fact, for any x, y ∈ A we will show that there exists a ∈ U for which k ( u a , ω a ) k ϕ − a ( x ) ,ϕ − a ( y ) ≤ max (cid:26) R , σ (cid:27) , (24)where u a = u ◦ ϕ a and ω a = ϕ ∗ a ω . Once we prove (24), the theorem easily follows: Theright-hand side of (24) depends on the point p and on the function u , but not on the choice f x, y ∈ A . Thus condition (11) of Corollary 2.15 is satisfied. From the conclusion ofCorollary 2.15, there exists a C , -function ˜ u : U → R with ˜ u | A = u | A , d ˜ u | A = ω = du | A . (25)Since U ⊇ B M ( p, δ ) , the theorem follows from (23) and (25). Therefore, all that remains isto show that for any x, y ∈ A there exists a ∈ U for which (24) holds true.Let us fix x, y ∈ A . Since ℓ u ( x ) > ε ≥ σ and also ℓ u ( y ) > σ then by Corollary 2.6there exist minimizing geodesics γ x , γ y : ( − σ, σ ) → M with γ x (0) = x, γ y (0) = y suchthat u ( γ x ( t )) = u ( x ) + t, u ( γ y ( t )) = u ( y ) + t, for t ∈ ( − σ, σ ) , (26)and such that ∇ u ( γ x ( t )) = ˙ γ x ( t ) , ∇ u ( γ y ( t )) = ˙ γ y ( t ) for t ∈ ( − σ, σ ) . (27)Recall that x, y ∈ A ⊆ B M ( p, δ ) . Denote ε := d ( x, y ) < δ ≤ σ. (28)Set t = u ( y ) − u ( x ) . Since u is -Lipschitz, then (28) implies that | t | < σ . We now define x i = γ x ( t + ( i − σ ) , y i = γ y (( i − σ ) for i = 0 , , . (29)Since | t | < σ then t + ( i − σ ∈ ( − σ, σ ) and the points x , x , x , y , y , y arewell-defined. Since t = u ( y ) − u ( x ) then (26) and (29) yield u ( x i ) = u ( y i ) = u ( x ) + iσ for i = 0 , , . (30)Recall that k u k Lip ≤ and that γ x , γ y are minimizing geodesics. We deduce from (29) and(30) that for i, j ∈ { , , } , d ( x i , x j ) = d ( y i , y j ) = σ | i − j | = | u ( x i ) − u ( y j ) | ≤ d ( x i , y j ) . (31)Since γ x (0) = x and | t | < σ , then by (29) the points x , x , x are of distance at most σ from x . Similarly, the points y , y , y are of distance at most σ from y = y . Since x, y ∈ B M ( p, δ ) we obtain x, x , x , x , y , y , y ∈ B ( p, δ ) ⊆ U, (32)as δ ≤ σ/ ≤ δ / . Recall from (29) that x = γ x ( t − σ ) and x = γ x ( t + σ ) . Since γ x (0) = x and | t | < σ , the point x lies on the geodesic arc from x to x . Furthermore, x
6∈ { x , x } . Thus all of the requirements of Lemma 2.23 are satisfied: This follows from(28), (31) and (32), as y = y . We are therefore permitted to use the conclusions of Lemma2.23. Denote a = x . s in Lemma 2.23 we define X, X , X , X , Y , Y , Y ∈ U a via x = exp a ( X ) and x i =exp a ( X i ) , y i = exp a ( Y i ) for i = 0 , , . Thus X = 0 . According to (28) and Lemma2.17(v), ε = d ( x, y ) = d ( x, y ) ≤ | X − Y | . (33)The four points a = x , x , x , x lie on the minimizing geodesic γ x , according to (29).Therefore the four vectors X , X , X , X lie on a line through the origin in T a M .Furthermore, since x and x lie on the geodesic arc between x and x , then X and X belong to the line segment between X and X . Since x is the midpoint of the geodesicbetween x and x , then x = exp a ( X ) = exp a (2 X ) . Hence, X = 2 X . (34)Since X lies on the line segment between the point X and the point X = 2 X while X
6∈ { X , X } , then there exists t ∈ (0 , σ ) such that X = X + ( t/σ ) · X . We claim that γ x ( t ) = exp a ( X + ( t/σ ) · X ) for t ∈ ( − σ, σ ) . (35)Indeed, since exp a ( X ) = x then | X | = d ( a, x ) = d ( x , x ) = σ according to (31) andthe strong convexity of U . Therefore t exp a ( X + ( t/σ ) · X ) is a geodesic of unit speed.Since γ x (0) = x = exp a ( X ) , then the equality in (35) holds true when t = 0 . The two unitspeed geodesics t γ x ( t ) and t exp a ( X + ( t/σ ) · X ) visit the point x at time t = 0 , andat a later time t ∈ (0 , σ ) they visit the point x . By strong convexity, these two geodesicscoincide, and (35) is proven. Next, from (31), (34) and Lemma 2.19, −−−→ XX = X − X = | X − X | · X | X | = | X − X | · X − X | X | = d ( x, x ) · −−−→ X X σ . (36)From (27) we see that ∇ u ( x ) is the unit tangent to the geodesic from x to x . Similarly, ∇ u ( y ) is the unit tangent to the geodesic from y = y to y . Thus, ∇ u ( x ) = F p ( X, X ) d ( x, x ) , ∇ u ( y ) = F p ( Y , Y ) d ( y , y ) = F p ( Y , Y ) σ , (37)where we used (31) in the last equality. Recall that u a ( Z ) = u ( ϕ a ( Z )) = u (exp a ( Z )) for Z ∈ U a . According to Lemma 2.18, (36) and (37), ∇ u a ( X ) = −−−→ XX d ( x, x ) = −−−→ X X σ = X σ , ∇ u a ( Y ) = −−→ Y Y σ . (38)From (26) and (35), the function u a = u ◦ exp a satisfies that u a (( t/σ ) X ) = u a (0) + t for all t ∈ [0 , σ ] . Since both X and X belong to the line segment between X = 0 and X = 2 X then u a ( X ) − u a ( X ) = (cid:28) X , X σ (cid:29) − (cid:28) X, X σ (cid:29) = (cid:28) X − X, X σ (cid:29) . (39)According to (38) and conclusion (5) of Lemma 2.23, |∇ u a ( X ) − ∇ u a ( Y ) | = 1 σ · |−−−→ X X − −−→ Y Y | ≤ ε/σ ≤ σ · | X − Y | , (40) here we used (33) in the last passage. Furthermore, conclusion (6) of Lemma 2.23 impliesthat |h X , Y − X i| ≤ ε ≤ | X − Y | , (41)where again we used (33) in the last passage. From (30) we know that u a ( X ) = u ( x ) = u ( y ) = u a ( Y ) . According to (38), (39) and (41), | u a ( X ) + h∇ u a ( X ) , Y − X i − u a ( Y ) | (42) = | u a ( X ) + h∇ u a ( X ) , X − X i + h∇ u a ( X ) , Y − X i − u a ( Y ) | = (cid:12)(cid:12)(cid:12)(cid:12) u a ( X ) + (cid:28) X σ , Y − X (cid:29) − u a ( Y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) X σ , Y − X (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ σ | X − Y | . From (22), (31) and (38), | u a ( X ) | ≤ R , |∇ u a ( X ) | = | X − X | σ = d ( x , x ) σ = 1 ≤ R . (43)Recall that ϕ a = exp a and that ω a = ϕ ∗ a ω = ϕ ∗ a ( du | A ) = du a | ϕ − a ( A ) . The inequalities(40), (42) and (43) mean precisely that k ( u a , ω a ) k ϕ − a ( x ) ,ϕ − a ( y ) = k ( u a , ω a ) k X,Y = k ( u a , ∇ u a ) k X,Y ≤ max (cid:26) R , σ (cid:27) . To summarize, given the arbitrary points x, y ∈ A , we found a ∈ U for which (24) holdstrue. The proof is thus complete.By using a partition of unity and a standard argument, we may deduce from Theorem2.10 the following corollary (which will not be needed here): Corollary 2.24.
Let M be a geodesically-convex Riemannian manifold. Let u : M → R satisfy k u k Lip ≤ and let ε > . Then there exists a C , -function ˜ u : M → R such thatfor any x ∈ M , x ∈ Strain ε [ u ] = ⇒ ˜ u ( x ) = u ( x ) , ∇ ˜ u ( x ) = ∇ u ( x ) . Let ( M , d, µ ) be a weighted Riemannian manifold of dimension n which is geodesically-convex. In this section we describe the conditioning of µ with respect to the partition T ◦ [ u ] associated with a given -Lipschitz function u . The conditioning is based on “ray clusters”which are defined in Section 3.1. Analogous constructions appear in Caffarelli, Feldman andMcCann [9], Evans and Gangbo [17], Feldman and McCann [18] and Trudinger and Wang[36]. Section 3.2 explains that the set Strain [ u ] may be partitioned into countably many rayclusters. The connection with curvature appears on Section 3.3. .1 Geodesics emanating from a C , -hypersurface In what follows we prefer to work with a slightly different normalization of the exponentialmap. For t ∈ R set Exp t ( v ) = exp p ( tv ) ( p ∈ M , v ∈ T p M ) . Then
Exp t : T M → M is a partially-defined map which is well-defined and smoothon a maximal open set containing the zero section. That is, for any v ∈ T M there is amaximal connected set I ⊆ R containing the origin such that Exp t ( v ) is well-defined for t ∈ I . This maximal connected subset I is always open, and if t ∈ I , then Exp s ( w ) iswell-defined for any ( w, s ) ∈ T M × R which is sufficiently close to ( v, t ) ∈ T M × R .Write dExp t : T ( T M ) → T M for the differential of the map Exp t : T M → M . Themaps
Exp t and dExp t are smooth in all of their variables, including the t -variable.Let γ : ( a, b ) → M be a smooth curve with a, b ∈ R ∪ {±∞} . We say that J : ( a, b ) → T M is a smooth vector field along γ if J is smooth and J ( t ) ∈ T γ ( t ) M for any t ∈ ( a, b ) . Asin Cheeger and Ebin [12, Section 1.1], we may use the Riemannian connection and considerthe covariant derivative of J along γ , denoted by J ′ = ∇ ˙ γ J. Then J ′ : ( a, b ) → T M is a well-defined, smooth vector field along γ . Assume that γ :( a, b ) → M is a geodesic. We say that a smooth vector field J along γ is a Jacobi field if J ′′ ( t ) = R ( ˙ γ ( t ) , J ( t )) ˙ γ ( t ) for t ∈ ( a, b ) , (1)where R is the Riemann curvature tensor . We refer the reader to Cheeger and Ebin [12,Chapter I] for background on the Jacobi equation (1). The space of Jacobi fields along thefixed geodesic curve γ is a linear space of dimension n . In fact, we may parameterizethe space of Jacobi fields along γ by the (2 n ) -dimensional vector space T ˙ γ (0) ( T M ) . Theparametrization is defined as follows: For ξ ∈ T ˙ γ (0) ( T M ) we define a Jacobi field J via J ( t ) = dExp t ( ξ ) for t ∈ ( a, b ) . (2)Let V : M → T M be a vector field on M , i.e., V ( p ) ∈ T p M for any p ∈ M .Assume that V is differentiable at the point p ∈ M . For w ∈ T p M we write ∂ w V ∈ T V ( p ) ( T M ) for the usual directional derivative of the map V : M → T M . We write ∇ w V ∈ T p M for the covariant derivative of V with respect to the Riemannian connection.Note the formal difference between the directional derivative ∂ w V ∈ T V ( p ) ( T M ) and thecovariant derivative ∇ w V ∈ T p M . In the case where M = R , the relation between ∂ w V and ∇ w V is rather like the relation between the tangent to the plane curve t ( t, f ( t )) andthe derivative of the scalar-valued function t f ( t ) . Lemma 3.1.
Let a ∈ [ −∞ , , b ∈ (0 , + ∞ ] , let γ : ( a, b ) → M be a geodesic and let ξ ∈ T ˙ γ (0) ( T M ) . Let J ( t ) be the Jacobi field along γ that is given by (2). Assume that V isa vector field on M that is differentiable at the point γ (0) ∈ M and satisfies ∂ J (0) V = ξ .Then, J ′ (0) = ∇ J (0) V. roof. Let β : ( − , → T M be a smooth, one-to-one curve satisfying β (0) = ˙ γ (0) and ˙ β (0) = ξ = ∂ J (0) V . A moment of contemplation reveals that ∇ J (0) β = ∇ J (0) V, where we use the conventions from [12, Section 1.1] regarding vector fields along a smoothmap and their covariant derivatives. Set α ( s, t ) = Exp t ( β ( s )) . Then α is smooth in ( s, t ) ∈ R near the origin, while J ( t ) = ∂α∂s (0 , t ) and β ( s ) = ∂α∂t ( s, . As in [12, Section 1.5] weabbreviate S = dα ( ∂∂s ) and T = dα ( ∂∂t ) , which are smooth vector fields along the map α with S (0 , t ) = J ( t ) and T ( s,
0) = β ( s ) . Then, J ′ (0) = ∇ T S | t,s =0 , ∇ J (0) V = ∇ J (0) β = ∇ S T | t,s =0 . (3)Since (cid:2) ∂∂s , ∂∂t (cid:3) = 0 then [ S, T ] = 0 and consequently ∇ S T = ∇ T S . The lemma thusfollows from (3).We say that a C -function f : M → R is twice differentiable with a symmetric Hessian at the point p ∈ M if the vector field ∇ f is differentiable at p and h∇ v ( ∇ f ) , w i = h∇ w ( ∇ f ) , v i for v, w ∈ T p M . The notation of the next lemma will accompany us now for several pages. We willconsider geodesics orthogonal to the level set { ˜ u = r } , where ˜ u : M → R is usuallytwice differentiable with a symmetric Hessian. This level set is locally parameterized by a C -function f : Ω → M where Ω ⊆ R n − is an open set. The geodesics are denoted by ˜ F ( y, t ) = Exp t ( ∇ ˜ u ( f ( y )) . Later on, the restriction of ˜ F to a certain set will be denoted by F , while ˜ u will be the function provided by Theorem 2.10. By differentiating ˜ F ( y, t ) withrespect to y i we obtain a Jacobi field J i , as is precisely stated in the following lemma: Lemma 3.2.
Let r ∈ R and let ˜ u : M → R be a C -function. Let Ω ⊆ R n − be an openset and let y ∈ Ω . Let f : Ω → M be a C -map, and assume that the function ˜ u is twicedifferentiable with a symmetric Hessian at the point f ( y ) . For y ∈ Ω and t ∈ R set ˜ F ( y, t ) = Exp t ( ∇ ˜ u ( f ( y ))) , N ( y, t ) = ∂ ˜ F∂t ( y, t ) . Our Riemannian manifold is not necessarily complete, and we assume that t ˜ F ( y, t ) iswell-defined in a maximal subset ( a y , b y ) ⊆ R containing the origin. Suppose that B ⊆ Ω is a measurable set containing y , such that y is a Lebesgue density point of B ⊆ R n − ,and ˜ u ( f ( y )) = r , |∇ ˜ u ( f ( y )) | = 1 for y ∈ B . (4) Then,(i) For any t ∈ ( a y , b y ) the map ˜ F is differentiable at the point ( y , t ) ∈ Ω × R . (Wenote that ˜ F is well-defined in an open neighborhood of ( y , t ) in R n − × R ). ii) There exist Jacobi fields J ( y , t ) , . . . , J n − ( y , t ) along the geodesic curve t ˜ F ( y , t ) , which are well-defined in the entire interval t ∈ ( a y , b y ) , such that J i ( y , t ) = ∂ ˜ F∂y i ( y , t ) for all i = 1 , . . . , n − , t ∈ ( a y , b y ) . (iii) At the point ( y , ∈ Ω × R we have h J i , N i = h J ′ i , N i = 0 ( i = 1 , . . . , n − , (5) and h J ′ i , J k i = h J ′ k , J i i ( i, k = 1 , . . . , n − . (6) Here, J ′ i ( y , t ) is the covariant derivative of the Jacobi field t J i ( y , t ) along thegeodesic curve t ˜ F ( y , t ) for t ∈ ( a y , b y ) .Proof. The curve t ˜ F ( y , t ) is a geodesic curve of speed one since |∇ ˜ u ( f ( y )) | = 1 asfollows from (4) and the fact that y ∈ B . The vector field t N ( y , t ) is the unit tangentalong this geodesic, with N ( y ,
0) = ∇ ˜ u ( f ( y )) . The equation ˜ F ( y, t ) = Exp t ( ∇ ˜ u ( f ( y ))) (7)is valid in an open set in Ω × R containing { y } × ( a y , b y ) . Note also that ˜ F ( y,
0) = f ( y ) for y ∈ Ω . Since f is a C -function, ∂f∂y i ( y ) = ∂ ˜ F∂y i ( y , for i = 1 , . . . , n − . Differentiating (7) at the point y = y yields J i ( y , t ) := ∂ ˜ F∂y i ( y , t ) = dExp t ( ξ y ,i ) for t ∈ ( a y , b y ) , i = 1 , . . . , n − , (8)where ξ y ,i = ∂ [( ∇ ˜ u ) ◦ f ] ∂y i ( y ) = ∂ J i ( y , ∇ ˜ u ∈ T N ( y , ( T M ) for i = 1 , . . . , n − . (9)This differentiation is legitimate since f is a C -map and since the vector field ∇ ˜ u : M → T M is differentiable at the point f ( y ) . We conclude that for any t ∈ ( a y , b y ) , the map ˜ F is differentiable at ( y , t ) , and (i) is proven. From (8) we learn that the vector fields J ( y , t ) , . . . , J n − ( y , t ) have the form (2), and hence they are Jacobi fields along thegeodesic t ˜ F ( y , t ) . This proves (ii). Thanks to (8) and (9) we may apply Lemma3.1 with V = ∇ ˜ u, ξ = ξ y ,i and J ( t ) = J i ( y , t ) , and conclude that J ′ i ( y ,
0) = ∇ J i ( y , ∇ ˜ u for i = 1 , . . . , n − . (10)Since y is a Lebesgue density point of B , then (4) entails that for i = 1 , . . . , n − , ∂ ˜ u ( f ( y )) ∂y i (cid:12)(cid:12)(cid:12)(cid:12) y = y = 0 and ∂ |∇ ˜ u ( f ( y )) | ∂y i (cid:12)(cid:12)(cid:12)(cid:12) y = y = 0 . (11) ince J i ( y ,
0) = ∂ ˜ F∂y i ( y ,
0) = ∂f∂y i ( y ) and N ( y ,
0) = ∇ ˜ u ( f ( y )) , we may rewrite (11) as h N ( y , , J i ( y , i = 0 and h∇ J i ( y , ∇ ˜ u, N ( y , i = 0 , (12)for i = 1 , . . . , n − . Now (5) follows from (10) and (12). As for the proof of (6): in viewof (10) we actually need to prove that h∇ J i ( y , ∇ ˜ u, J k ( y , i = h∇ J k ( y , ∇ ˜ u, J i ( y , i for i, k = 1 , . . . , n − . The latter relations hold as ˜ u is twice differentiable with a symmetric Hessian at the point f ( y ) = ˜ F ( y , .Recall the definitions of Strain [ u ] , Strain ε [ u ] and α u , β u from Section 2.1. Definition 3.3.
Let u : M → R satisfy k u k Lip ≤ and let R ⊆ M be a Borel set. We saythat R is a “seed of a ray cluster” associated with u if there exist numbers r ∈ R , ε > ,open sets U ⊆ M , Ω ⊆ R n − and C , -functions ˜ u : U → R , f : Ω → M for which thefollowing hold:(i) For any x ∈ U ∩ Strain ε [ u ] we have that ˜ u ( x ) = u ( x ) and ∇ ˜ u ( x ) = ∇ u ( x ) .(ii) The C , -map f : Ω → M is one-to-one with f (Ω ) = { x ∈ U ; ˜ u ( x ) = r } . Theinverse map f − : f (Ω ) → Ω is continuous.(iii) For almost any point y ∈ Ω , the function ˜ u is twice differentiable with a symmetricHessian at the point f ( y ) .(iv) R ⊆ { x ∈ U ∩ Strain ε [ u ] ; ˜ u ( x ) = r } .If the functions α u , β u : R → R ∪ {±∞} are continuous, then we say that R is a“seed of a ray cluster of continuous length”. Note that any Borel set which is contained in a seed of a ray cluster, is in itself a seed ofa ray cluster. Recall from Lemma 2.8 that T ◦ [ u ] is the collection of all relative interiors ofnon-degenerate transport rays associated with u , and that T ◦ [ u ] is a partition of Strain [ u ] . Definition 3.4.
Let u : M → R satisfy k u k Lip ≤ . A subset R ⊆ Strain [ u ] is a “raycluster” associated with u if there exists R ⊆ M which is a seed of a ray cluster such that R = { x ∈ M ; ∃I ∈ T ◦ [ u ] such that x ∈ I and I ∩ R = ∅} . (13) We say that R is a “ray cluster of continuous length” if R is a seed of a ray cluster ofcontinuous length. When A ⊆ R n is a measurable set and f : A → R m is locally-Lipschitz, the function f maps measurable sets to measurable sets: Indeed, any measurable set equals the unionof a Lebesgue-null set and countably many compacts, hence also its image under a locally-Lipschitz map is the union of a Lebesgue-null set and countably many compacts. Therefore,the concept of a measurable subset of a differentiable manifold M is well-defined. Similarly,the concepts of a Lebesgue-null set and a Lebesgue density point of a measurable set in adifferentiable manifold M are well-defined. The Lebesgue theorem, stating that almost any oint of a measurable set A is a Lebesgue density point of A , also applies in the context ofan abstract differentiable manifold.For a subset A ⊆ R n , a function f : A → R m and a point x ∈ A , we say that f isdifferentiable at x if there is a unique linear map T : R n → R m such that lim A ∋ x → x | f ( x ) + T ( x − x ) − f ( x ) | / | x − x | = 0 . In this case we may speak of the differential of f at x . For instance, if f : A → R m isdifferentiable at the point x ∈ A ⊆ R n , and B ⊆ A is a measurable set containing x suchthat x is a Lebesgue density point of B , then f | B is differentiable at x . In what follows wewill usually consider the differential of a function f : A → R m only at Lebesgue densitypoints of A .Similarly, given differentiable manifolds M and N , a subset A ⊆ M and a function f : A → N , we may speak about the differentiability of f at the point p ∈ A . When f isdifferentiable at p , we may consider the differential of f at p , and we may also consider thedirectional derivatives ∂ v f for v ∈ T p M . A function defined in a subset of a differentiablemanifold is said to be locally-Lipschitz when it is locally-Lipschitz in any chart. By theRademacher theorem and the Kirszbraun theorem (see, e.g., Evans and Gariepy [17, Section3.1]), any locally-Lipschitz function defined on a measurable subset A of a differentiablemanifold, is differentiable almost-everywhere in A .A parallel line-cluster is a subset B ⊆ R n − × R of the following form: There exista measurable set B ⊆ R n − and continuous functions a : B → [ −∞ , and b : B → (0 , + ∞ ] such that B = (cid:8) ( y, t ) ∈ R n − × R ; y ∈ B , a y < t < b y (cid:9) , (14)where a y = a ( y ) and b y = b ( y ) for y ∈ B . Note that when y ∈ B is a Lebesgue densitypoint of B , the point ( y, t ) ∈ B is a Lebesgue density point of B for any t ∈ ( a y , b y ) .An almost line-cluster is a subset B ⊆ R n − × R of the form (14) where B ⊆ R n − ismeasurable and the functions a : B → [ −∞ , and b : B → (0 , + ∞ ] are only assumedto be measurable, and not continuous. Note that a parallel line-cluster is always measurable,as well as an almost line-cluster. We say that a map F is invertible if it is one-to-one andonto. Proposition 3.5.
Let u : M → R satisfy k u k Lip ≤ . Suppose that R ⊆ Strain [ u ] is anon-empty ray cluster of continuous length. Then there exist a parallel line-cluster B ⊆ R n − × R , a measurable set B ⊆ R n − , functions a, b : B → R ∪ {±∞} and a locally-Lipschitz, invertible map F : B → R with the following properties:(i) The relation (14) holds true. Write f ( y ) = F ( y, for y ∈ B . Then the set R = f ( B ) is a seed of a ray cluster satisfying (13). Additionally, a y = − α u ( f ( y )) , b y = β u ( f ( y )) for all y ∈ B . (15) (ii) For any y ∈ B , the curve t F ( y, t ) t ∈ ( a y , b y ) s a minimizing geodesic whose image is the relative interior of a transport ray asso-ciated with u . Furthermore, there exists r ∈ R such that u ( F ( y, t )) = t + r for all ( y, t ) ∈ B. (16) (iii) For almost any Lebesgue density point y ∈ B the following hold: The map F isdifferentiable at ( y , t ) for all t ∈ ( a y , b y ) , and there exist Jacobi fields J ( y , t ) , . . . ,J n − ( y , t ) along the geodesic t F ( y , t ) in the entire interval t ∈ ( a y , b y ) suchthat for i = 1 , . . . , n − , J i ( y , t ) = ∂F∂y i ( y , t ) for all t ∈ ( a y , b y ) . (17) Denoting N ( y , t ) = ∂F∂t ( y , t ) we have, at the point ( y , ∈ B , h J i , N i = h J ′ i , N i = 0 ( i = 1 , . . . , n − , (18) and h J ′ i , J k i = h J ′ k , J i i ( i, k = 1 , . . . , n − . (19) Here, J ′ i ( y , is the covariant derivative at t = 0 of the Jacobi field t J i ( y , t ) along the geodesic curve t F ( y , t ) .(iv) For ( y, t ) ∈ B denote T ( y, t ) = {h J i ( y, t ) , J k ( y, t ) i} i,k =1 ,...,n , where J n := N .Then the symmetric matrix T ( y, t ) is well-defined and positive semi-definite almosteverywhere in B , and for any Borel set A ⊆ R , λ M ( A ) = Z F − ( A ) p det T ( y, t ) dydt, (20) where λ M is the Riemannian volume measure in M .Proof. Let R ⊆ M be the seed of a ray cluster of continuous length given by Definition3.4. Then R is a Borel set with R = { x ∈ M ; ∃I ∈ T ◦ [ u ] such that x ∈ I and I ∩ R = ∅} . (21)Since R is a seed of a ray cluster, Definition 3.3 provides us with certain numbers r ∈ R , ε > , open sets U ⊆ M , Ω ⊆ R n − and C , -functions ˜ u : U → R , f : Ω → M such that R ⊆ { x ∈ U ∩ Strain ε [ u ] ; ˜ u ( x ) = r } . (22)Additionally, f is a one-to-one map with f (Ω ) = { x ∈ U ; ˜ u ( x ) = r } . In particular, R ⊆ f (Ω ) . Denote B := f − ( R ) ⊆ Ω . Since R ⊆ f (Ω ) then f ( B ) = R . (23)Since B is the preimage of the Borel set R under the continuous map f , then B ⊆ R n − is measurable. According to (22) and (23), for each y ∈ B , the point f ( y ) belongs to train ε [ u ] ⊆ Strain [ u ] . Since T ◦ [ u ] is a partition of Strain [ u ] , then for any y ∈ B thereexists a unique I = I ( y ) ∈ T ◦ [ u ] for which f ( y ) ∈ I . In view of (23), we may rewrite (21)as follows: R = [ y ∈ B I ( y ) . (24)For any y ∈ B , the set I ( y ) is the relative interior of a non-degenerate transport ray. Ac-cording to Corollary 2.6 there exists an open set ( a y , b y ) ⊆ R containing the origin, with a y = − α u ( f ( y )) , b y = β u ( f ( y )) , such that I ( y ) = { Exp t ( ∇ u ( f ( y ))) ; t ∈ ( a y , b y ) } for y ∈ B , (25)and such that t Exp t ( ∇ u ( f ( y ))) is a minimizing geodesic in t ∈ ( a y , b y ) with u (Exp t ( ∇ u ( f ( y )))) = u ( f ( y )) + t for y ∈ B , t ∈ ( a y , b y ) . (26)The curve t Exp t ( ∇ u ( f ( y ))) is a geodesic of speed one, so |∇ u ( f ( y )) | = 1 for y ∈ B . (27)Since R is a seed of a ray cluster of continuous length, then the functions α u , β u : R → (0 , + ∞ ] are continuous. Therefore b y = β u ( f ( y )) and a y = − α u ( f ( y )) are continuousfunctions of y ∈ B , thanks to (23) and the continuity of f . Consequently, B = (cid:8) ( y, t ) ∈ R n − × R ; y ∈ B , a y < t < b y (cid:9) (28)is a parallel line-cluster. According to (22), (23) and item (i) of Definition 3.3, u ( f ( y )) = ˜ u ( f ( y )) = r , ∇ ˜ u ( f ( y )) = ∇ u ( f ( y )) for y ∈ B . (29)For y ∈ Ω and t ∈ R define ˜ F ( y, t ) = Exp t ( ∇ ˜ u ( f ( y ))) , N ( y, t ) = ∂ ˜ F∂t ( y, t ) . (30)Since M is not necessarily complete, then ( y, t ) ˜ F ( y, t ) and ( y, t ) N ( y, t ) are well-defined on a maximal open subset of Ω × R that contains Ω × { } . The functions ˜ u and f are C , -maps, and hence Ω ∋ y
7→ ∇ ˜ u ( f ( y )) ∈ T M is locally-Lipschitz. The exponential map is smooth, and from (30) we learn that ˜ F is locally-Lipschitz. According to (25), the map ˜ F is well-defined on the entire set B . Set F = ˜ F | B , a well-defined, locally-Lipschitz map. From (28), (29) and (30), F ( y, t ) = ˜ F ( y, t ) = Exp t ( ∇ ˜ u ( f ( y ))) = Exp t ( ∇ u ( f ( y ))) for all ( y, t ) ∈ B. (31)We conclude from (24), (25), (28) and (31) that R = F ( B ) . hus F : B → R is onto. We argue that for any y , y ∈ B , y = y = ⇒ f ( y )
6∈ I ( y ) . (32)Indeed, u ( f ( y )) = u ( f ( y )) = r according to (29). Hence, if f ( y ) ∈ I ( y ) then by (25)and (26) necessarily f ( y ) = Exp t ( ∇ u ( f ( y ))) for t = 0 . Therefore f ( y ) = f ( y ) andconsequently y = y as the function f is one-to-one. This establishes (32). Recalling that T ◦ [ u ] is a partition, we deduce from (32) that the union in (24) is a disjoint union. Glancingat (25) and (31), we see that the locally-Lipschitz map F : B → R is one-to-one and henceinvertible, as required.Let us verify conclusion (i) of the proposition: The relation (14) holds true in view of(28). It follows from (31) that F ( y,
0) = f ( y ) for all y ∈ B . By (21) and (23), the set R = f ( B ) is a seed of a ray cluster satisfying (13). The definition of a y and b y aboveimplies (15), and (i) is proven. We move on to the proof of conclusion (ii) of the proposition:The fact that t F ( y, t ) is a minimizing geodesic whose image is the relative interior of atransport ray follows from (25) and (31). The relation (16) follows from (26), (29) and (31).Thus conclusion (ii) is proven as well.In order to obtain conclusion (iii) we would like to apply Lemma 3.2. To this end, observethat our definition (30) of ˜ F ( y, t ) and N ( y, t ) coincides with that of Lemma 3.2. Accordingto Definition 3.3(iii), for almost any y ∈ B ⊆ Ω , the function ˜ u is twice differentiablewith a symmetric Hessian at f ( y ) . Note that the requirement (4) of Lemma 3.2 is satisfiedin view of (27) and (29). Thus, from conclusion (ii) of Lemma 3.2, for almost any Lebesguedensity point y ∈ B , J ( y , t ) = ∂ ˜ F∂y ( y , t ) , . . . , J n − ( y , t ) = ∂ ˜ F∂y n − ( y , t ) , (33)are well-defined Jacobi fields along the entire geodesic t ˜ F ( y , t ) for t ∈ ( a y , b y ) .In fact, ( y , t ) is a Lebesgue density point of B for any t ∈ ( a y , b y ) . Recalling that F = ˜ F | B we conclude from Lemma 3.2(i) that the map F : B → R is differentiable at ( y , t ) whenever t ∈ ( a y , b y ) . The relation (17) thus follows from the validity of (33) forall t ∈ ( a y , b y ) . The Jacobi fields t J ( y , t ) , . . . , t J n − ( y , t ) also satisfy (18) and(19), thanks to Lemma 3.2(iii), and the proof of (iii) is complete.We continue with the proof of (iv). First of all, the function F is locally-Lipschitzand hence differentiable almost everywhere in B . According to conclusion (iii) which wasproven above, for almost any ( y, t ) ∈ B , T ( y, t ) = {h J i ( y, t ) , J k ( y, t ) i} i,k =1 ,...,n = (cid:26)(cid:28) ∂F∂y i ( y, t ) , ∂F∂y k ( y, t ) (cid:29)(cid:27) i,k =1 ,...,n (34)where ∂F/∂y n := ∂F/∂t . We will use the area formula for Lipschitz maps from Evans andGariepy [17]. Let us recall the relevant theory. Let H : R n → R n be a Lipschitz function.The Jacobian of H , denoted by J H , is well-defined almost everywhere. According to [17,Section 3.3.3], for any measurable function g : R n → [0 , ∞ ) and a measurable set D ⊆ R n , Z D g ( x ) J H ( x ) dx = Z R n X x ∈ D ∩ H − ( y ) g ( x ) dy, (35) here an empty sum is defined to be zero. We claim that in order to define the left-hand sideand the right-hand side of (35), it suffices to know the values of H in the set D alone. Indeed,the Jacobian J H ( x ) is determined by H | D at any Lebesgue density point x ∈ D in which H is differentiable. The Kirszbraun theorem [17, Section 3.3.1] states that any Lipschitz mapfrom D to R n may be extended to a Lipschitz map from R n to R n . It therefore suffices toassume that H : D → R n is a Lipschitz function in order for (35) to hold true. In fact, it isenough to assume that H : D → R n is only locally-Lipschitz. Indeed, there exist compacts K ⊆ K ⊆ . . . that are contained in D with m D \ ∞ [ i =1 K i ! = 0 , where m is the Lebesgue measure on R n . We now apply (35) with the compact set K i playing the role of D and use the monotone convergence theorem. This yields (35) for theoriginal set D , even though H is only locally-Lipschitz. To summarize, when D ⊆ R n isa measurable set and H : D → R n is a locally-Lipschitz, one-to-one map, then for anymeasurable function g : R n → [0 , ∞ ) , Z D g ( x ) J H ( x ) dx = Z H ( D ) g ( H − ( y )) dy. (36)Next, what happens if the range of H is not a Euclidean space, but a Riemannian manifold M ? In this case, we claim that for any measurable set D ⊆ R n and a locally-Lipschitz map H : D → M which is one-to-one, Z D ϕ ( x ) p det T ( x ) dx = Z H ( D ) ϕ ( H − ( y )) dλ M ( y ) , (37)for any measurable ϕ : R n → [0 , ∞ ) . Here, T ( x ) = ( h ∂H/∂x i , ∂H/∂x j i ) i,j =1 ,...,n . Notethat (iv) follows from (34) and (37), with D = B, H = F and ϕ = 1 H − ( A ) . In order todeduce (37) from (36) we need to work in a local chart, and observe that p det T ( x ) is theRiemannian volume of the parallelepiped spanned by the tangent vectors ∂H∂x , . . . , ∂H∂x n . The usual Jacobian J H ( x ) is the Euclidean volume of this parallelepiped in our local chart.We conclude that p det T ( x ) /J H ( x ) is precisely the density of the Riemannian volumemeasure λ M at the point H ( x ) in our local chart. By setting g ( x ) = ϕ ( x ) p det T ( x ) /J H ( x ) , we deduce (37) from (36). Remark 3.6.
It suffices to assume that A ⊆ R is a measurable set in order for (20) to holdtrue. In fact, denote by θ the complete measure on the set B whose density is ( y, t ) p det T ( y, t ) . Note also that the restriction of λ M to R is a complete measure on R . Thevalidity of (20) for all Borel subsets of R and a standard measure-theoretic argument showthat a subset A ⊆ R is λ M -measurable if and only if F − ( A ) is θ -measurable. Therefore, F pushes forward the measure θ to the restriction of λ M to the ray cluster R . emark 3.7. What happens if the ray cluster R from Proposition 3.5 is not assumed to be of continuous length ? The assumption that the ray cluster R is of continuous length was mainlyused to prove that the set B defined in (28) is a parallel line-cluster. Without the assumptionthat R is of continuous length, the functions b y = β u ( f ( y )) , a y = − α u ( f ( y )) are still measurable functions of y ∈ B , thanks to Lemma 2.9 and the continuity of f .Therefore B is an almost line-cluster . We thus see that only minor changes will occur in theconclusion of the proposition, if the ray cluster R is not assumed to be of continuous length.One obvious change would be that B becomes an almost line-cluster, and not a parallelline-cluster. The only additional change is that“for all t ∈ ( a y , b y ) ”in the second line of (iii) and also in (17) will be replaced by“for almost all t ∈ ( a y , b y ) ”.Indeed, the function F = ˜ F | B is differentiable at ( y , t ) and it satisfies the equality in (17)at any point ( y , t ) ∈ B which is a Lebesgue density point of B . By the Lebesgue densitytheorem, for almost any y ∈ B and for almost any t ∈ ( a y , b y ) , the point ( y , t ) ∈ B is aLebesgue density point of B . To conclude, we are allowed to apply Proposition 3.5, with theaforementioned tiny changes, even if the ray cluster R is not assumed to be of continuouslength.For a subset A ⊆ M define Ends ( A ) ⊆ M to be the union of all relative boundaries of transport rays intersecting A . In other words, a point x ∈ M belongs to Ends ( A ) if andonly if there exists a transport ray I ∈ T [ u ] , whose relative boundary contains x , such that A ∩ I 6 = ∅ . Lemma 3.8.
Let u : M → R satisfy k u k Lip ≤ and let R ⊆ Strain [ u ] be a ray cluster.Then, λ M ( Ends ( R )) = 0 . Proof.
We can assume that R = ∅ . We may apply Proposition 3.5(ii) thanks to Remark 3.7.Whence, R = { F ( y, t ) ; y ∈ B , a y < t < b y } . (38)Furthermore, F = ˜ F | B where ˜ F as defined in (30) is a locally-Lipschitz map which iswell-defined in a maximal open subset of Ω × R containing Ω × { } . We claim that Ends ( R ) = n ˜ F ( y, t ) ; y ∈ B , t ∈ R ∩ { a y , b y } , ˜ F ( y, t ) is well-defined o . (39)Indeed, fix an arbitrary point x ∈ R . Since R ⊆ Strain [ u ] , then according to Lemma 2.5,there is a unique transport ray I ∈ T [ u ] containing x . The relative interior of I contains thepoint x . By Proposition 3.5(ii), the relative interior of I must take the form { F ( y, t ) ; t ∈ ( a y , b y ) } (40) or a certain y ∈ B . The transport ray I ⊆ M is a closed set. Recall that F = ˜ F | B , andthat the curve t F ( y, t ) is a minimizing geodesic in t ∈ ( a y , b y ) . We thus deduce from(30), (40) and Lemma 2.3 that I = n ˜ F ( y, t ) ; t ∈ R ∩ [ a y , b y ] , ˜ F ( y, t ) is well-defined o . (41)Since x ∈ R was an arbitrary point, the relation (39) follows from the representation (41) ofthe unique transport ray I containing x . Consider the set n ( y, t ) ∈ B × R ; t ∈ { a y , b y } , ˜ F ( y, t ) is well-defined o . (42)This set is contained in the union of two graphs of measurable functions, and hence it is aset of measure zero in R n − × R . Since Ends ( R ) is the image of the set in (42) under thelocally-Lipschitz map ˜ F , then Ends ( R ) is a null-set in the n -dimensional manifold M . As before, we write λ M for the Riemannian volume measure on the geodesically-convex,Riemannian manifold M . Our main result in this subsection is the following: Proposition 3.9.
Let u : M → R satisfy k u k Lip ≤ . Then there exists a countable family { R i } i =1 ,..., ∞ of disjoint ray clusters of continuous length such that λ M Strain [ u ] \ ∞ [ i =1 R i !! = 0 . We begin the proof of Proposition 3.9 with the following lemma.
Lemma 3.10.
Let u : M → R satisfy k u k Lip ≤ . Let R ⊆ Strain [ u ] be any ray clusterassociated with u . Then R is a Borel subset of M .Proof. We may assume that R = ∅ . According to Proposition 3.5 and Remark 3.7 we knowthat R = F ( B ) where B is an almost-line cluster. Let R ⊆ M and r ∈ R be as inProposition 3.5. We claim that a given point x ∈ Strain [ u ] belongs to R if and only if thefollowing two conditions are met:(A) r − u ( x ) ∈ ( − α u ( x ) , β u ( x )) .(B) Exp r − u ( x ) ( ∇ u ( x )) ∈ R .In order to prove this claim, assume that x ∈ Strain [ u ] satisfies conditions (A) and (B).Since T ◦ [ u ] is a partition of Strain [ u ] , there exists I ∈ T ◦ [ u ] such that x ∈ I . From (A) andCorollary 2.6 the point Exp r − u ( x ) ( ∇ u ( x )) belongs to I , while condition (B) shows thatthis point belongs to R . Hence I ∩ R = ∅ . From Definition 3.4 we obtain that I ⊆ R andconsequently x ∈ R . Conversely, assume that x ∈ R . According to Proposition 3.5 thereexists ( y, t ) ∈ B for which F ( y, t ) = x and u ( x ) = t + r . Additionally, α u ( x ) = t − a y , β u ( x ) = b y − t, n the notation of Proposition 3.5. Since B is an almost-line cluster, then ∈ ( a y , b y ) and consequently r − u ( x ) = − t ∈ ( a y − t, b y − t ) = ( − α u ( x ) , β u ( x )) . We have thusverified condition (A). By Proposition 3.5 and Corollary 2.6, we have R ∋ F ( y,
0) =Exp r − u ( x ) ( ∇ u ( x )) , and (B) follows as well.Recall that the set Strain [ u ] is Borel according to Lemma 2.9, as well as the functions α u , β u : M → R ∪ {±∞} . Since u is continuous, then the collection of all x ∈ Strain [ u ] satisfying condition (A) is a Borel set. As for condition (B), the set R is a seed of a raycluster and by definition it is a Borel set. Consider the partially-defined function Strain [ u ] ∋ x Exp r − u ( x ) ( ∇ u ( x )) ∈ M . (1)We claim that this function is well-defined on a Borel subset of Strain [ u ] , and that it is aBorel map. Indeed, Lemma 2.4 shows that the Lipschitz function u is differentiable in theBorel set Strain [ u ] . Consequently ∇ u : Strain [ u ] → T M is a well-defined Borel map, as itmay be represented as a pointwise limit of Borel maps. The exponential map is continuousand the domain of definition of the partially-defined map T M × R ∋ ( v, t ) Exp t ( v ) ∈ M is an open set. Hence the map in (1) is a Borel map which is defined on a Borel subset of Strain [ u ] . We conclude that the collection of all x ∈ Strain [ u ] satisfying condition (B) isBorel, being the preimage of the Borel set R under the Borel map (1). Therefore the set R ⊆ Strain [ u ] , which is defined by conditions (A) and (B), is a Borel set. Lemma 3.11.
Let u : M → R satisfy k u k Lip ≤ . Assume that R, R , R , . . . , R L ⊆ Strain [ u ] are ray clusters. Then also R \ ( S Li =1 ) R i is a ray cluster.Proof. Denote by R the seed of the ray cluster R provided by Definition 3.4. Then R isa Borel set. Lemma 3.10 implies that ˜ R = R \ ( ∪ Li =1 R i ) is a Borel set as well. By theremark following Definition 3.3, the set ˜ R is a seed of a ray cluster associated with u . Infact, the set ˜ R is the seed of the ray cluster R \ ( ∪ Li =1 R i ) , as follows from Definition 3.4and the fact that T ◦ [ u ] is a partition of Strain [ u ] .The equality of the mixed second derivatives of C , -functions, stated in the followinglemma, is of great importance to us. Lemma 3.12.
Let U ⊆ R n be an open set and let f : U → R be a C , -function. Then for i, j = 1 , . . . , n , the functions ∂ i f and ∂ j f are differentiable almost everywhere in U , with ∂ i ( ∂ j f ) = ∂ j ( ∂ i f ) almost everywhere in U. (2) Proof.
Let x ∈ U . It suffices to prove the lemma in an open neighborhood of x , in which f and ∂ f, . . . , ∂ n f are Lipschitz functions. By the Rademacher theorem, the functions ∂ f, . . . , ∂ n f are differentiable almost everywhere in U . By considering slices of U , we see hat it suffices to prove (2) assuming that n = 2 and that U is a rectangle parallel to the axes,of the form U = (cid:8) ( x, y ) ∈ R ; a < x < b, c < y < d (cid:9) . Denote h = ∂∂x (cid:18) ∂f∂y (cid:19) . Since ∂f /∂y is Lipschitz, then h is an L ∞ -function. Furthermore, for any ( x, y ) ∈ U , ∂f∂y ( x, y ) = ∂f∂y ( a, y ) + Z xa h ( t, y ) dt. Integrating with respect to the y -variable we see that for any ( x, y ) ∈ U , f ( x, y ) = f ( x, c ) + Z yc ∂f∂y ( x, s ) ds = f ( x, c ) + Z yc ∂f∂y ( a, s ) ds + Z [ a,x ] × [ c,y ] h, (3)where the use of Fubini’s theorem is legitimate as h is an L ∞ -function on U . Differentiating(3) with respect to x , we deduce that the Lipschitz function ∂f /∂x satisfies ∂f∂x ( x, y ) = ∂f∂x ( x, c ) + Z yc h ( x, s ) ds (4)almost everywhere in U . Both the left-hand side and the right-hand side of (4) are differ-entiable with respect to y almost everywhere in U . Therefore, by differentiating (4) withrespect to y we obtain ∂∂y (cid:18) ∂f∂x (cid:19) = h almost everywhere in U . Thus (2) is proven. Corollary 3.13.
Let f : M → R be a C , -function. Then the vector field ∇ f is differen-tiable almost-everywhere in M , and for almost any p ∈ M , h∇ v ( ∇ f ) , w i = h∇ w ( ∇ f ) , v i for v, w ∈ T p M . (5) Here, by “almost-everywhere” we refer to the Riemannian volume measure λ M .Proof. Working in a local chart, we may replace M by an open set U ⊆ R n equipped witha Riemannian metric tensor. Since f : U → R is a C , -function, Lemma 3.12 implies thatthe functions ∂ f, . . . , ∂ n f are differentiable almost everywhere, and ∂ i ( ∂ j f ) = ∂ j ( ∂ i f ) (6)almost everywhere in U . The Leibnitz rule applies at any point where the involved functionsare differentiable and hence, h∇ ∂ i ( ∇ f ) , ∂ j i − h∇ ∂ j ( ∇ f ) , ∂ i i = ∂ i h∇ f, ∂ j i − ∂ j h∇ f, ∂ i i − h∇ f, ∇ ∂ i ∂ j − ∇ ∂ j ∂ i i = ∂ i ( ∂ j f ) − ∂ j ( ∂ i f ) at any point in which ∂ f, . . . , ∂ n f are differentiable. Now (5) follows from the validity of(6) almost everywhere in U . emma 3.14. Let u : M → R satisfy k u k Lip ≤ and let ε > and p ∈ Strain ε [ u ] . Thenthere exist an open set V ⊆ M containing p and a ray cluster R ⊆ M such thatStrain ε [ u ] ∩ V ⊆ R. (7) Proof.
Set ε = ε/ . Applying Theorem 2.10, we find δ > and a C , -function ˜ u : B M ( p, δ ) → R such that x ∈ B M ( p, δ ) ∩ Strain ε [ u ] = ⇒ ˜ u ( x ) = u ( x ) , ∇ ˜ u ( x ) = ∇ u ( x ) . (8)We would like to apply the implicit function theorem, in the form of Lemma 2.11(iii). De-creasing δ if necessary, we may assume that B M ( p, δ ) is contained in a single chart of thedifferentiable manifold M . Since p ∈ Strain ε [ u ] ⊆ Strain ε [ u ] then p belongs to the relativeinterior of some transport ray. From (8) and Lemma 2.4, ∇ ˜ u ( p ) = ∇ u ( p ) = 0 and ˜ u ( p ) = u ( p ) . (9)We may apply Lemma 2.11(iii) in the local chart, thanks to (9). We conclude from Lemma2.11(iii) that there exist an open set U ⊆ B M ( p, δ ) (10)containing p , an open set Ω = Ω × ( a, b ) ⊆ R n − × R and a C , -diffeomorphism G :Ω → U with ˜ u ( G ( y, t )) = t for ( y, t ) ∈ Ω × ( a, b ) . (11)Since p ∈ U and G : Ω → U is onto, then (9) and (11) imply that u ( p ) = ˜ u ( p ) ∈ ( a, b ) . (12)The set U is an open neighborhood of p , hence there exists < η < ε with B M ( p, η ) ⊆ U. (13)According to Corollary 3.13, for almost any x ∈ U , the C , -function ˜ u is twice differen-tiable with a symmetric Hessian at x . Since G is a C -diffeomorphism, then for almost any ( y, t ) ∈ Ω × ( a, b ) , the function ˜ u is twice differentiable with a symmetric Hessian at thepoint G ( y, t ) . From the latter fact and from (12) we conclude that there exists t ∈ ( a, b ) ∩ (cid:16) u ( p ) − η , u ( p ) + η (cid:17) (14)with the following property: For almost any y ∈ Ω ⊆ R n − , the function ˜ u is twicedifferentiable with a symmetric Hessian at the point G ( y, t ) . Denote R = { x ∈ U ∩ Strain ε [ u ] ; ˜ u ( x ) = t } . (15)Lemma 2.9 implies that Strain ε [ u ] = { x ∈ M ; ℓ u ( x ) > ε } is a Borel set. From (15),the set R ⊆ M is also Borel. We claim that R is a seed of a ray cluster in the sense ofDefinition 3.3. In order to prove our claim we define r := t and set f ( y ) := G ( y, t ) ( y ∈ Ω ) . ince G is a C , -diffeomorphism onto U , then the C , -function f is one-to-one with acontinuous inverse. The relation (11) implies that f (Ω ) = { x ∈ U ; ˜ u ( x ) = t } = { x ∈ U ; ˜ u ( x ) = r } . (16)Let us verify that the numbers r ∈ R , ε > , the open sets U ⊆ M , Ω ⊆ R n − and the C , -functions ˜ u : U → R , f : Ω → M satisfy the requirements of Definition 3.3. Indeed,by the choice of t we verify requirement (iii) of Definition 3.3. By using (16) and thepreceding sentence we obtain Definition 3.3(ii). The relation (15) and the fact that r = t show that Definition 3.3(iv) holds as well. From (8) and (10) we deduce Definition 3.3(i).Thus R is a seed of a ray cluster associated with u . Set R = { x ∈ M ; ∃I ∈ T ◦ [ u ] such that x ∈ I and I ∩ R = ∅} . (17)Then R ⊆ Strain [ u ] is a ray cluster, according to Definition 3.4. We still need to find anopen set V ⊆ M containing p for which (7) holds true. Let us define V = n x ∈ B M (cid:16) p, η (cid:17) ; | u ( x ) − t | < η/ o , (18)which is an open set containing p in view of (14). In order to prove (7), we recall that ε = 2 ε and let x ∈ Strain ε [ u ] ∩ V be an arbitrary point. Since ℓ u ( x ) > ε , then Corollary 2.6 impliesthat there exist I ∈ T ◦ [ u ] and a minimizing geodesic γ : [ − ε, ε ] → M with γ (0) = x (19)such that γ ([ − ε, ε ]) ⊆ I , (20)and such that u ( γ ( t )) = u ( x ) + t for t ∈ [ − ε, ε ] . (21)It follows from (21) and the definition of α u , β u and ℓ u in Section 2.1 that ℓ u ( γ ( t )) ≥ ε − | t | for t ∈ ( − ε, ε ) . (22)Since x ∈ V , then | u ( x ) − t | < η/ according to (18). Denoting t = t − u ( x ) , we have | t | = | u ( x ) − t | < η/ < ε = ε/ , (23)where η < ε according to the line before (13). From (21) and (23) we see that u ( γ ( t )) = u ( x ) + t = t . From (22) and (23) it follows that ℓ u ( γ ( t )) > ε/ ε . Therefore, γ ( t ) ∈ Strain ε [ u ] ∩ { x ∈ M ; u ( x ) = t } . (24)Furthermore, x ∈ V and hence d ( x, p ) < η/ by (18). Since γ is a unit speed geodesic, thenfrom (19) and (23), d ( γ ( t ) , p ) ≤ d ( γ (0) , p ) + | t | = d ( x, p ) + | t | < η/ η/ η. (25) e learn from (13) and (25) that γ ( t ) ∈ U . From (8), (10) and (24), we thus obtain that ˜ u ( γ ( t )) = u ( γ ( t )) = t . By using (15) and (24), we finally obtain that γ ( t ) ∈ R . Note also that γ ( t ) ∈ I , thanks to (20) and (23). We have thus found a point γ ( t ) ∈I ∩ R , and hence I ∩ R = ∅ . Recalling that I ∈ T ◦ [ u ] we learn from (17) that I ⊆ R .Since x = γ (0) ∈ I by (19) and (20), then x ∈ R . However, x was an arbitrary point in Strain ε [ u ] ∩ V , and hence the proof of (7) is complete. Lemma 3.15.
Let u : M → R satisfy k u k Lip ≤ . Then there exists a countable family { R i } i =1 , ,... of disjoint ray clusters associated with u such thatStrain [ u ] = ∞ [ i =1 R i . (26) Proof.
In order to prove the lemma, it suffices to find ray clusters ˜ R i ⊆ M for i = 1 , , . . . which are not necessarily disjoint, such that Strain [ u ] ⊆ ∞ [ i =1 ˜ R i . (27)Indeed, any ray cluster R is automatically contained in Strain [ u ] . By setting R i = ˜ R i \∪ j and to find ray clusters R , R , . . . with Strain ε [ u ] ⊆ ∞ [ i =1 R i . (28)Let us fix ε > . We need to find ray clusters R , R , . . . satisfying (28). For p ∈ Strain ε [ u ] let us write V p,ε = V ⊆ M for the open set containing p that is provided by Lemma 3.14.Then for any p ∈ Strain ε [ u ] there is a ray cluster R = R p,ε ⊆ M such that Strain ε [ u ] ∩ V p,ε ⊆ R p,ε . (29)Consider all open sets of the form V p,ε where p ∈ Strain ε [ u ] . This collection is an open coverof Strain ε [ u ] . Recall that M is second-countable. Hence we may find an open sub-cover of Strain ε [ u ] which is countable. That is, there exist points p , p , . . . ∈ Strain ε [ u ] such that Strain ε [ u ] ⊆ ∞ [ i =1 V p i ,ε . (30)From (29) and (30) we conclude that the ray clusters R i = R p i ,ε satisfy (28), and the lemmais proven. roof of Proposition 3.9. In view of Lemma 3.10 and Lemma 3.15, all that remains is toprove the following: For any ray cluster R ⊆ M with λ M ( R ) > , there exist disjoint rayclusters of continuous length { R i } i =1 ,..., ∞ , all contained in R , such that λ M R \ ∞ [ i =1 R i !! = 0 . (31)According to Remark 3.7, we may apply Proposition 3.5 for the ray cluster R . Let B be thealmost-line cluster that is provided by Remark 3.7 and Proposition 3.5, and let F, f, B , a, b be as in Proposition 3.5. From Proposition 3.5(i), the set R = f ( B ) is a seed of a ray clus-ter. The set B ⊆ R n − is a measurable set, and a : B → [ −∞ , and b : B → (0 , + ∞ ] are measurable functions. By Luzin’s theorem from real analysis, there exist disjoint σ -compact subsets ˜ B ( k )0 ⊆ B for k = 1 , , . . . such that m B \ ∞ [ k =1 ˜ B ( k )0 !! = 0 , (32)while for any k ≥ , the functions a | ˜ B ( k )0 and b | ˜ B ( k )0 are continuous. Here, m is the Lebesguemeasure on R n − . Note that ˜ R ( k ) := f ( ˜ B ( k )0 ) is a σ -compact set for any k ≥ , being theimage of a σ -compact set under a continuous map. By the remark following Definition 3.3,the set ˜ R ( k ) ⊆ R is a seed of a ray cluster.From our construction the functions a y = − α u ( f ( y )) and b y = β u ( f ( y )) are continuousfunctions of y ∈ ˜ B ( k )0 , for any k ≥ . From Definition 3.3(ii), the function f − is continuouson R , and therefore the functions α u , β u are continuous on ˜ R ( k ) = f ( ˜ B ( k )0 ) for any k ≥ .This shows that ˜ R ( k ) is actually a seed of a ray cluster of continuous length. The function f is one-to-one, and therefore ˜ R (1) , ˜ R (2) , . . . are pairwise-disjoint.For k ≥ , define R k to be the union of all relative interiors of transport rays intersecting ˜ R ( k ) . The sets R , R , . . . are pairwise-disjoint and are contained in R , according to Propo-sition 3.5(ii). From Definition 3.4, the sets R , R , . . . are ray clusters of continuous length,while Lemma 3.10 implies the measurability of these sets. The desired relation (31) holdstrue in view of (32) and Proposition 3.5(iv). This completes the proof. We begin this section with an addendum to Proposition 3.5.
Lemma 3.16.
We work under the notation and assumptions of Proposition 3.5. Let y = y ∈ B be a Lebesgue density point of B for which the conclusions of Proposition 3.5(iii)hold true. Then either for all t ∈ ( a y , b y ) the vectors J ( y, t ) , . . . , J n − ( y, t ) ∈ T F ( y,t ) M are linearly independent, or else for all t ∈ ( a y , b y ) , these vectors are linearly dependent. roof. Fix λ , . . . , λ n − ∈ R and denote J ( y, t ) = n − X i =1 λ i J i ( y, t ) for t ∈ ( a y , b y ) . We would like to show that the set { t ∈ ( a y , b y ) ; J ( y, t ) = 0 } is an open set. Assume that t ∈ ( a y , b y ) satisfies J ( y, t ) = 0 . (1)We need to prove that J ( y, t ) = 0 for t in a small neighborhood of t . To this end, denote v = ( λ , . . . , λ n − ) ∈ R n − . Since y ∈ B is a Lebesgue density point of B ⊆ R n − , thenthere exists a C -curve γ : ( − , → R n − with γ (0) = y and ˙ γ (0) = v , such that the set I = { s ∈ ( − ,
1) ; γ ( s ) ∈ B } has an accumulation point at zero. We are going to view γ as a map from I to B , and wewill never use the values of γ outside I . Thus, from now on when we write ˙ γ (0) = v , weactually mean that lim I ∋ s → γ ( s ) − γ (0) s = v. We plan to apply the geometric lemma of Feldman and McCann, which is Lemma 2.22above. Set p = F ( y, t ) ∈ M . (2)Let δ = δ ( p ) > be the parameter provided by Lemma 2.22. Fix ε > with ε < min { δ , b y − t , t − a y } . (3)Then a y < t − ε while b y > t + ε . Since B is a parallel line cluster, then the functions a and b are continuous on B . Since γ is continuous with γ (0) = y , then for some η > , a γ ( s ) < t − ε, b γ ( s ) > t + ε for all s ∈ I ∩ ( − η, η ) . (4)According to Proposition 3.5(iii) and the chain rule, for any t ∈ ( t − ε, t + ε ) , J ( y, t ) = n − X i =1 λ i ∂F∂y i ( y, t ) = dds F ( γ ( s ) , t ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (5)where we only consider values s ∈ I when computing the limit defining the derivative withrespect to s . Note that the use of the chain rule is legitimate, as F is differentiable at ( y, t ) while γ (0) = y and ˙ γ (0) = v = ( λ , . . . , λ n − ) . From (5), for any t ∈ ( t − ε, t + ε ) , | J ( y, t ) | = lim I ∋ s → d ( F ( γ (0) , t ) , F ( γ ( s ) , t )) | s | = lim I ∋ s → d ( F ( y, t ) , F ( γ ( s ) , t )) | s | . (6)Fix < δ < ε . For s ∈ ( − η, η ) ∩ I and i = 0 , , define x i = F ( y, t + δ ( i − , z i ( s ) = F ( γ ( s ) , t + δ ( i − . (7) he points x , x , x , z ( s ) , z ( s ) , z ( s ) ∈ M are well-defined due to (3) and (4). Accord-ing to Proposition 3.5(ii), u ( x i ) = t + δ ( i −
1) + r = u ( z i ( s )) for i = 0 , , , s ∈ I ∩ ( − η, η ) . (8)Recall that k u k Lip ≤ and that t F ( y, t ) is a minimizing geodesic, as well as t F ( γ ( s ) , t ) . We thus conclude from (7) and (8) that for any s ∈ I ∩ ( − η, η ) and i, j = 0 , , , d ( x i , x j ) = d ( z i ( s ) , z j ( s )) = δ | i − j | = | u ( x i ) − u ( z j ( s )) | ≤ d ( x i , z j ( s )) . (9)Furthermore, since d ( x i , x ) ≤ δ < ε for i = 0 , , , then thanks to (2) and (3), x , x , x ∈ B M ( x , ε ) = B M ( p, ε ) ⊆ B M ( p, δ ) . (10)The map F is continuous, while γ ( s ) → y as I ∋ s → . Therefore, for i = 0 , , wehave that z i ( s ) → x i as I ∋ s → . From (10) we thus conclude that z ( s ) , z ( s ) , z ( s ) ∈B M ( p, δ ) for any s ∈ I ∩ ( − ˜ η, ˜ η ) for some < ˜ η < η . Thanks to (9) we may apply Lemma2.22 for the six points x , x , x , z ( s ) , z ( s ) , z ( s ) ∈ B M ( p, δ ) , when s ∈ I ∩ ( − ˜ η, ˜ η ) . From the conclusion of Lemma 2.22, lim sup I ∋ s → d ( x , z ( s )) + d ( x , z ( s )) | s | ≤ · lim sup I ∋ s → d ( x , z ( s )) | s | . (11)By using (6), (7) and (11) we obtain | J ( y, t − δ ) | + | J ( y, t + δ ) | ≤ · | J ( y, t ) | . (12)However, δ > was an arbitrary number in (0 , ε ) . From (1) and (12) we therefore concludethat | J ( y, t ) | = 0 for all t ∈ ( t − ε, t + ε ) . This completes the proof that the set { t ∈ ( a y , b y ) ; J ( y, t ) = 0 } is an open set. Since J is asmooth Jacobi field, then this set is also closed. Therefore, either t J ( y, t ) never vanisheson ( a y , b y ) , or else it is the zero function. In other words, for any λ , . . . , λ n − ∈ R , ∃ t ∈ ( a y , b y ) , n − X i =1 λ i J i ( y, t ) = 0 = ⇒ ∀ t ∈ ( a y , b y ) , n − X i =1 λ i J i ( y, t ) = 0 . By linear algebra, either J ( y, t ) , . . . , J n − ( y, t ) are linearly independent for all t ∈ ( a y , b y ) ,or else they are linearly dependent for all t ∈ ( a y , b y ) .Recall that ( M , d, µ ) is an n -dimensional weighted Riemannian manifold which is geodesically-convex. Recall also that λ M is the Riemannian volume measure on the Riemannian manifold M . Let ρ : M → R be the smooth function for which dµdλ M = e − ρ . (13) efinition 3.17. A measure ν on M is called a “needle candidate” of the weighted Rie-mannian manifold ( M , d, µ ) and the Lipschitz function u if there exist a non-empty sub-set ( a, b ) ⊆ R with a, b ∈ R ∪ {±∞} , a measure θ on ( a, b ) , a minimizing geodesic γ : ( a, b ) → M and Jacobi fields J ( t ) , . . . , J n − ( t ) along γ with the following proper-ties:(i) The measure ν is the push-forward of θ under the map γ .(ii) Denote J n = ˙ γ . Then the measure θ is absolutely-continuous with respect to theLebesgue measure in ( a, b ) ⊆ R , and its density is proportional to t e − ρ ( γ ( t )) · q det ( h J i ( t ) , J k ( t ) i ) i,k =1 ,...,n . (14) (iii) There exists t ∈ ( a, b ) with h J i ( t ) , ˙ γ ( t ) i = h J ′ i ( t ) , ˙ γ ( t ) i = 0 ( i = 1 , . . . , n − , (15) and h J ′ i ( t ) , J k ( t ) i = h J ′ k ( t ) , J i ( t ) i ( i, k = 1 , . . . , n − . (16) (iv) Either for all t ∈ ( a, b ) the vectors J ( t ) , . . . , J n − ( t ) ∈ T γ ( t ) M are linearly independent, or else for all t ∈ ( a, b ) these vectors are linearly dependent.(v) Denote A = ( a, b ) ⊆ R . Then the set γ ( A ) is the relative interior of a transport rayassociated with u and u ( γ ( t )) = t for all t ∈ A. Assume that Ω , Ω , . . . are certain disjoint sets. Let ν i be a measure defined on Ω i for i ≥ . We may clearly consider the measure ν = P i ≥ ν i defined on Ω = ∪ i ≥ Ω i . A subset A ⊆ Ω is ν -measurable if and only if A ∩ Ω i is ν i -measurable for any i ≥ .Recall that T ◦ [ u ] is a partition of Strain [ u ] and that π : Strain [ u ] → T ◦ [ u ] is the partitionmap, i.e., x ∈ π ( x ) ∈ T ◦ [ u ] for any x ∈ Strain [ u ] . According to Lemma 2.5, for any x ∈ Strain [ u ] , the set π ( x ) is the relative interior of the unique transport ray containing x . Lemma 3.18.
Let u : M → R satisfy k u k Lip ≤ . Then there exist a measure ν on T ◦ [ u ] and a family { µ I } I∈ T ◦ [ u ] of measures on M , such that the following hold true:(i) If G ⊆ T ◦ [ u ] is ν -measurable then π − ( G ) ⊆ Strain [ u ] is a measurable subset of M . For any measurable set A ⊆ M , the map I 7→ µ I ( A ) is well-defined ν -almosteverywhere and is a ν -measurable map.(ii) For any measurable set A ⊆ M , µ ( A ∩ Strain [ u ]) = Z T ◦ [ u ] µ I ( A ) dν ( I ) . (17) iii) For ν -almost any I ∈ T ◦ [ u ] , the measure µ I is a needle candidate of ( M , d, µ ) and u that is supported on I and it satisfies µ I ( M ) > . Furthermore, A and γ fromDefinition 3.17 satisfy I = γ ( A ) .Proof. The measure µ is assumed to be absolutely-continuous with respect to λ M . Ac-cording to Proposition 3.9, there exist disjoint ray clusters of continuous length { R i } i =1 , ,... with µ Strain [ u ] \ ∞ [ i =1 R i !! = 0 . (18)Recall from Definition 3.4 and Lemma 3.10 that each ray cluster R i is a measurable setcontained in Strain [ u ] of the form R i = ∪ I∈ S i I for some subset S i ⊆ T ◦ [ u ] . Fix i ≥ . Letus apply Proposition 3.5 for R i , which is a ray cluster of continuous length. Proposition 3.5provides us with a certain parallel line cluster B ⊆ R n − × R , a locally-Lipschitz, invertiblemap F : B → R i , and also with vector fields J ( y, t ) , . . . , J n − ( y, t ) . Let J n , r , B , a y and b y be as in Proposition 3.5. Then for almost any Lebesgue densitypoint y ∈ B , the vector fields J ( y, t ) , . . . , J n − ( y, t ) are well-defined Jacobi fields alongthe entire geodesic t F ( y, t ) for t ∈ ( a y , b y ) . Consider the measure on B whose densitywith respect to the Lebesgue measure on B is ( y, t ) q det ( h J ℓ ( y, t ) , J k ( y, t ) i ) ℓ,k =1 ,...,n . (19)According to Proposition 3.5(iv) and Remark 3.6, the map F pushes forward the measurewhose density is given by (19) to the restriction of λ M to the ray cluster R i . Next, considerthe measure on B with density ( y, t ) e − ρ ( F ( y,t )) · q det ( h J ℓ ( y, t ) , J k ( y, t ) i ) ℓ,k =1 ,...,n . (20)Glancing at (13), we see that the map F pushes forward the measure whose density is givenby (20) to the restriction of µ to R i . From Proposition 3.5(ii), for any y ∈ B there exists I ( y ) ∈ T ◦ [ u ] such that I ( y ) = { F ( y, t ) ; a y < t < b y } . Furthermore, I ( y ) ⊆ R i , and since F is invertible then I ( y ) ∩ I ( y ) = ∅ for y = y . ByProposition 3.5(ii), for all y ∈ B the map t F ( y, t ) is a minimizing geodesic. Define themeasure ˜ µ I ( y ) to be the push-forward under the map t F ( y, t ) of the measure on ( a y , b y ) whose densityis given by (20). Then ˜ µ I ( y ) is a well-defined measure supported on I ( y ) for almost any y ∈ B . Recall that the map F pushes forward the measure whose density is given by (20)to the restriction of µ to R i . By Fubini’s theorem, for any measurable set A ⊆ R i , µ ( A ) = Z B ˜ µ I ( y ) ( A ) dy = Z B µ I ( y ) ( A ) e −| y | dy, (21) here µ I ( y ) := e | y | ˜ µ I ( y ) . Denote ˜ B = (cid:8) y ∈ B ; µ I ( y ) ( M ) > (cid:9) , (22)which is a measurable subset of B ⊆ R n − . Define the measure ν i to be the push-forwardunder the map y
7→ I ( y ) of the measure on ˜ B whose density is y e −| y | . Then ν i is afinite measure supported on T ◦ [ u ] . In fact, ν i is supported on S i ⊆ T ◦ [ u ] since I ( y ) ∈ S i for all y ∈ B . From (21) and (22), for any measurable set A ⊆ M , µ ( A ∩ R i ) = Z S i µ I ( A ∩ R i ) dν i ( I ) = Z S i µ I ( A ) dν i ( I ) . (23)Furthermore, µ I ( M ) > for ν i -almost any I ∈ S i , by the definition of ˜ B . Recall thatwhen we push-forward a measure, we also push-forward its σ -algebra. Therefore if a subset G ⊆ S i is ν i -measurable, then { y ∈ ˜ B ; I ( y ) ∈ G } is a measurable subset of B . Since B is a parallel line cluster, then also { ( y, t ) ∈ B ; I ( y ) ∈ G } is measurable in R n − × R .The image of the latter measurable set under F equals π − ( G ) . Since F is locally-Lipschitz,then π − ( G ) is a measurable subset of Strain [ u ] , whenever G ⊆ S i is ν i -measurable.Let us show that µ I is a needle-candidate for ν i -almost any I ∈ S i . Since µ I is pro-portional to ˜ µ I , it suffices to prove that ˜ µ I ( y ) is a needle-candidate for almost any y ∈ ˜ B .Properties (i) and (ii) from Definition 3.17 hold by the definition of ˜ µ I ( y ) , where we set J i ( t ) = J i ( y, t − r ) , γ ( t ) = F ( y, t − r ) , a = a y + r , b = b y + r . Property (v) follows from Proposition 3.5(ii). We deduce property (iii) of Definition 3.17(with t = r ) from Proposition 3.5(iii). Property (iv) follows from Lemma 3.16. Note alsothat setting A = ( a, b ) we have I ( y ) = γ ( A ) . (24)Hence ˜ µ I ( y ) is a needle-candidate supported on I ( y ) for almost any y ∈ ˜ B , and conse-quently µ I is a needle-candidate supported on I for ν i -almost any I ∈ S i . Write ˜ S i ⊆ S i for the collection of all I ∈ S i for which µ I is a needle-candidate supported on I with µ I ( M ) > . Then ν i ( S i \ ˜ S i ) = 0 . For completeness, let us redefine µ I ≡ for I ∈ S i \ ˜ S i .Note that (23) still holds true for any measurable set A ⊆ M , since we altered the definitionof µ I only on a ν i -null set.To summarize, we found a family of measures { µ I } I∈ S i such that (23) holds true forany measurable set A ⊆ M . We now let i vary. Since the ray clusters { R i } i =1 , ,... aredisjoint, then S , S , . . . ⊆ T ◦ [ u ] are also disjoint. Denoting ν = P i ν i , we deduce (17)from (18) and (23). This completes the proof of (ii), and also of the second assertion in (i).Furthermore, for ν -almost any I ∈ T ◦ [ u ] , we have that I ∈ S i for some i , and the measure µ I is a needle-candidate supported on I with µ I ( M ) > . It thus follows from (24) thatconclusion (iii) holds true. Note that if a subset G ⊆ T ◦ [ u ] is ν -measurable, then G ∩ S i is ν i -measurable for any i , and hence π − ( G ∩ S i ) ⊆ R i is measurable in M . Consequently π − ( G ) is λ M -measurable whenever G ⊆ T ◦ [ u ] is ν -measurable. This completes the proofof (i). The lemma is therefore proven. ecall from Section 1 the definition of the generalized Ricci tensor Ric µ,N of the weightedRiemannian manifold ( M , d, µ ) . Definition 3.19.
Let n ≥ , N ∈ ( −∞ , ∪ [ n, + ∞ ] and let ( M , d, µ ) be an n -dimensionalweighted Riemannian manifold. We say that a measure ν on the Riemannian manifold M isan “ N -curvature needle” if there exist a non-empty, connected open set A ⊆ R , a smoothfunction Ψ : A → R and a minimizing geodesic γ : A → M such that:(i) Denote by θ the measure on A ⊆ R whose density with respect to the Lebesgue mea-sure is e − Ψ . Then ν is the push-forward of θ under the map γ .(ii) The following inequality holds in the entire set A : Ψ ′′ ≥ Ric µ,N ( ˙ γ, ˙ γ ) + (Ψ ′ ) N − , (25) where in the case N = ∞ , we interpret the term (Ψ ′ ) / ( N − as zero. The following proposition asserts that any needle-candidate in the sense of Definition3.17 is in fact an N -curvature needle . Proposition 3.20.
Let n ≥ , N ∈ ( −∞ , ∪ [ n, + ∞ ] and let ( M , d, µ ) be an n -dimensionalweighted Riemannian manifold which is geodesically-convex. Let u : M → R satisfy k u k Lip ≤ . Let ν be a needle-candidate of ( M , d, µ ) and u . Then either ν is the zeromeasure, or else ν is an N -curvature needle. The proof of Proposition 3.20 essentially boils down to a classical estimate in Rieman-nian geometry from Heintze and Karcher [25] that was generalized to the case of weighted
Riemannian manifolds by Bayle [5, Appendix E.1] and by Morgan [33]. According to Gro-mov [22], the estimate stems from the work of Paul Levy on the isoperimetric inequality in1919. We begin the proof of Proposition 3.20 with the following trivial lemma:
Lemma 3.21.
Let a, b ∈ R with b > and a [ − b, . Then, x a + y b ≥ ( x − y ) a + b ( x, y ∈ R ) . Proof.
We use the inequality | b/a | · x ± xy + | a/b | · y ≥ to deduce that x a + y b − ( x − y ) a + b = 1 a + b (cid:18) ba x + 2 xy + ab y (cid:19) ≥ , whenever b > and a [ − b, .Let us recall the familiar formulas for differentiating a determinant. If A t is an invertible n × n matrix that depends smoothly on t ∈ R , then ddt log | det( A t ) | = Trace [ A − t · ˙ A t ] , (26)and d dt log | det( A t ) | = Trace [ A − t · ¨ A t ] − Trace (cid:20)(cid:16) A − t · ˙ A t (cid:17) (cid:21) . (27) roof of Proposition 3.20. Let ν be a needle-candidate of ( M , d, µ ) and u . We may assumethat ν is not the zero measure. Let a, b, θ, γ and J , . . . , J n − be as in Definition 3.17. For t ∈ ( a, b ) denote f ( t ) = e − ρ ( γ ( t )) · q det ( h J i ( t ) , J k ( t ) i ) i,k =1 ,...,n (28)where J n = ˙ γ . According to Definition 3.17(ii), the density of the measure θ on ( a, b ) ⊆ R is proportional to the function f . We will prove that f is smooth and positive in ( a, b ) , andthat Ψ := − log f satisfies Ψ ′′ ≥ Ric µ,N ( ˙ γ, ˙ γ ) + (Ψ ′ ) N − , (29)where in the case N = + ∞ we interpret the term (Ψ ′ ) / ( N − as zero. ComparingDefinition 3.19 of N -curvature needles and Definition 3.17 of needle-candidates, we seethat the proposition would follow from (29). The rest of the proof is therefore devoted toestablishing (29). The Jacobi fields J , . . . , J n − satisfy the Jacobi equation: J ′′ i ( t ) = R ( ˙ γ ( t ) , J i ( t )) ˙ γ ( t ) for t ∈ ( a, b ) , i = 1 , . . . , n − . (30)Since γ is a geodesic then ∇ ˙ γ ˙ γ = 0 , and for any i = 1 , . . . , n − and t ∈ ( a, b ) , ddt h J i , ˙ γ i = h J ′ i , ˙ γ i , d dt h J i , ˙ γ i = h J ′′ i , ˙ γ i . (31)From (30) and the symmetries of the Riemann curvature tensor we deduce that h J ′′ i , ˙ γ i ≡ .Therefore h J i ( t ) , ˙ γ ( t ) i is an affine function of t ∈ ( a, b ) . It thus follows from (15) and (31)that for any t ∈ ( a, b ) , J ( t ) , . . . , J n − ( t ) ⊥ ˙ γ ( t ) . (32)From (28) and (32) we obtain f ( t ) = e − ρ ( γ ( t )) · q det ( h J i ( t ) , J k ( t ) i ) i,k =1 ,...,n − . (33)(The indices run only up to n − , as ˙ γ = J n is a unit vector orthogonal to J , . . . , J n − ).Since θ is not the zero measure, there exists t ∈ ( a, b ) for which f ( t ) = 0 . From (33) welearn that the vectors J ( t ) , . . . , J n − ( t ) ∈ T γ ( t ) M are linearly independent. According to Definition 3.17(iv), the vectors J ( t ) , . . . , J n − ( t ) are linearly independent for all t ∈ ( a, b ) . Hence, (33) yields ∀ t ∈ ( a, b ) , f ( t ) > . (34)From the Jacobi equation (30), for any t ∈ ( a, b ) and i, k = 1 , . . . , n − , ddt (cid:0) h J ′ i , J k i − h J i , J ′ k i (cid:1) = h J ′′ i , J k i − h J i , J ′′ k i = h R ( ˙ γ, J i ) ˙ γ, J k i − h J i , R ( ˙ γ, J k ) ˙ γ i = 0 , (35) y the symmetries of the Riemann curvature tensor. By using (16) and (35) we deduce thatin the entire interval ( a, b ) ⊆ R , h J ′ i , J k i = h J i , J ′ k i for i, k = 1 , . . . , n. (36)Let G t = ( G t ( i, k )) i,k =1 ,...,n − be the symmetric, positive-definite ( n − × ( n − matrixwhose entries are G t ( i, k ) = h J i ( t ) , J k ( t ) i . According to (33) and (34), the function Ψ = − log f satisfies, Ψ( t ) = ρ ( γ ( t )) −
12 log det G t for t ∈ ( a, b ) . (37)Denote H ( t ) = ˙ γ ( t ) ⊥ ⊂ T ˙ γ ( t ) M , the orthogonal complement to the vector ˙ γ ( t ) . From (31)and (32), J i ( t ) , J ′ i ( t ) ∈ H ( t ) for all t ∈ ( a, b ) , i = 1 , . . . , n − . (38)For any t ∈ ( a, b ) the linearly-independent vectors J ( t ) , . . . , J n − ( t ) ∈ H ( t ) constitute abasis of the ( n − -dimensional space H ( t ) . In view of (38), we may define an ( n − × ( n − matrix A t = ( A t ( i, k )) i,k =1 ,...,n − by requiring that J ′ i ( t ) = n − X k =1 A t ( i, k ) J k ( t ) for t ∈ ( a, b ) , i = 1 , . . . , n − . (39)Recall that G t ( i, k ) = h J i ( t ) , J k ( t ) i . From (36) and (39), for any t ∈ ( a, b ) , ˙ G t ( i, k ) = h J ′ i , J k i + h J i , J ′ k i = 2 h J ′ i , J k i = 2 * n − X ℓ =1 A t ( i, ℓ ) J ℓ , J k + = 2 n − X ℓ =1 A t ( i, ℓ ) G t ( ℓ, k ) . Equivalently, ˙ G t = 2 A t G t . Since G t is a symmetric matrix then also A t G t = ˙ G t / is asymmetric matrix. Since G t is a positive-definite matrix, from (26), then ddt log det( G t ) = Trace h G − t ˙ G t i = 2 Trace (cid:2) G − t A t G t (cid:3) = 2 Trace [ A t ] . (40)As for the second derivative, we use (39) and the Jacobi equation (30) and obtain, ¨ G t ( i, k ) = h J ′′ i , J k i + 2 h J ′ i , J ′ k i + h J ′′ k , J i i (41) = 2 h R ( ˙ γ, J i ) ˙ γ, J k i + 2 n − X ℓ,m =1 A t ( i, ℓ ) A t ( k, m ) G t ( ℓ, m ) where we used the symmetries of the Riemann curvature tensor in the last passage. Recallthat Ric M ( ˙ γ, ˙ γ ) is the trace of the linear transformation V
7→ − R ( ˙ γ, V ) ˙ γ in the linear space H ( t ) . By linear algebra, (41) entails that Trace h G − t ¨ G t i = − Ric M ( ˙ γ ( t ) , ˙ γ ( t )) + Trace (cid:2) G − t A t G t (cid:3) , (42) here we used the fact that A t G t A ∗ t = A t ( A t G t ) ∗ = A t G t in the last passage, as A t G t issymmetric. Since ˙ G t = 2 A t G t then from (27) and (42), d dt log det( G t ) = − Ric M ( ˙ γ ( t ) , ˙ γ ( t )) + 2 Trace (cid:2) A t (cid:3) − Trace (cid:2) G − t A t G t (cid:3) . (43)Applying (37) and (40) yields Ψ ′ ( t ) = ∂ ˙ γ ( t ) ρ − Trace [ A t ] . (44)Since γ is a geodesic, the equations (37) and (43) lead to Ψ ′′ ( t ) = Hess ρ ( ˙ γ ( t ) , ˙ γ ( t )) + Ric M ( ˙ γ ( t ) , ˙ γ ( t )) + Trace (cid:2) A t (cid:3) . (45)We will now utilize the definition of the generalized Ricci tensor with parameter N . There-fore, from (45), Ψ ′′ ( t ) ≥ Ric µ,N ( ˙ γ ( t ) , ˙ γ ( t )) + ( ∂ ˙ γ ( t ) ρ ) N − n + Trace (cid:2) A t (cid:3) , (46)where in the case where N = ∞ we interpret the term ( ∂ ˙ γ ( t ) ρ ) / ( N − n ) as zero. In the casewhere N = n , we require ρ to be a constant function and the latter term is again interpretedas zero. The matrix ˙ G t = 2 A t G t is symmetric, and hence G − / t A t G / t is also symmetric.Thus the ( n − × ( n − matrix A t is conjugate to a symmetric matrix and consequently ithas n − real eigenvalues (repeated according to their multiplicity). The Cauchy-Schwartzinequality yields [ Trace ( A t )] ≤ ( n − Trace [ A t ] and therefore, for any t ∈ ( a, b ) , Ψ ′′ ( t ) ≥ Ric µ,N ( ˙ γ ( t ) , ˙ γ ( t )) + ( ∂ ˙ γ ( t ) ρ ) N − n + ( Trace [ A t ]) n − . (47)In the case where N = ∞ or N = n , we deduce (29) from (44) and (47). Otherwise, wehave N ∈ R \ [1 , n ] and from (47) and Lemma 3.21, Ψ ′′ ( t ) ≥ Ric µ,N ( ˙ γ ( t ) , ˙ γ ( t )) + ( ∂ ˙ γ ( t ) ρ − Trace [ A t ]) N − . (48)From (44) and (48) we conclude that (29) holds true for any t ∈ ( a, b ) , and the proof of theproposition is complete. Example 3.22.
Consider the example where ρ ≡ Const and where
M ⊆ R n is an open,convex set. Equations (44) and (45) along with simple manipulations show that here, Ψ ′ ( t ) = − Trace [ A t ] , Ψ ′′ ( t ) = Trace [ A t ] and ˙ A t = − A t . (49)The eigenvalues of A t may be viewed as “principal curvatures” or as “eigenvalues of thesecond fundamental form” of a level set of u . Solving (49), we see that the density f ( t ) = e − Ψ( t ) is proportional to the function t k Y i =1 | t − λ i | for t ∈ ( a, b ) , (50)where k ≤ n − and λ , . . . , λ k ∈ R \ ( a, b ) are some numbers. An empty product is definedto be one. We learn from (50) that the positive function f : ( a, b ) → R is a polynomial ofdegree at most n − , all of whose roots lie in R \ ( a, b ) . heorem 3.23. Let n ≥ and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) is an n -dimensional weighted Riemannian manifold which is geodesically-convex. Let u : M → R satisfy k u k Lip ≤ . Then there exist a measure ν on the set T ◦ [ u ] and a family { µ I } I∈ T ◦ [ u ] of measures on M such that:(i) For any Lebesgue-measurable set A ⊆ M , the map I 7→ µ I ( A ) is well-defined ν -almost everywhere and is a ν -measurable map. When a subset S ⊆ T ◦ [ u ] is ν -measurable then π − ( S ) ⊆ Strain [ u ] is a measurable subset of M .(ii) For any Lebesgue-measurable set A ⊆ M , µ ( A ∩ Strain [ u ]) = Z T ◦ [ u ] µ I ( A ) dν ( I ) . (iii) For ν -almost any I ∈ T ◦ [ u ] , the measure µ I is an N -curvature needle supported on I ⊆ M . Furthermore, the set A ⊆ R and the minimizing geodesic γ : A → M fromDefinition 3.19 may be selected so that I = γ ( A ) and so that u ( γ ( t )) = t for all t ∈ A. Proof.
Apply Lemma 3.18 to obtain certain measures ν and { µ I } I∈ T ◦ [ u ] . Applying Lemma3.18(iii) and Proposition 3.20, we learn that µ I is an N -curvature needle supported on I for ν -almost any I ∈ T ◦ [ u ] . Together with Definition 3.17(v), this proves conclusion (iii).Conclusions (i) and (ii) follow from Lemma 3.18(i) and Lemma 3.18(ii), respectively. Proof of Theorem 1.4.
Recall from Section 1 that the weighted Riemannian manifold ( M , d, µ ) satisfies the curvature-dimension condition CD ( κ, N ) when Ric µ,N ( v, v ) ≥ κ for any p ∈ M , v ∈ T p M , | v | = 1 . Glancing at Definition 1.1 and Definition 3.19, we see that under curvature-dimension con-dition CD ( κ, N ) , any N -curvature needle is in fact a CD ( κ, N ) -needle. The theorem thusfollows from Theorem 3.23. In this section we prove Theorem 1.5, following the approach of Evans and Gangbo [17].We assume that ( M , d, µ ) is an n -dimensional, geodesically-convex, weighted Riemannianmanifold of class CD ( κ, N ) , where n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Supposethat f : M → R is a µ -integrable function with Z M f dµ = 0 . (1) ssume also that there exists a point x ∈ M with Z M | f ( x ) | · d ( x , x ) dµ ( x ) < ∞ . (2)It follows from (2) that for any -Lipschitz function v : M → R , Z M | f v | dµ ≤ | v ( x ) | Z M | f | dµ + Z M | f ( x ) | d ( x , x ) dµ ( x ) < ∞ , as | v ( x ) | ≤ | v ( x ) | + d ( x , x ) for all x ∈ M . Conclusion (A) of Theorem 1.5 follows fromthe following standard lemma: Lemma 4.1.
There exists a -Lipschitz function u : M → R with Z M uf dµ = sup (cid:26)Z M vf dµ ; v : M → R , k v k Lip ≤ (cid:27) . (3) Proof.
Recall that ( M , d ) is a locally-compact, separable, metric space (see, e.g., Section2.1). For k = 1 , , . . . let v k : M → R be a -Lipschitz function such that Z M v k f dµ k →∞ −→ sup k v k Lip ≤ Z M vf dµ. Since R M f dµ = 0 , then we may add a constant to v k and assume that v k ( x ) = 0 forall k . By the Arzela-Ascoli theorem, there exists a subsequence v k i that converges locally-uniformly to a -Lipschitz function u : M → R with u ( x ) = 0 . Since | v k ( x ) | ≤ d ( x , x ) for all x ∈ M and k ≥ , then we may apply the dominated convergence theorem thanks to(2). We conclude that Z M uf dµ = lim i →∞ Z M v k i f dµ = sup k v k Lip ≤ Z M vf dµ. The maximization problem in Lemma 4.1 is dual to the L -Monge-Kantorovich problemin the theory of optimal transportation. For information about the Monge-Kantorovich L -transportation problem, we refer the reader to the book by Kantorovich and Akilov [27,Section VIII.4] and to the papers by Ambrosio [1], Evans and Gangbo [17] and Gangbo[20].Most of the remainder of this section is devoted to the proof of conclusions (B) and (C)of Theorem 1.5. To that end, let us fix a -Lipschitz function u : M → R such that Z M uf dµ = sup k v k Lip ≤ Z M vf dµ. (4)Recall the definition of a transport ray from Section 2.1. The set T [ u ] is the collection ofall transport rays associated with u . From the definition of a transport ray, for any x, y ∈ M , | u ( x ) − u ( y ) | = d ( x, y ) ⇐⇒ ∃I ∈ T [ u ] , x, y ∈ I . (5) transport ray is called degenerate when it is a singleton. By the maximality property oftransport rays (see Definition 2.2), for any x ∈ M , { x } ∈ T [ u ] ⇐⇒ ∀ x = y ∈ M , | u ( y ) − u ( x ) | < d ( x, y ) . (6)Define Loose [ u ] ⊆ M to be the union of all degenerate transport rays associated with u .Thus, Loose [ u ] = { x ∈ M ; { x } ∈ T [ u ] } . By the maximality property of transport rays, for any
I ∈ T [ u ] , I ∩
Loose [ u ] = ∅ ⇐⇒ ∃ x ∈ Loose [ u ] , I = { x } . (7)From Lemma 2.3, any transport ray I ∈ T [ u ] is the image of a minimizing geodesic. Therelative interior of I ∈ T [ u ] is empty if and only if I is a singleton. Recall from Lemma 2.8that T ◦ [ u ] is the collection of all relative interiors of non-degenerate transport rays associatedwith u , while Strain [ u ] = [ I∈ T ◦ [ u ] I . (8)It follows from (7) and (8) that Strain [ u ] ∩ Loose [ u ] = ∅ . (9)Finally, let us set Ends [ u ] = M \ ( Loose [ u ] ∪ Strain [ u ]) . Thus, Strain [ u ] , Ends [ u ] and Loose [ u ] are three disjoint sets whose union equals M . Lemma 4.2. µ ( Ends [ u ]) = λ M ( Ends [ u ]) = 0 .Proof. Recall from Section 3.1 that for a subset A ⊆ M , we define Ends ( A ) ⊆ M to bethe union of all relative boundaries of transport rays intersecting A . We claim that Ends [ u ] ⊆ Ends ( Strain [ u ]) . (10)Indeed, if x ∈ Ends [ u ] , then { x } is not a transport ray as x Loose [ u ] . From Definition 2.2,there exists a non-degenerate transport ray I ∈ T [ u ] that contains x . Since x Strain [ u ] ,then the point x ∈ I does not belong to the relative interior of I . Consequently, x belongs tothe relative boundary of I . Since the relative interior of I is non-empty, then I ∩
Strain [ u ] = ∅ and consequently x ∈ Ends ( Strain [ u ]) . Thus (10) is proven. Next, according to Lemma3.15, there exist ray clusters R , R , . . . such that Strain [ u ] = ∪ i R i . Hence, Ends ( Strain [ u ]) = ∞ [ i =1 Ends ( R i ) . (11)However, Lemma 3.8 asserts that λ M ( Ends ( R i )) = 0 for any i ≥ . Consequently, from(10) and (11) we conclude that λ M ( Ends [ u ]) = 0 . Since µ is absolutely-continuous with respect to λ M , the lemma is proven. he following lemma, just like our entire proof of conclusion (B), is similar to the massbalance lemma of Evans and Gangbo [17, Lemma 5.1]. For a set K we write K for thefunction that equals one on K and vanishes elsewhere. Lemma 4.3.
Let K ⊆ M be a compact set. For δ > denote u δ ( x ) = inf y ∈M [ u ( y ) + d ( x, y ) − δ · K ( y )] for x ∈ M . (12) Let A ⊆ M be the union of all transport rays I ∈ T [ u ] that intersect K . Then there existsa function v : M → [0 , such that lim δ → + u ( x ) − u δ ( x ) δ = x ∈ M \ Av ( x ) x ∈ A \ K x ∈ K (13) Moreover, for any x ∈ M and δ > we have that ≤ u ( x ) − u δ ( x ) ≤ δ .Proof. Since k u k Lip ≤ then for all x ∈ M , u δ ( x ) = inf y ∈M [ u ( y ) + d ( x, y ) − δ · K ( y )] ≥ inf y ∈M [ u ( y ) + d ( x, y )] − δ ≥ u ( x ) − δ. (14)The “Moreover” part of the lemma follows from (14) and from the simple inequality u δ ( x ) ≤ u ( x ) . For any x, y ∈ M we have that u ( x ) − u ( y ) − d ( x, y ) ≤ as u is -Lipschitz.Therefore, for any x ∈ M , the function δ u ( x ) − u δ ( x ) δ = sup y ∈M (cid:20) u ( x ) − u ( y ) − d ( x, y ) δ + 1 K ( y ) (cid:21) is non-decreasing in δ > . Hence the limit in (13) exists and belongs to [0 , for all x ∈ M .Next, fix a point x ∈ M \ A . Then for any y ∈ K , the points x and y do not belong to thesame transport ray. Therefore | u ( x ) − u ( y ) | < d ( x, y ) and hence u ( y ) + d ( x, y ) > u ( x ) forany y ∈ K . By the compactness of K , there exists δ x > such that inf y ∈ K [ u ( y ) + d ( x, y )] = min y ∈ K [ u ( y ) + d ( x, y )] > u ( x ) + δ x . (15)Since u is -Lipschitz, then u ( y ) + d ( x, y ) ≥ u ( x ) for all y ∈ M . Consequently, from (12)and (15), u δ ( x ) = u ( x ) when < δ < δ x . This proves (13) in the case where x ∈ M \ A . Consider now the case where x ∈ K . Then, u δ ( x ) = inf y ∈M [ u ( y ) + d ( x, y ) − δ · K ( y )] ≤ u ( x ) + d ( x, x ) − δ = u ( x ) − δ. (16)From (14) and (16) we learn that u δ ( x ) = u ( x ) − δ for any x ∈ K and δ > . This proves(13) for the case where x ∈ K . ollowing Evans and Gangbo [17, Lemma 5.1], we say that a measurable subset A ⊆ M is a transport set associated with u if for any x ∈ A \ Ends [ u ] and I ∈ T [ u ] , x ∈ I = ⇒ I ⊆ A. (17)In other words, a transport set A is a measurable set that contains all transport rays intersect-ing A \ Ends [ u ] . Lemma 4.4.
Let A ⊆ M be a transport set associated with u . Then, Z A f dµ ≥ . Proof.
It suffices to prove that R A f dµ > − ε for any ε > . To this end, let us fix ε > .According to Lemma 4.2, the set Ends [ u ] is of µ -measure zero. Therefore, Z A \ Ends [ u ] | f | dµ = Z A | f | dµ < ∞ . (18)Since µ is a Borel measure, it follows from (18) that there exists a compact K ⊆ A \ Ends [ u ] such that Z A \ K | f | dµ < ε. (19)For δ > we define u δ : M → R as in (12). Then u δ is a -Lipschitz function, since it isthe infimum of a family of -Lipschitz functions. From (4), Z M u − u δ δ · f · dµ ≥ for all δ > . (20)For k = 1 , , . . . denote v k ( x ) = u ( x ) − u /k ( x )1 /k ( x ∈ M ) . (21)From the “Moreover” part of Lemma 4.3 we know that ≤ v k ( x ) ≤ for all x ∈ M and k ≥ . According to Lemma 4.3, there exists a function v : M → [0 , such that v k ( x ) −→ v ( x ) for all x ∈ M . Furthermore, by (13), v ( x ) = (cid:26) x ∈ M \ A x ∈ K (22)where we used the fact that A is a transport set and hence A contains all transport raysintersecting K ⊆ A \ Ends [ u ] . Since f is µ -integrable and | v k ( x ) | ≤ for all k and x , thenwe may use the dominated convergence theorem and conclude from (20) and (22) that ≤ Z M v k f dµ k →∞ −→ Z M vf dµ = Z A vf dµ = Z A \ K vf dµ + Z K f dµ. (23)Since v ( x ) ∈ [0 , for all x ∈ M , then according to (19) and (23), Z K f dµ ≥ − Z A \ K vf dµ ≥ − Z A \ K | f | dµ > − ε, and the lemma is proven. orollary 4.5. Let A ⊆ M be a transport set associated with u . Then, Z A f dµ = 0 . Proof.
In view of Lemma 4.4 we only need to prove that R A f dµ ≤ . Note that the supre-mum of R v ( − f ) dµ over all -Lipschitz functions v is attained for v = − u . Furthermore, T [ u ] = T [ − u ] and Ends [ u ] = Ends [ − u ] . Therefore A is also a transport set associated with − u . We may therefore apply Lemma 4.4 with f replaced by − f and with u replaced by − u .By the conclusion of Lemma 4.4, R A ( − f ) dµ ≥ , and the corollary is proven.Recall that T ◦ [ u ] is a partition of Strain [ u ] , and that π : Strain [ u ] → T ◦ [ u ] is the partitionmap, i.e., x ∈ π ( x ) ∈ T ◦ [ u ] for all x ∈ Strain [ u ] . Lemma 4.6.
Let S ⊆ T ◦ [ u ] . Assume that π − ( S ) ⊆ Strain [ u ] is a measurable subset of M .Then, Z π − ( S ) f dµ = 0 . Proof.
Recall that
Strain [ u ] , Loose [ u ] and Ends [ u ] are three disjoint sets whose union equals M . In view of Lemma 4.2 and Corollary 4.5, it suffices to show that there exists a transportset A ⊆ M with π − ( S ) ⊆ A and A \ π − ( S ) ⊆ Ends [ u ] . (24)Any J ∈ T ◦ [ u ] is the relative interior of a non-degenerate transport ray. Since transportrays are closed sets, it follows from Lemma 2.3 that the closure J of any J ∈ T ◦ [ u ] is atransport ray. We claim that for any J ∈ T ◦ [ u ] , J \ J ⊆ M \ ( Loose [ u ] ∪ Strain [ u ]) = Ends [ u ] . (25)Indeed, it follows from (7) that J is contained in M \
Loose [ u ] since it is a transport raywhose relative interior is non-empty. Any point x ∈ J belonging to Strain [ u ] must lie in J ,according to Lemma 2.5. Hence J \ J is disjoint from
Strain [ u ] , and (25) is proven. Denote A = [ J ∈ S J . (26)Clearly A ⊇ S J ∈ S J = π − ( S ) . It follows from (25) that A \ π − ( S ) = ( [ J ∈ S J ) \ ( [ J ∈ S J ) ⊆ [ J ∈ S ( J \ J ) ⊆ Ends [ u ] . (27)Now (24) follows from (27) and from the fact that A ⊇ S J ∈ S J = π − ( S ) . All thatremains is to show that A ⊆ M is a transport set. Since π − ( S ) is assumed to be measurableand Ends [ u ] is a null set, then the measurability of A follows from (24). In order to prove ondition (17) and conclude that A is a transport set, we choose x ∈ A \ Ends [ u ] and I ∈ T [ u ] with x ∈ I . (28)Since x ∈ A \ Ends [ u ] , then necessarily x ∈ π − ( S ) ⊆ Strain [ u ] according to (27). Denoteby J the relative interior of the transport ray I . From (28) and Lemma 2.5 we deduce that I is the unique transport ray containing x , and that x ∈ J . Since x ∈ π − ( S ) , we learn that J ∈ S . From (26) we conclude that I = J ⊆ A . We have thus verified condition (17) andproved that A is a transport set associated with u . The lemma is proven. Proof of Theorem 1.5(B).
The measurability of
Strain [ u ] follows from Lemma 2.9. Wewould like to show that f ( x ) = 0 for µ -almost any point x ∈ M \ Strain [ u ] . (29)We learn from (7) and from the definition (17) that any measurable set S ⊆ Loose [ u ] is atransport set associated with u . From Corollary 4.5, for any measurable set S ⊆ Loose [ u ] , Z S f dµ = 0 . This implies that f vanishes µ -almost everywhere in Loose [ u ] . Recall that M \
Strain [ u ] = Loose [ u ] ∪ Ends [ u ] . In view of Lemma 4.2, we conclude (29).Next, let ν and { µ I } I∈ T ◦ [ u ] be measures on T ◦ [ u ] and M , respectively, satisfying con-clusions (i), (ii) and (iii) of Theorem 1.4. Thus, for ν -almost any I ∈ T ◦ [ u ] , the measure µ I is a CD ( κ, N ) -needle supported on I . Additionally, for any measurable set A ⊆ M , µ ( A ∩ Strain [ u ]) = Z T ◦ [ u ] µ I ( A ) dν ( I ) , (30)and in particular, the map I 7→ µ I ( A ) is ν -measurable. It follows from (30) that for any µ -integrable function g : M → R , Z Strain [ u ] gdµ = Z T ◦ [ u ] (cid:18)Z I g ( x ) dµ I ( x ) (cid:19) dν ( I ) . (31)In order to complete the proof, we need to show that Z I f dµ I = 0 for ν -almost any I ∈ T ◦ [ u ] . (32)Since f is µ -integrable, from (31) the map I 7→ R I f dµ I is ν -integrable, and in particular, itis well-defined for ν -almost any I ∈ T ◦ [ u ] . The desired conclusion (32) would follow oncewe show that for any ν -measurable subset S ⊆ T ◦ [ u ] , Z S (cid:18)Z I f dµ I (cid:19) dν ( I ) = 0 . (33) hus, let us fix a ν -measurable subset S ⊆ T ◦ [ u ] . From Theorem 1.4(i), the set π − ( S ) is ameasurable subset of M . According to Lemma 4.6, Z π − ( S ) f dµ = Z Strain [ u ] f ( x ) · π − ( S ) ( x ) dµ ( x ) . (34)By using (31) and (34), Z Strain [ u ] f · π − ( S ) dµ = Z T ◦ [ u ] S ( I ) · (cid:18)Z I f dµ I (cid:19) dν ( I ) = Z S (cid:18)Z I f dµ I (cid:19) dν ( I ) . Recalling that S ⊆ T ◦ [ u ] was an arbitrary ν -measurable set, we see that (33) is proven. Theproof is complete. Proof of Theorem 1.2.
From Theorem 1.4, Theorem 1.5(A) and Theorem 1.5(B) we obtaina -Lipschitz function u : M → R , a certain measure ν on T ◦ [ u ] and a family of measures { µ I } I∈ T ◦ [ u ] on the manifold M . We make the following formal manipulations: Let Ω be thepartition of M obtained by adding the singletons {{ x } ; x ∈ M \ Strain [ u ] } to the partition T ◦ [ u ] of Strain [ u ] . Let ˜ ν be the push-forward of µ | M\ Strain [ u ] under the map x
7→ { x } to theset Ω . Define ν = ν + ˜ ν, a measure on Ω . Finally, for x ∈ M \ Strain [ u ] write µ { x } for Dirac’s delta measure at x .From Theorem 1.4, for any measurable subset A ⊆ M , µ ( A ) = µ ( A ∩ Strain [ u ]) + µ ( A \ Strain [ u ])= Z T ◦ [ u ] µ I ( A ) dν ( I ) + Z M\ Strain [ u ] µ { x } ( A ) dµ ( x ) = Z Ω µ I ( A ) dν ( I ) . Thus conclusion (i) holds true with ν replaced by ν . For ν -almost any I ∈ Ω , we havethat either I is a singleton, or else I is the relative interior of a transport ray on which the CD ( κ, N ) -needle µ I is supported. We have thus verified conclusion (ii). Theorem 1.5(B)shows that f vanishes almost everywhere in M \
Strain [ u ] . Conclusion (iii) thus followsfrom Theorem 1.5(B). Proof of Theorem 1.5(C).
This follows from Theorem 1.4(iii) and the previous proof.
Corollary 4.7 (“Uniqueness of maximizer”) . Let ( M , d, µ ) be an n -dimensional, geodesically-convex, weighted Riemannian manifold. Suppose that f : M → R is a µ -integrable functionwith R M f dµ = 0 and that there exists x ∈ M with R M d ( x , x ) | f ( x ) | dµ ( x ) < + ∞ . As-sume furthermore that µ ( { x ∈ M ; f ( x ) = 0 } ) = 0 . (35) Let u , u : M → R be -Lipschitz functions with Z M u f dµ = Z M u f dµ = sup (cid:26)Z M uf dµ ; u : M → R , k u k Lip ≤ (cid:27) . (36) Then u − u is a constant function. roof. A -Lipschitz function u : M → R for which the supremum in (36) is attained iscalled here a maximizer . According to (35) and Theorem 1.5(B), the set M \
Strain [ u ] is aLebesgue-null set for any maximizer u . From Lemma 2.4 we deduce that for any maximizer u : M → R , |∇ u ( x ) | = 1 for almost any x ∈ M . Suppose now that u and u are two maximizers. Then also ( u + u ) / is a maximizer.Therefore for almost any x ∈ M , |∇ u ( x ) | = |∇ u ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) + ∇ u ( x )2 (cid:12)(cid:12)(cid:12)(cid:12) = 1 . Consequently ∇ u = ∇ u almost everywhere, and hence u − u ≡ Const .The CD ( κ, N ) curvature-dimension condition was used in our argument only in order todeduce that N -curvature needles are CD ( κ, N ) -needles. The “ N -curvature needle” variantof Theorem 1.4 is rendered as Theorem 3.23 above. Next we formulate an N -curvaturevariant of Theorem 1.5: Theorem 4.8.
Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) is an n -dimensional weighted Riemannian manifold which is geodesically-convex. Let f : M → R be a µ -integrable function with R M f dµ = 0 . Assume that there exists a point x ∈ M with R M | f ( x ) | · d ( x , x ) dµ ( x ) < ∞ . Then,(A) There exists a -Lipschitz function u : M → R such that Z M uf dµ = sup k v k Lip ≤ Z M vf dµ. (B) For any such function u , the function f vanishes µ -almost everywhere in M\ Strain [ u ] .Furthermore, let ν and { µ I } I∈ T ◦ [ u ] be measures on T ◦ [ u ] and M , respectively, satis-fying conclusions (i), (ii) and (iii) of Theorem 3.23. Then for ν -almost any I ∈ T ◦ [ u ] , Z I f dµ I = 0 . The proof of Theorem 4.8 is almost identical to the proof of Theorem 1.5. The onlydifference is that one needs to appeal to Theorem 3.23 rather than to Theorem 1.4 ratherthan, and to replace the words “ CD ( κ, N ) -needle” by “ N -curvature needle” throughout theproof. Remark 4.9.
Similarly, Theorem 1.2 and Theorem 1.3 remain valid without the CD ( κ, N ) -assumption, yet one has to replace the words “ CD ( κ, N ) -needle” by “ N -curvature needle”. One-dimensional log-concave needles are quite well-understood. Theorem 1.2 allows usto reduce certain questions pertaining to Riemannian manifolds whose Ricci curvature isnon-negative, to analogous questions for one-dimensional log-concave needles. .1 The inequalities of Buser, Ledoux and E. Milman Let M be a Riemannian manifold with distance function d . For a subset S ⊆ M and ε > denote S ε = (cid:26) x ∈ M ; inf y ∈ S d ( x, y ) < ε (cid:27) , the ε -neighborhood of the set S . The next proposition was proven by E. Milman [32], im-proving upon earlier results by Buser [8] and by Ledoux [28]: Proposition 5.1.
Let n ≥ , R > . Assume that ( M , d, µ ) is an n -dimensional weightedRiemannian manifold of class CD (0 , ∞ ) which is geodesically-convex with µ ( M ) = 1 .Assume that for any -Lipschitz function u : M → R , inf α ∈ R Z M | u ( x ) − α | dµ ( x ) < R. (1) Then for any measurable set S ⊆ M and < ε < R , µ ( S ε \ S ) ≥ c · εR · µ ( S ) · (1 − µ ( S )) , where c > is a universal constant. It is well-known that the optimal choice of α in (1) is the median of the function u . Theexpectation E = R M udµ is also a reasonable choice for the parameter α , since R M | u − E | dµ is at most twice as large as the actual infimum in (1). We begin the proof of Proposition5.1 with the following standard estimate from the theory of one-dimensional log-concavemeasures: Lemma 5.2.
Let
R > , let A ⊆ R be a non-empty, open connected set, let Ψ : A → R be aconvex function with R A e − Ψ < ∞ , and let η be the measure supported on A whose densityis e − Ψ . Suppose that R = R A | t | dη ( t ) /η ( R ) . Then for any < t < , < ε < R and ameasurable subset S ⊆ R , η ( S ) = t · η ( R ) = ⇒ η ( S ε \ S ) ≥ c · εR · t (1 − t ) · η ( R ) , (2) where c > is a universal constant.Proof. We may add a constant to Ψ and stipulate that η ( R ) = 1 . We may rescale andassume furthermore that R = R A | t | dη ( t ) = 1 . According to Bobkov [6, Proposition 2.1], itsuffices to prove (2) under the additional assumption that S is a half-line in R with η ( S ) = t .Reflecting Ψ if necessary, we may suppose that S takes the form S = ( −∞ , a ) for some a ∈ A . Furthermore, we may assume that η (( a, a + ε )) ≤ min { t, − t } / . (3)Indeed, if (3) fails then η ( S ε \ S ) = η (( a, a + ε )) ≥ ( ε/R ) · t (1 − t ) / and (2) holds true.For x ∈ R and < s < denote Φ( x ) = Z x −∞ e − Ψ , I ( s ) = exp( − Ψ(Φ − ( s ))) . ince Ψ is convex, then I : (0 , → (0 , ∞ ) is a well-defined concave function accordingto Bobkov [7, Lemma 3.2]. Furthermore, since R A | t | dη ( t ) = 1 then I (1 / ≥ c where c > is a universal constant, as is shown in [7, Section 3]. Therefore, by the concavity ofthe non-negative function I : (0 , → R , I ( t ) ≥ c · min { t, − t } for all < t < . (4)According to (3) and (4), η (( a, a + ε )) ≥ ε · inf x ∈ ( a,a + ε ) ∩ A e − Ψ( x ) ≥ ε · inf s ∈ [ t,t +min { t, − t } / I ( s ) ≥ ε · c · min { t, − t } , and (2) is proven. Proof of Proposition 5.1.
Denote t = µ ( S ) ∈ [0 , . We may assume that t ∈ (0 , , asotherwise there is nothing to prove. Set f ( x ) = 1 S ( x ) − t for x ∈ M . Then R M f dµ = 0 ,and certainly for any x ∈ M , Z M | f ( x ) | · d ( x , x ) dµ ( x ) ≤ | t + 1 | · Z M d ( x , x ) dµ ( x ) < ∞ , where the integrability of the -Lipschitz function x d ( x , x ) follows from (1). Ap-plying Theorem 1.5, we obtain a certain -Lipschitz function u : M → R and measures ν and { µ I } I∈ T ◦ [ u ] on T ◦ [ u ] and M respectively. It follows from (1) that after adding anappropriate constant to the -Lipschitz function u , we have Z M | u | dµ ≤ R. (5)For ν -almost any I ∈ T ◦ [ u ] we know that R I f dµ I = 0 . Consequently, for ν -almost any I ∈ T ◦ [ u ] , µ I ( S ) = t · µ I ( M ) < ∞ . (6)From Theorem 1.5(B), the function f vanishes µ -almost everywhere outside Strain [ u ] , butour function f ( x ) = 1 S ( x ) − t never vanishes in M . Hence Strain [ u ] is a set of a full µ -measure. From Theorem 1.4(ii) and from (5) we thus obtain that Z T ◦ [ u ] (cid:18)Z I | u | dµ I (cid:19) dν ( I ) = Z Strain [ u ] | u | dµ = Z M | u | dµ ≤ R. (7)Denote B = (cid:26) I ∈ T ◦ [ u ] ; Z I | u | dµ I ≤ R · µ I ( M ) (cid:27) . (8)Since µ ( M ) = µ ( Strain [ u ]) = 1 then R T ◦ [ u ] µ I ( M ) dν ( I ) = 1 . From (7) and the Markov-Chebyshev inequality, Z B µ I ( M ) dν ( I ) ≥ . (9)Furthermore, µ I is a log-concave needle (i.e., a CD (0 , ∞ ) -needle) for ν -almost any I ∈ B .We would like to show that for ν -almost any I ∈ B and any < ε < R , µ I ( S ε \ S ) ≥ c · εR · t (1 − t ) · µ I ( M ) , (10) or a universal constant c > . Let us fix I ∈ B such that µ I is a log-concave needle forwhich (6) holds true. Let A ⊆ R , Ψ : A → R and γ : A → M be as in Definition 1.1. Then A ⊆ R is a non-empty, open, connected set and Ψ : A → R is smooth and convex. FromTheorem 1.4(iii) we know that I = γ ( A ) and u ( γ ( t )) = t for all t ∈ A. (11)Since I ∈ B , we may apply Lemma 5.2 thanks to (6), (8) and (11). The conclusion ofLemma 5.2 implies (10). Consequently, for any < ε < R , µ ( S ε \ S ) = Z T ◦ [ u ] µ I ( S ε \ S ) dν ( I ) ≥ Z B µ I ( S ε \ S ) dν ( I ) ≥ c εR · t (1 − t ) · Z B µ I ( M ) dν ( I ) . The proposition now follows from (9).Proposition 5.1 is stated and proved in the particular case where κ = 0 and N = ∞ .For general κ and N , an appropriate CD ( κ, N ) -variant of the one-dimensional Lemma 5.2would lead to a CD ( κ, N ) -variant of the n -dimensional Proposition 5.1. For κ ∈ R , = N ∈ R ∪ { + ∞} and D ∈ (0 , + ∞ ) write F κ,N,D for the collection of allmeasures ν supported on the interval (0 , D ) ⊆ R which are CD ( κ, N ) -needles. Accordingto Definition 1.1, a measure ν belongs to F κ,N,D if and only if ν is supported on a non-empty, open interval A ⊆ (0 , D ) with density e − Ψ , where Ψ : A → R is a smooth functionthat satisfies Ψ ′′ ≥ κ + (Ψ ′ ) N − . (1)The term (Ψ ′ ) / ( N − in (1) is interpreted as zero when N = + ∞ . In order to includethe case D = + ∞ , we write F κ,N, + ∞ for the collection of all measures ν on R which are CD ( κ, N ) -needles. Define λ κ,N,D = inf (cid:26) R R | u ′ | dν R R u dν ; ν ∈ F κ,N,D , u ∈ C ∩ L ∩ ( ν ) , Z R udν = 0 , Z R u dν > (cid:27) , where L ∩ ( ν ) is an abbreviation for L ( ν ) ∩ L ( ν ) . There are some cases where λ κ,N,D may be computed explictely. For example, for N ∈ ( −∞ , − ∪ (1 , + ∞ ] , the simple one-dimensional lemma of Payne and Weinberger [34] shows that λ ,N,D = π D . (2)We refer the reader to Bakry and Qian [3] and references therein for generalizations of thefollowing proposition: roposition 5.3. Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) isan n -dimensional weighted Riemannian manifold of class CD ( κ, N ) which is geodesically-convex. Denote D = Diam ( M ) = sup x,y ∈M d ( x, y ) ∈ (0 , + ∞ ] the diameter of M . Then for any C -function f : M → R with f ∈ L ( µ ) ∩ L ( µ ) , Z M f dµ = 0 = ⇒ λ κ,N,D · Z M f dµ ≤ Z M |∇ f | dµ. (3) Proof.
Let f : M → R be a C -function with f ∈ L ∩ ( µ ) and R M f dµ = 0 . ApplyingTheorem 1.2, we see that (3) would follow from the following inequality: for any measure ν on M which is a CD ( κ, N ) -needle, (cid:20) f ∈ L ∩ ( ν ) and Z M f dν = 0 (cid:21) = ⇒ λ κ,N,D · Z M f dν ≤ Z M |∇ f | dν. (4)Thus, let us fix a CD ( κ, N ) -needle ν for which f ∈ L ∩ ( ν ) and R M f dν = 0 . Let A ⊆ R , Ψ : A → R and γ : A → M be as in Definition 1.1. Denoting g = f ◦ γ , we seethat | g ′ ( t ) | ≤ |∇ f ( γ ( t )) | for t ∈ A, as γ is a unit speed geodesic. Hence (4) would follow from the inequality Z A ge − Ψ = 0 = ⇒ λ κ,N,D · Z A g e − Ψ ≤ Z A ( g ′ ) e − Ψ , (5)where g : A → R is a C -function with R A (cid:0) | g | + g (cid:1) e − Ψ < ∞ . The set A is open andconnected, and since γ : A → M is a minimizing geodesic then A is an open interval whoselength is at most D . The smooth function Ψ : A → R satisfies (1), and the desired inequality(5) holds in view of the definition of λ κ,N,D . This completes the proof.The case κ = 0 of Proposition 5.3, with the constant λ ,N,D given by (2), appears inPayne-Weinberger [34] in the Euclidean case, and in Li-Yau [29] and Yang-Zhong [38] inthe Riemannian case. Recall the definition of F κ,N,D from the previous subsection. Recall that A ε stands for the ε -neighborhood of the set A . For < t < and ε > define I κ,N,D ( t, ε ) = inf { ν ( A ε ) ; ν ∈ F κ,N,D , A ⊆ R , ν ( R ) = 1 , ν ( A ) = t } . (1)That is, I κ,N,D ( t, ε ) is the infimal measure of an ε -neighborhood of a subset of measure t .There are cases where the function I κ,N,D may be computed explicitly. For example, when κ > , N = D = ∞ , the infimum in (1) is attained when A is a half-line and ν is a Gaussianmeasure on the real line of variance /κ . See E. Milman [31] and references therein for moreinformation about the function I κ,N,D . roposition 5.4. Let n ≥ , κ ∈ R and N ∈ ( −∞ , ∪ [ n, + ∞ ] . Assume that ( M , d, µ ) isan n -dimensional weighted Riemannian manifold of class CD ( κ, N ) which is geodesically-convex. Assume that µ ( M ) = 1 . Denote D = Diam ( M ) , the diameter of M . Then for anymeasurable set A ⊆ M and ε > , denoting t = µ ( A ) , µ ( A ε ) ≥ I κ,N,D ( t, ε ) . Proof.
Denote f ( x ) = 1 A ( x ) − t . Then R M f dµ = 0 . The proposition follows by applyingTheorem 1.2 and arguing similarly to the proof of Proposition 5.3.Similarly, one may reduce the proof of log-Sobolev or transportation-cost inequalities tothe one-dimensional case by using Theorem 1.2, as well as the proof of the inequalities ofCordero-Erausquin, McCann and Schmuckenschl¨ager [13, 14]. By using Theorem 4.8, it isalso straightforward to reduce the proof of the Brascamp-Lieb inequality and its dimensionalvariants to the one-dimensional case. We will end this section with the proof of the fourfunctions theorem , rendered as Theorem 1.3 above. Proof of Theorem 1.3.
By approximation, we may assume that the function f : M → [0 , + ∞ ) does not vanish in M (for example, replace f by f + εg where g is a positivefunction with suitable integrability properties, and then let ε tend to zero). We claim that forany CD ( κ, N ) -measure η on the Riemannian manifold M for which f , f , f , f ∈ L ( η ) , (cid:18)Z M f dη (cid:19) α (cid:18)Z M f dη (cid:19) β ≤ (cid:18)Z M f dη (cid:19) α (cid:18)Z M f dη (cid:19) β . (2)Indeed, inequality (2) appears in the assumptions of the theorem, but under the additionalassumption that η is a probability measure. By homogeneity, (2) holds true under the addi-tional assumption that η is a finite measure. In the general case, we may select a sequence offinite CD ( κ, N ) -measures η ℓ such that η ℓ ր η , and use the monotone convergence theorem.Thus (2) is proven.Next, denote λ = R M f dµ/ R M f dµ , define f = f − λf , and apply Theorem 1.2.Let Ω , { µ I } I∈ Ω , ν be as in Theorem 1.2. Then for ν -almost any I ∈ Ω we have that f , f , f , f ∈ L ( µ I ) and (cid:18)Z I f dµ I (cid:19) α (cid:18)Z I f dµ I (cid:19) β ≤ (cid:18)Z I f dµ I (cid:19) α (cid:18)Z I f dµ I (cid:19) β (3)as follows from (2) and from the pointwise inequality f α f β ≤ f α f β that holds almost-everywhere in M . However, R I f dµ I = λ R I f dµ I for ν -almost any I ∈ Ω . Thus (3)implies that for ν -almost any I ∈ Ω , λ α/β Z I f dµ I ≤ Z I f dµ I . (4)Integrating (4) with respect to the measure ν yields λ α/β Z M f dµ = λ α/β Z Ω (cid:18)Z I f dµ I (cid:19) dν ( I ) ≤ Z Ω (cid:18)Z I f dµ I (cid:19) dν ( I ) = Z M f dµ. rom the definition of λ we thus obtain (cid:18)Z M f dµ (cid:19) α (cid:18)Z M f dµ (cid:19) β ≤ (cid:18)Z M f dµ (cid:19) α (cid:18)Z M f dµ (cid:19) β , and the theorem is proven. This section contains ideas and conjectures for possible extensions of the results in thismanuscript. First, we conjecture that the results and the arguments presented above maybe generalized to the case of a smooth Finsler manifold. Another interesting generalizationinvolves several constraints . That is, suppose that we are given a weighted Riemannianmanifold ( M , d, µ ) and a µ -integrable function f : M → R k with Z M f dµ = 0 . We would like to understand whether the measure µ may be decomposed into k -dimensionalpieces in a way analogous to Theorem 1.2. Definition 6.1.
Let M and N be geodesically-convex Riemannian manifolds. We declarethat “ M → N has the isometric extension property” if for any subset A ⊆ M and adistance-preserving map f : A → N , there exists a geodesically-convex subset B ⊆ M containing A and an extension of f to a distance-preserving map f : B → N . Lemma 2.1 shows that R → M has the isometric extension property whenever M is ageodesically-convex Riemannian manifold. If M ⊆ R n is a convex set then for any k ≤ n , R k → M has the isometric extension property. Also S k → S n has the isometric extension property,as well as S k → M when M is a geodesically-convex subset of the sphere S n . These factshave direct proofs which do not rely on the Kirszbraun theorem. Let us discuss in greaterdetail the case where M ⊆ R n is an open, convex set. Suppose that u : M → R k is a -Lipschitz map. We may generalize Definition 2.2 as follows: A subset S ⊆ M is a leaf associated with u if | u ( x ) − u ( y ) | = | x − y | for all x, y ∈ S , and if for any S ) S there exist x, y ∈ S with | u ( x ) − u ( y ) | < | x − y | . For any leaf S ⊆ M , the set u ( S ) = { u ( x ) ; x ∈ S} is a closed, convex subset of R k . This follows from the isometric extension property of R k → M . Let us define Strain [ u ] to be the union of all relative interiors of leafs. Write T ◦ [ u ] for the collection of all non-empty relative interiors of leafs. Suppose that µ is ameasure on the convex set M ⊆ R n such that ( M , | · | , µ ) is an n -dimensional weighted iemannian manifold of class CD ( κ, N ) . We conjecture that there exists a measure ν on T ◦ [ u ] and a family of measures { µ S } S∈ T ◦ [ u ] such that µ ( A ∩ Strain [ u ]) = Z T ◦ [ u ] µ S ( A ) dν ( S ) for any measurable A ⊆ M . Additionally, for ν -almost any S ∈ T ◦ [ u ] , the measure µ S is supported on S and ( S , | · | , µ S ) is a weighted Riemannian manifold of class CD ( κ, N ) . In other words, at least in theEuclidean setting, we conjecture that Theorem 1.4 admits a direct generalization to functions u : M → R k . Perhaps the generalization works whenever u : M → N is -Lipschitz,where N → M has the isometric extension property, and we require certain bounds onsectional curvatures. Moreover, in the Euclidean setting, we believe that Theorem 1.5 maybe generalized as follows: Assume that f : M → R k satisfies R M f dµ = 0 and also R M | f ( x ) | · d ( x , x ) dµ ( x ) < + ∞ for a certain x ∈ M . Let us maximize Z M h f, u i dµ (1)among all -Lipschitz functions u : M → R k . One may use Kirszbraun’s theorem andprove that for any maximizer u : M → R k and for ν -almost any leaf S ∈ T ◦ [ u ] , Z S f dµ S = 0 and Z M h f, u i dµ S = sup (cid:26)Z S h f, v i dµ S ; v : S → R k , k v k Lip ≤ (cid:27) . Remark 6.2.
The bisection method outlined in Section 1 has one significant advantage com-pared to our results. The methods discussed in this manuscript are very much linear , as weobtain a geodesic foliation from the linear maximization problem (1). In comparison, thebisection method works only in symmetric spaces such as R n or S n , but in these spaces itoffers more flexibility, since one may devise various linear and non-linear rules for the bisec-tion procedure. This flexibility is exploited artfully by Gromov [24]. It is currently unclear tous whether one may arrive at an integrable foliation in the situations considered by Gromov[24].Another possible research direction is concerned with CD ( κ, N + 1) -needles in onedimension. It seems that many concepts and results from convexity theory admit generaliza-tions to the class of CD ( κ, N + 1) -needles. For example, when = N ∈ R and κ/N > ,we may define a Legendre-type transform of a function f : R → [0 , + ∞ ] by setting f ∗ ( s ) = inf t ; f ( t ) < + ∞ g ( s + t ) f ( t ) for s ∈ R , (2)where g ( t ) = (cid:26) sin (cid:18)r κN · t (cid:19) · [0 ,π ] (cid:18)r κN · t (cid:19)(cid:27) N nd we agree that g ( s + t ) / ≡ + ∞ and that N = 0 when N ∈ (0 , + ∞ ) and N = + ∞ when N ∈ ( −∞ , . It seems that the function f ∗ is either a density of a CD ( κ, N + 1) -needle in R , or else it is a limit of such densities. We say that a function f : R → [0 , + ∞ ] is ( κ, N + 1) -concave if the set { t ∈ R ; f ( t ) > R · g ( s + t ) } is connected for all R > , s ∈ R . Perhaps the transform (2) is an order-reversing involutionon the class of upper semi-continuous ( κ, N + 1) -concave functions on R .One reason for investigating one-dimensional CD ( κ, N ) -needles is that CD ( κ, N ) -needles may be further decomposed into needles of a simpler form that satisfy a certainlinear constraint. This was already discovered by Lov´asz and Simonovits [30] in the mostinteresting case κ = 0 , N = n . Definition 6.3.
Let κ ∈ R , = N ∈ R ∪ {∞} and let ν be a measure on a certainRiemannian manifold M which is a CD ( κ, N ) -needle. Let A, Ψ and γ be as in Definition1.1. We say that ν is a “ CD ( κ, N ) -affine needle” if the following inequality holds true inthe entire set A : Ψ ′′ = κ + (Ψ ′ ) N − , where in the case N = ∞ , we interpret the term (Ψ ′ ) / ( N − as zero. For x ∈ R write x + = max { x, } . The class of CD ( κ, N ) -affine needles may bedescribed explicitly, as follows:1. The exponential needles are CD (0 , ∞ ) -affine needles, for which the function e − Ψ isan exponential function restricted to the open, connected set A . That is, the function e − Ψ takes the form A ∋ t α · e β · t for certain β ∈ R , α > . The κ -log-affine needles are CD ( κ, ∞ ) -affine needles, forwhich Ψ( t ) − κt / is an affine function in the open, connected set A .2. The N -affine needles are CD (0 , N + 1) -affine needles with = N ∈ R , for which f /N is an affine function in the open, connected set A .3. For = κ ∈ R and = N ∈ R , the CD ( κ, N + 1) -affine needles satisfy, for all t ∈ A , e − Ψ( t ) = (cid:8) α · sin (cid:0)p κN t − β (cid:1) · [0 ,π ] (cid:0)p κN t − β (cid:1)(cid:9) N + κ/N > α + tβ ) N + κ = 0 (cid:0) α · sinh (cid:0)p | κN | · t (cid:1) + β · cosh (cid:0)p | κN | · t (cid:1)(cid:1) N + κ/N < for some α, β ∈ R .In the case where N ∈ (0 , + ∞ ] and κ ≥ it seems pretty safe to make the following: Conjecture 6.4.
Let µ be a probability measure on R which is a CD ( κ, N + 1) -needle.Let ϕ : R → R be a continuous, µ -integrable function with R R ϕdµ = 0 . Then there existprobability measures { µ α } α ∈ Ω on R and a probability measure ν on the set Ω such that: i) For any Lebesgue-measurable set A ⊆ R we have µ ( A ) = R Ω µ α ( A ) dν ( α ) .(ii) For ν -almost any α ∈ Ω , the measure µ α is either supported on a singleton, or else itis a CD ( κ, N + 1) -affine needle with R R ϕdµ α = 0 . Conjecture 6.4 reduces certain questions on CD ( κ, N + 1) -needles to an inequality in-volving only two or three real parameters. A proof of Conjecture 6.4 in the case where N = + ∞ or κ = 0 follows from Choquet’s integral representation theorem and the resultsof Fradelizi and Gu´edon [19]. We are not sure what should be the correct formulation ofConjecture 6.4 in the case where N < and κ < . Appendix: The Feldman-McCann proof of Lemma 2.22
In this appendix we describe the Feldman-McCann proof of Lemma 2.22. Let M be aRiemannian manifold with distance function d . Fix p ∈ M and let δ = δ ( p ) > be theconstant provided by Lemma 2.17. Thus, U = B M ( p, δ / is a strongly-convex set. As inSection 2.3, for a ∈ U we write U a = exp − a ( U ) ⊆ T a M , a convex subset of T a M . For a ∈ U and X, Y ∈ U a , denoting x = exp a ( X ) , y = exp a ( Y ) we set F a ( X, Y ) = exp − x ( y ) ∈ T x M , and also Φ a ( X, Y ) =
The parallel translate of F a ( X, Y ) along the unique geodesic from x to a. The map Φ a : U a × U a → T a M satisfies | Φ a ( X, Y ) | = | F a ( X, Y ) | = d (exp a X, exp a Y ) . (1)The behavior of Φ a on lines through the origin is quite simple: Since exp a ( sX ) and exp a ( tX ) lie on the same geodesic emanating from a , then for any X ∈ T a M and s, t ∈ R , Φ a ( sX, tX ) = ( t − s ) X when sX, tX ∈ U a . (2)See [18, Section 3.2] for more details about Φ a . Our next lemma is precisely Lemma 14in [18]. The proof given in [18, Lemma 14] is very simple and uses essentially the samenotation as ours, and it is not reproduced here. In fact, the argument is similar to theproof of Lemma 2.21 above, and it relies only on the smoothness of Φ a and on the rela-tion Φ a (0 , Y ) = Y that follows from (2). Lemma A.1.
Let a ∈ U and X, Y , Y ∈ U a . Then, | Φ a ( X, Y ) − Φ a ( X, Y ) − ( Y − Y ) | ≤ ¯ C p · | X | · | Y − Y | , where ¯ C p > is a constant depending only on p . roof of Lemma 2.22 (due to Feldman and McCann [18]). Define δ = δ ( p ) = min (cid:26) · ¯ C p , δ (cid:27) , (3)where ¯ C p > is the constant from Lemma A.1. Both the assumptions and the conclusionof the lemma are not altered if we replace x i , y i by x − i , y − i for i = 0 , , . Applying thisreplacement if necessary, we assume from now on that d ( x , y ) ≤ d ( x , y ) . (4)The points x , x , x , y , y , y belong to B M ( p, δ ) ⊆ U . Recall that the main assumptionof the Lemma is that d ( x i , x j ) = d ( y i , y j ) = σ | i − j | ≤ d ( x i , y j ) for i, j = 0 , , . (5)Define ε := d ( x , y ) . (6)Denote a = x and let X , X , X , Y , Y , Y ∈ U a be such that x i = exp a ( X i ) and y i = exp a ( Y i ) for i = 0 , , . Since a = x then X = 0 . For i = 0 , , we know that x i , y i ∈ B M ( p, δ ) and X i , Y i ∈ U a . It follows from (1), (2)and (5) that | X i | = | Φ a ( X , X i ) | = d ( x , x i ) ≤ δ , | Y i | = | Φ a ( X , Y i ) | = d ( x , y i ) ≤ δ . (7)By using (7) and Lemma A.1, for any R, Z, W ∈ { X , X , X , Y , Y , Y } , | Φ a ( R, Z ) − Φ a ( R, W ) − ( Z − W ) | ≤ ¯ C p · | R | · | Z − W | ≤ C p δ | Z − W | ≤ | Z − W | , (8)where we used (3) in the last passage. By using (1), (6) and also (8) with R = Z = X and W = Y , | Y − X | ≤ · | Φ a ( X , Y ) − Φ a ( X , X ) | = 109 · | Φ a ( X , Y ) | = 109 · d ( x , y ) = 109 · ε, (9)where Φ a ( X , X ) = 0 by (2). From (2), (5) and the fact that X = 0 , σ ≤ d ( x , y ) = | Φ a ( X , Y ) | = | Y | = | ( Y − X ) + ( X − X ) | . (10)Note that | X − X | = | Φ a ( X , X ) | = 2 σ from (1), (2) and (5). Hence, by squaring (10), (2 σ ) ≤ | Y − X | + 2 h Y − X , X − X i + (2 σ ) . (11)According to (5), the point x is the midpoint of the geodesic between x = a and x .Therefore x = exp a ( X ) = exp a (2 X ) and by strong-convexity X = X . Consequently X − X = 2( X − X ) , and from (11) we deduce that h Y − X , X − X i = 12 h Y − X , X − X i ≥ − | Y − X | . (12) ur next goal, like in [18, Lemma 16], is to prove that h Y − X , Y − Y i ≥ − | Y − X | . (13)Begin by applying (2) and (5), in order to obtain σ ≤ d ( y , x ) = | Φ a ( Y , X ) | = | (Φ a ( Y , X ) − Φ a ( Y , Y )) + Φ a ( Y , Y ) | . (14)From (5), the point y is the midpoint of the geodesic between y and y . This implies that F a ( Y , Y ) = 2 F a ( Y , Y ) and therefore Φ a ( Y , Y ) = 2Φ a ( Y , Y ) . Recall that | Φ a ( Y , Y ) | = d ( y , y ) = 2 σ , according to (5). Thus, by squaring (14) and rearranging, −| Φ a ( Y , X ) − Φ a ( Y , Y ) | ≤ h Φ a ( Y , X ) − Φ a ( Y , Y ) , Φ a ( Y , Y ) i = 4 h Φ a ( Y , X ) − Φ a ( Y , Y ) , Φ a ( Y , Y ) − Φ a ( Y , Y ) i . (15)The deduction of (13) from (15) involves several approximations. Begin by using (15) andalso (8) with R = Y , Z = X , W = Y , to obtain − (11 / · | X − Y | ≤ h Φ a ( Y , X ) − Φ a ( Y , Y ) , Φ a ( Y , Y ) − Φ a ( Y , Y ) i . (16)Applying (4), together with (8) for R = Z = X , W = Y , we obtain | Y | = | Φ a ( X , Y ) | ≤ | Φ a ( X , Y ) | = | Φ a ( X , Y ) − Φ a ( X , X ) | ≤ ·| Y − X | . (17)According to Lemma A.1 and (17), for any Z, W ∈ { X , X , X , Y , Y , Y } , | Φ a ( Y , Z ) − Φ a ( Y , W ) − ( Z − W ) | ≤ ¯ C p · | Y | · | Z − W | ≤ C p · | Y − X | · | Z − W | . (18)It follows from (16) and from the case Z = X , W = Y in (18) that − (11 / · | X − Y | (19) ≤ h X − Y , Φ a ( Y , Y ) − Φ a ( Y , Y ) i + 8 ¯ C p | X − Y | · | Φ a ( Y , Y ) − Φ a ( Y , Y ) | . Note that | Φ a ( Y , Y ) − Φ a ( Y , Y ) | ≤ | Y − Y | , as follows from an application of (8) with R = Y , Z = Y , W = Y . We now use (18) with Z = Y and W = Y , and upgrade (19)to − (11 / | X − Y | ≤ h X − Y , Y − Y i + 30 · ¯ C p | X − Y | · | Y − Y | . (20)The next step is to use that | Y − Y | ≤ | Y | + | Y | ≤ δ ≤ / (300 ¯ C p ) according to (3) and(7). Thus (20) implies − (11 / · | X − Y | ≤ h X − Y , Y − Y i + | X − Y | , and (13) follows. From (12) and (13), h Y − X ,Y − X i = h Y − X , ( Y − Y ) + ( Y − X ) + ( X − X ) i≥ − | X − Y | | X − Y | − | X − Y | ≥ · | Y − X | . (21) ccording to (9), (21) and the Cauchy-Schwartz inequality, · ε · | Y − X | ≥ | Y − X | · | Y − X | ≥ h Y − X , Y − X i ≥ · | Y − X | . (22)From (22), | Y − X | ≤ ε. (23)We may summarize (9), (17) and (23) by | Y i − X i | ≤ ε ( i = 0 , , . (24)For i = 0 , , , we use (1), (24) and also (8) with R = Z = X i and W = Y i . This yields d ( x i , y i ) = | Φ a ( X i , Y i ) | = | Φ a ( X i , Y i ) − Φ a ( X i , X i ) | ≤ (11 / · | Y i − X i | ≤ ε, (25)where Φ a ( X i , X i ) = 0 according to (2). The lemma follows from (6) and (25). References [1] Ambrosio, L.,
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