A differential perspective on Gradient Flows on {\sf CAT}(κ)-spaces and applications
AA DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ ) -SPACES AND APPLICATIONS NICOLA GIGLI AND FRANCESCO NOBILI
Abstract.
We review the theory of Gradient Flows in the framework of convex and lowersemicontinuous functionals on
CAT ( κ )-spaces and prove that they can be characterized by thesame differential inclusion y (cid:48) t ∈ − ∂ − E ( y t ) one uses in the smooth setting and more preciselythat y (cid:48) t selects the element of minimal norm in − ∂ − E ( y t ). This generalizes previous results inthis direction where the energy was also assumed to be Lipschitz.We then apply such result to the Korevaar-Schoen energy functional on the space of L and CAT (0) valued maps: we define the Laplacian of such L map as the element of minimal normin − ∂ − E ( u ), provided it is not empty. The theory of gradient flows ensures that the set of mapsadmitting a Laplacian is L -dense. Basic properties of this Laplacian are then studied. Contents
1. Introduction 12. Calculus on
CAT ( κ )-spaces 32.1. CAT ( κ )-spaces 32.2. Tangent cone 52.3. Weak convergence 72.4. Geometric tangent bundle 83. Gradient flows on CAT ( κ )-spaces 93.1. Metric approach 93.2. The object − ∂ − E ( y ) 113.3. Subdifferential formulation 134. Laplacian of CAT (0)-valued maps 154.1. Pullback geometric tangent bundle 154.1.1. The general non-separable case 154.1.2. The separable setting 194.2. The Korevaar-Schoen energy 214.3. The Laplacian of a
CAT (0)-valued map 22References 271.
Introduction
The theory of gradient flows in metric spaces has been initiated by De Giorgi and collaborators[10], [9] (see also the more recent [2]): a basic feature of the approach is to provide a very generalexistence theory - at this level uniqueness is typically lost - without neither curvature assumptionson the space nor semiconvexity of the functional.In this setting gradient flow trajectories ( x t ) of E (or curves of maximal slopes) are defined byimposing the maximal rate of dissipationdd t E ( x t ) = −| ˙ x t | = −| ∂ − E | ( x t ) , a.e. t, a r X i v : . [ m a t h . M G ] D ec NICOLA GIGLI AND FRANCESCO NOBILI where here | ˙ x t | is the metric speed of the curve (see Theorem 2.2) and | ∂ − E | is the slope of E (see(3.1)). It has been later understood ([2], [3], [17], [30], [29]) that if E is λ -convex and the metricspace has some form of some Hilbert-like structure at small scales, then an equivalent formulationcan be given via the so-called Evolution Variational Inequality(EVI) dd t d ( x t , y )2 + E ( x t ) + λ d ( x t , y ) ≤ E ( y ) a.e. t for any choice of point y on the space. See Theorem 3.2 for the precise definitions and [29] fora thorough study of the EVI condition.The geometry of the metric space and the convexity properties of the functional under consid-eration greatly affect the kind of results one can obtain for gradient flows. For the purpose of thismanuscript, the works [28], [22] are particularly relevant: it is showed that the classical Crandall-Liggett generation theorem can be generalized to the metric setting of CAT (0) spaces to producea satisfactory theory of gradient flows for semi-convex and lower semicontinuous functionals.If the metric space one is working on admits some nicely-behaved tangent spaces/cones, onemight hope to give a meaning to the classical defining formula x (cid:48) t ∈ − ∂ − E ( x t ) a.e. t or to its more precise variant(1.1) x (cid:48) + t = the element of minimal norm in − ∂ − E ( x t ) ∀ t > . This has been done in [26], where previous approaches in [32] have been generalized. Here, notably,the basic assumptions on the metric space are of first order in nature (and refer precisely to thestructure of tangent cones) and the energy functional is assumed to be semiconvex and locallyLipschitz. While the convexity assumption is very natural when studying gradient flows (all in all,even in the Hilbert setting many fundamental results rely on such hypothesis), asking for Lipschitzcontinuity is a bit less so: it certainly covers many concrete examples, for instance of functionalsbuilt upon distance functions on spaces satisfying some one-sided curvature bound, but from theanalytic perspective it may be not satisfying: already the Dirichlet energy as a functional on L is not Lipschitz, and the same holds for the Korevaar-Schoen energy we aim to study here.Our motivation to study this topic comes from the desire of providing a notion of Laplacian for CAT (0)-valued Sobolev maps, where here ‘Sobolev’ is intended in the sense of Korevaar-Schoen[24] (see also the more recent review of their theory done in [21]). Denoting by E KS the underlyingnotion of energy and imitating one of the various equivalent definitions for the Laplacian in theclassical smooth and linear setting, one is lead to define the Laplacian of u as the element ofminimal norm in − ∂ − E KS ( u ). This approach of course carries at least two tasks: to define what − ∂ − E is and to show that it is not empty for a generic convex and lower semicontinuous functional E . Providing a reasonable definition for − ∂ − E is not that hard (see Definition 3.6), but is lessobvious to show that this object is not-empty (in particular, minimizing E ( · ) + d ( · ,x )2 τ is of no helphere, see the discussion in Remark 3.7). It is here that the theory of gradient flows comes to help:our main result is that, for semiconvex and lower semicontinuous functions on a CAT ( κ ) space, the analogue of (1.1) holds, see Theorem 3.10.As a byproduct, we deduce that the domain of − ∂ − E is dense in the one of E . A result similarto ours has been obtained in [8] under some additional geometric assumptions on the base space,which in some sense tell that there is the opposite of any tangent vector.As said, we then apply this result to study the Laplacian of CAT (0)-valued Sobolev maps. Letus remark that in this case the relevant metric space L (Ω , Y ¯ y ) is that of L maps from some opensubset Ω of a metric measure space X to a pointed CAT (0) space (Y , ¯ y ) and the energy functionalis the Korevaar-Schoen energy E KS : it is well known that L (Ω , Y ¯ y ) is a CAT (0) space and that E KS is convex and lower semicontinuous, but certainly not Lipschitz, whence the need to generalizeLytchak results to cover also this case.Once we have a notion for − ∂ − E KS we enrich the paper with: DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 3 i) the actual definition of Laplacian ∆ u of a CAT (0)-valued map u (Definition 4.12), whichpays particular attention to the link between the tangent cones in L (Ω , Y ¯ y ), where − ∂ − E KS lives, and the tangent cones in Y, where we think ‘variations’ of u should live,see in particular Propositions 4.5 and 4.8,ii) a basic, weak, integration by parts formula, see Proposition 4.13, which is sufficient toshow that our approach is compatible with the classical one valid in the smooth category,iii) a presentation of a simple and concrete example (Example 4.20) showing why ∆ u seemsto be very much linked to the geometry of Y, but less so to Sobolev calculus on it.Finally, we point out that this note is part of a larger program aiming at stating and provingthe Eells-Sampson-Bochner inequality [13] for Sobolev maps from (open subsets of a) RCD spaceX to a CAT (0) space Y (see [11, 18, 21] for partial results in this direction): knowing what theLaplacian of a
CAT (0)-valued map is, is a crucial step for this program.
Acknowledgement.
We thank A. Lytchak and M. Baˇc´ak for comments on an preliminaryversion of this manuscript. 2.
Calculus on
CAT ( κ ) -spaces CAT ( κ ) -spaces. Let us briefly recall some useful tools in metric spaces (Y , d Y ). Definition 2.1 (Locally AC curve) . Let (Y , d Y ) be a metric space and let I ⊂ R be an interval.A curve I (cid:51) t (cid:55)→ γ t ∈ Y is absolutely continuous if there exists a function g : I (cid:55)→ R + in L ( I ) s.t. (2.1) d Y ( γ t , γ s ) ≤ ˆ ts g ( r ) d r ∀ s ≤ t in I. Moreover, γ is said to be locally absolutely continuous if every point admits a neighbourhood whereit is absolutely continuous. Next, we state the existence of the metric counterpart of ‘modulus of velocity’ of a curve.
Theorem 2.2 (Metric speed) . Let (Y , d Y ) be a metric space and let I ⊂ R be an interval. Then,for every AC curve I (cid:51) t (cid:55)→ γ t ∈ Y , there exists the limit lim h ↓ d Y ( γ t + h , γ t ) h a.e. t ∈ I, which we denote by | ˙ γ t | and call metric speed . Moreover, it is the least, in the a.e. sense, function L ( I ) that can be taken in (2.1). See, for the proof, [2, Theorem 1.1.2]. A curve [0 , (cid:51) t (cid:55)→ γ t ∈ Y is a minimizing constantspeed geodesic (or simply a geodesic) if d Y ( γ t , γ s ) = | t − s | d Y ( γ , γ ), for every t, s ∈ [0 , y to z is unique, we shall denoteit by G zy .For κ ∈ R , we call M κ , the model space of curvature κ , i.e. the simply connected, complete2-dimensional manifold with constant curvature κ , and d κ the distance induced by the metrictensor. This restricts ( M κ , d κ ) to only three possibilities: the hyperbolic space H κ of constantsectional curvature κ , if κ <
0, the plane R with usual euclidean metric, if κ = 0, and the sphere S κ of constant sectional curvature κ , if κ >
0. Also, set D κ := diam( M κ ), i.e. D κ = (cid:26) ∞ is κ ≤ , π √ κ if κ > . We refer to [7, Chapter I.2] for a detailed study of the model spaces M κ .In order to speak of κ -upper bound of the sectional curvature in a geodesic metric space (Y , d Y ),we shall enforce a metric comparison property to geodesic triangles of Y, the intuition being thatthey are ‘thinner’ than in M κ . To define them we start by recalling that if a, b, c ∈ Y is a triple of
NICOLA GIGLI AND FRANCESCO NOBILI points satisfying d Y ( a, b )+ d Y ( b, c )+ d Y ( c, a ) < D κ , then there are points, unique up to isometriesof the ambient space and called comparison points , ¯ a, ¯ b, ¯ c ∈ M κ such that d κ (¯ a, ¯ b ) = d Y ( a, b ) , d κ (¯ b, ¯ c ) = d Y ( b, c ) , d κ (¯ c, ¯ a ) = d Y ( c, a ) . In the case where Y is geodesics (and this will be always assumed), we refer to (cid:52) ( a, b, c ) asthe geodesic triangle in Y consisting in three points a, b, c , the vertices , and a choice of threecorresponding geodesics, the edges , linking pairwise the points. By (cid:52) κ (¯ a, ¯ b, ¯ c ) we denote the sobuilt geodesic triangle in M κ , which from now on we call comparison triangle. A point d ∈ Y issaid to be intermediate between b, c ∈ Y provided d Y ( b, d ) + d Y ( d, c ) = d Y ( b, c ) (this means that d lies on a geodesic joining b and c ). The comparison point of d is the (unique, once we fix thecomparison triangle) point ¯ d ∈ M κ , such that d κ ( ¯ d, ¯ b ) = d Y ( d, b ) , d κ ( ¯ d, ¯ c ) = d Y ( d, c ) . Definition 2.3 ( CAT ( κ )-spaces) . A metric space (Y , d Y ) is called a CAT ( κ ) -space if it is com-plete, geodesic and satisfies the following triangle comparison principle: for any a, b, c ∈ Y satis-fying d Y ( a, b ) + d Y ( b, c ) + d Y ( c, a ) < D κ and any intermediate point d between b, c , denoting by (cid:52) κ (¯ a, ¯ b, ¯ c ) the comparison triangle and by ¯ d ∈ M κ the corresponding comparison point (as said, ¯ a, ¯ b, ¯ c, ¯ d are unique up isometries of M κ ), it holds (2.2) d Y ( a, d ) ≤ d κ (¯ a, ¯ d ) . A metric space (Y , d Y ) is said to be locally CAT ( κ ) if it is complete, geodesic and every point in Y has a neighbourhood which is a CAT ( κ ) -space with the inherited metric. Notice that balls of radius < D κ / M κ are convex, i.e. meaning thatgeodesics with endpoint them lies entirely inside. Hence the comparison property (2.2) grantsthat the same is true on CAT ( κ )-spaces (see [7, Proposition II.1.4.(3)] for the rigorous proof ofthis fact). It is then easy to see that, for the same reasons, (Y , d Y ) is locally CAT ( κ ) providedevery point has a neighbourhood U where the comparison inequality (2.2) holds for every tripleof points a, b, c ∈ U , where the geodesics connecting the points (and thus the intermediate points)are allowed to exit the neighbourhood U .Let us fix the following notation: if (Y , d Y ) is a local CAT ( κ )-space, for every y ∈ Y we set r y := sup (cid:8) r ≤ D κ / B r ( y ) is a CAT ( κ )-space (cid:9) . Notice that in particular B r y ( y ) is a CAT ( κ )-space. The definition trivially grants that r y ≥ r z − d ( y, z ) and thus in particular y (cid:55)→ r y is continuous.Finally, we remark the important fact which will be exploited in the sequelOn CAT ( κ )-spaces, geodesics with endpoint at distance < D κ are unique (up toreparametrization) and vary continuously with respect to the endpoints.For a quantitative version of this fact, see [11, Lemma 2.2]. Finally, it will be important to examinethe case of global CAT (0)-spaces, as they naturally arise as tangent structures of
CAT ( κ )-spaces(see Theorem 2.5 below) and also because we are going to examine CAT (0)-valued maps in Section4. Since M is the euclidean plane R equipped with the euclidean norm, for Y CAT (0) and a, b, c ∈ Y as in Definition 2.3, the defining inequality (2.2) reads d Y ( γ t , a ) ≤ (cid:107) (1 − t )¯ b + t ¯ c − ¯ a (cid:107) , for every t ∈ [0 , γ t is the constant speed geodesic connecting b to c and ¯ a, ¯ b, ¯ c ∈ R arecomparison points. By squaring and expanding the right hand side, we easily obtain the condition(2.3) d ( γ t , a ) ≤ (1 − t ) d ( γ , a ) + t d ( γ , a ) − t (1 − t ) d ( γ , γ ) , for every t ∈ [0 , CAT (0)-spaces) isto be understood as a synthetic deficit of the curvature of Y, with respect to the euclidean plane R (where it holds with equality). In other words, it quantifies how much the triangle (cid:52) ( a, b, c )is ‘thin’ compared to (cid:52) (¯ a, ¯ b, ¯ c ) in the euclidean plane. The advantage of (2.3) is to be morepractical to work with in convex analysis and optimization. DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 5 Tangent cone.
We recall here the notion of tangent cone on a
CAT ( κ )-space, referring tothe above-mentioned bibliography for a much more complete discussion.We define the tangent cone of a CAT ( κ )-space by means of geodesics. Let us start with someconsiderations valid in a general geodesic space Y: as we shall see, the construction is valid onthis generality, but it will benefit from the CAT ( κ ) condition making a suitable calculus possible.Let y ∈ Y, and denote by
Geo y Y the space of (constant speed) geodesics emanating from y and defined on some right neighbourhood of 0. We endow this space with the pseudo-distance d y defined as following:(2.4) d y ( γ, η ) := lim t ↓ d Y ( γ t , η t ) t ∀ γ, η ∈ Geo y Y . It is easy to see that d y naturally induces an equivalence relation on Geo y Y, by simply imposing γ ∼ η if d y ( γ, η ) = 0 . By construction, d y passes to the quotient Geo y Y / ∼ and with (a common)abuse of notation, we still denote d y the distance on the quotient space. The equivalence class ofthe geodesic γ under this relation will be denoted γ (cid:48) . In particular this applies to the geodesics G zy defined on [0 , Geo y Y / ∼ will be denoted by ( G zy ) (cid:48) . Definition 2.4 (Tangent cone) . Let Y be a geodesic space and y ∈ Y . The tangent cone (T y Y , d y ) ,is the completion of ( Geo y Y / ∼ , d y ) . Moreover, we call y ∈ T y Y , the equivalence class of thesteady geodesic at y . A direct consequence of the local
CAT ( κ ) condition is that, for every y ∈ Y , γ, η ∈ Geo y Y, thelimsup in (2.4) is actually a limit. It will be also useful to notice that(2.5) if Y is
CAT (0), t (cid:55)→ d Y ( γ t , η t ) t is non-decreasing ∀ γ, η ∈ Geo y Y , a property which is directly implied by (2.3). A well known (see e.g. [7, Theorem II-3.19]) anduseful fact is that tangent cones at local CAT ( κ ) spaces are CAT (0) spaces:
Theorem 2.5.
Let Y be locally CAT ( κ ) . Then, for every y ∈ Y , the tangent cone (T y Y , d y ) is a CAT (0)-space.
We now build a calculus on the tangent cone that resembles the one of Hilbert spaces. ◦ Multiplication by a positive scalar . Let λ ≥
0. Then the map sending t (cid:55)→ γ t to t (cid:55)→ γ λt is easily seen to pass to the quotient in Geo y Y / ∼ and to be λ -Lipschitz. Hence it can beextended by continuity to a map defined on T y Y, called multiplication by λ . ◦ Norm . | v | y := d y ( v, ◦ Scalar product . (cid:10) v, w (cid:11) y := (cid:2) | v | y + | w | y − d y ( v, w ) (cid:3) . ◦ Sum . v ⊕ w := 2 m , where m is the midpoint of v, w (well-defined because T y Y is a
CAT (0)-space).We report from [11, Theorem 2.9] the following fact:(2.6) for D dense in B r y ( y ) we have that { α ( G wy ) (cid:48) : α ∈ Q + , w ∈ D } is dense in T y Y . Moreover, we recall the following proposition:
Proposition 2.6 (Basic calculus on the tangent cone) . Let Y be locally CAT ( κ ) and y ∈ Y .Then, the four operations defined above are continuous in their variables. The ‘sum’ and the‘scalar product’ are also symmetric. Moreover: d y ( λv, λw ) = λ d y ( v, w ) , (2.7a) (cid:10) λv, w (cid:11) y = (cid:10) v, λw (cid:11) y = λ (cid:10) v, w (cid:11) y , (2.7b) | (cid:10) v, w (cid:11) y | ≤ | v | y | w | y , (2.7c) (cid:10) v, w (cid:11) y = | v | y | w | y if and only if | w | y v = | v | y w, (2.7d) d y ( v, w ) + | v ⊕ w | y ≤ | v | y + | w | y ) , (2.7e) (cid:10) v ⊕ v , w (cid:11) y ≥ (cid:10) v , w (cid:11) y + (cid:10) v , w (cid:11) y (2.7f) NICOLA GIGLI AND FRANCESCO NOBILI for any v, v , v , w ∈ T y Y and λ ≥ .Proof. The continuity of ‘norm’, ‘scalar product’ and ‘multiplication by a scalar’ are obvious bydefinition, the one of ‘sum’ then follows from the continuity of the midpoint of a geodesic as afunction of the extremal points.Points (2.7a), (2.7b), (2.7c), (2.7d), (2.7e) are well known and recalled, e.g., in [11, Proposition2.11]. The concavity property (2.7f) is also well known. A way to prove it is to notice that from(2.7b) and letting m be the midpoint of v , v we get that (cid:10) v ⊕ v , w (cid:11) y = 2 ε − (cid:10) εm, w (cid:11) y = ε − (cid:0) ε | m | y + | w | y − d y ( εm, w ) (cid:1) ∀ ε > . From the fact that T y Y is
CAT (0) and the fact that εm is the midpoint of εv , εv (consequenceof (2.7a)) we get that d y ( εm, w ) ≤ d y ( εv , w ) + d y ( εv , w ) and plugging this in the above weget (cid:10) v ⊕ v , w (cid:11) y ≥ ε − (cid:0) (cid:0) | w | y − d y ( εv , w ) (cid:1) + (cid:0) | w | y − d y ( εv , w ) (cid:1)(cid:1) = (cid:10) v , w (cid:11) y + (cid:10) v , w (cid:11) y − ε ( | v | y + | v | y ) ∀ ε > ε ↓ (cid:3) It will also be useful to know that(2.8) α ( G zy ) (cid:48) ⊕ β ( G wy ) (cid:48) = lim t ↓ ε ( G m t y ) (cid:48) , for z, w ∈ B r y ( y ) \ { y } , where m t is the midpoint of ( G zy ) αt and ( G zy ) βt , see for instance [7, II-Theorem 3.19] for the simple proof.We conclude recalling that on CAT ( κ )-spaces not only a notion of metric derivative is in placefor absolutely continuous curves, but it is possible to speak about right (or left) derivatives in thefollowing sense, as proved in [25]: Proposition 2.7 (Right derivatives) . Let Y be locally CAT ( κ ) and ( y t ) an absolutely continuouscurve. Then, for a.e. t , the tangent vectors h ( G y t + h y t ) (cid:48) ∈ T γ t Y have a limit y (cid:48) + t in T γ t Y as h ↓ . For us such concept will be useful in particular in connection with the well known first-ordervariation of the squared distance:
Proposition 2.8.
Let Y be a CAT ( κ ) -space, ( y t ) an absolutely continuous curve and z ∈ Y . Then: dd t d ( y t , z ) = − (cid:10) y (cid:48) + t , ( G zy t ) (cid:48) (cid:11) γ t a.e. t. To prove the above proposition, see e.g. [11, Propositions 2.17 and 2.20], one needs to introducethe notion of angle between geodesics and study its monotonicity properties, its behaviour alongabsolutely continuous curve and finally its connection with the inner product we introduced.Nevertheless, even if we omit the proof, in the sequel we shall use the following fact (see [11,Lemma 2.19]): let Y be
CAT ( κ ), ( y t ) be an absolutely continuous curve and z ∈ Y. Then, for thetime t s.t. | ˙ y t | exists and it is positive, we have − (cid:10) h ( G y t + h y t ) (cid:48) , ( G zy t ) (cid:48) (cid:11) y t ≤ − d Y ( y t , z ) d Y ( y t + h , y t ) h cos( ∠ κy t ( y t + h , z )) , ∀ h > y t + h ∈ B r yt ( y t )lim h ↓ − cos( ∠ κy t ( y t + h , z )) = lim h ↓ d Y ( y t + h , z ) − d Y ( y t , z ) h | ˙ y t | , (2.9)where ∠ κy t ( y t + h , z ) is the angle at ¯ y in M k of the comparison triangle (cid:52) κ (¯ y, ¯ y h , ¯ z ). The firstof these is an obvious consequence of the definition of ∠ κy ( z , z ) together with the fact that κ (cid:55)→ ∠ κy ( z , z ), and thus κ (cid:55)→ − cos( ∠ κy ( z , z )), is increasing, while the second one follows fromthe Taylor expansion of cos( ∠ κy ( z , z )) for d Y ( y, z ) small (notice that the explicit formula forcos( ∠ κy ( z , z )) in terms of d Y ( y, z ) , d Y ( y, z ) , d Y ( z , z ) can be obtained by the cosine rule). DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 7 Weak convergence.
In this section, we recall the concept of weak convergence in a
CAT (0)-space, highlighting the similarities with weak convergence on a Hilbert setting.Still, it is important to underline that although a well-behaved notion of ‘weakly convergingsequence’ exists, in [6] it is stressed that the existence of a well-behaved weak topology inducingsuch convergence is an open challenge. For the goal of this manuscript, here we just recall anoperative definition of weak convergence and its properties.Let us first clarify the notion of semiconvexity on a geodesic metric space.
Definition 2.9 (Semiconvex function) . Let Y be a geodesic space and E : Y → R ∪ { + ∞} . Wesay that E is λ -convex , λ ∈ R , if for any geodesic γ it holds E ( γ t ) ≤ (1 − t ) E ( γ ) + t E ( γ ) − λ t (1 − t ) d ( γ , γ ) ∀ t ∈ [0 , . If λ = 0 , then we simply speak of convex functions. We shall denote by D ( E ) ⊂ Y the set of y ’ssuch that E ( y ) < ∞ . Notice that, if Y is
CAT (0) and E : Y → R + ∪ { + ∞} is 2-convex and lower semicontinuous,then it admits a unique minimizer. To see this, we argue as for Proposition 3.5 and prove thatany minimizing sequence ( y n ) ⊂ Y is Cauchy: let I := inf E ≥ y n,m the midpoint of y n , y m andnotice that I ≤ E ( y n,m ) ≤ (cid:0) E ( y n ) + E ( y m ) (cid:1) − d ( y n , y m ) ∀ n, m ∈ N , so that rearranging and passing to the limit we get14 lim n,m →∞ d ( y n , y m ) ≤ lim n,m →∞ (cid:0) E ( y n ) + E ( y m ) (cid:1) − I = 0 , giving the claim. The first example of 2-convex functional we have encountered is the squareddistance from a point in a CAT (0)-space, as inequality (2.3) suggests. Hence, for ( y n ) ⊂ Y be abounded sequence, we can consider the mappingY (cid:51) y (cid:55)→ ω ( y ; ( y n )) := lim n d ( y, y n ) . and notice that, as a limsup of a sequence of 2-convex and locally equiLipschitz functions, it isstill 2-convex and locally Lipschitz. By the above remark, it has a unique minimizer. Definition 2.10 (Asymptotic center and weak convergence) . Let Y be CAT (0)-space and ( y n ) bea bounded sequence. We call the minimizer of ω ( · , ( y n )) the asymptotic center of ( y n ) .We say that a sequence ( y n ) ⊂ Y weakly converges to y , and write y n (cid:42) y , if y is the asymptoticcenter of every subsequence ( y n k ) of ( y n ) . In analogy with the Hilbert setting, we shall sometimes say that ( y n ) converges strongly to y if d Y ( y n , y ) →
0. The main properties of weak convergence are collected in the following statement:
Proposition 2.11.
Let Y be a CAT (0)-space. Then, the following holds: i) If ( y n ) converges to y strongly, then it converges weakly. ii) y n → y if and only if y n (cid:42) y and for some z ∈ Y we have d Y ( y n , z ) → d Y ( y, z ) . iii) Any bounded sequence admits a weakly converging subsequence. iv) If C ⊂ Y is convex and closed, then it is sequentially weakly closed. v) If E : Y → R ∪{ + ∞} is a convex and lower semicontinuous function, then it is sequentiallyweakly lower semicontinuous.Moreover, at the tangent cone T y Y at y ∈ Y (which is also a CAT (0)-space by Theorem 2.5) wealso have vi)
Let ( v n ) , ( w n ) ⊂ T y Y be such that v n → v and w n (cid:42) w for some v, w ∈ T y Y . Then lim n →∞ (cid:10) v n , w n (cid:11) ≤ (cid:10) v, w (cid:11) . NICOLA GIGLI AND FRANCESCO NOBILI
Proof. ( i ) is obvious, as a strong limit is trivially the asymptotic center of the full sequence. For( ii ) , ( iii ) , ( iv ) see [5, Proposition 3.1.6], [5, Proposition 3.1.2] and [5, Proposition 3.2.1] respectively.( v ) follows trivially from ( iv ) by considering the strongly closed and convex sublevels of E . Finally,for ( vi ) we let C := sup n | w n | y < ∞ and notice that for every ε > ε (cid:10) v n , w n (cid:11) y = (cid:10) v n , εw n (cid:11) y ≤ | v n | y + | εw | y − d y ( v, εw n ) + (cid:0) d y ( v, εw n ) − d y ( v n , εw n ) (cid:1) + 4 ε C ≤ | v n | y + | εw | y − d y ( v, εw n ) + 4 εC d y ( v, v n )( | v | y + | v n | y ) + 4 ε C and that εw n (cid:42) εw (by (2.7a)). Sending n → ∞ and using the sequential weak lower semiconti-nuity of d y ( v, · ) (consequence of ( v )) we obtain that2 ε lim n →∞ (cid:10) v n , w n (cid:11) y ≤ | v | y + | εw | y − d y ( v, εw ) + 4 ε C = 2 ε (cid:10) v, w (cid:11) y + 4 ε C and the claim follows dividing by ε > ε ↓ (cid:3) Geometric tangent bundle.
In this section we briefly recall some concepts from [11] aboutthe construction of the Geometric Tangent Bundle T G Y of a given separable local
CAT ( κ )-spaceY. From now on, B (Y) is the Borel σ -algebra on Y. As a set, the space T G Y is defined asT G Y := (cid:8) ( y, v ) : y ∈ Y , v ∈ T y Y (cid:9) . Such set is equipped with a σ -algebra B (T G Y), called Borel σ -algebra (with a slight abuse ofterminology, because there is no topology inducing it), defined as the smallest σ -algebra such thatthe following maps are measurable:i) the canonical projection π Y : T G Y → Yii) the maps π − ( B r ¯ y ¯ y ) (cid:51) ( y, v ) (cid:55)→ (cid:10) v, ( G zy ) (cid:48) (cid:11) y ∈ R for every ¯ y ∈ Y , z ∈ B r ¯ y (¯ y ).It turns out that B (T G Y) is countably generated and that, rather than asking ( ii ) for every z ∈ Y,one can require it only for a dense set of points (notice that in the axiomatization chosen in [11] onespeaks about the differential of the distance function rather than of scalar product with vectorsof the form ( G zy ) (cid:48) , but the two approaches are actually trivially equivalent thanks to the explicitexpression of the differential of the distance in terms of such scalar product which is hidden inProposition 2.8). We also recall that(2.10) the map T G Y (cid:51) ( y, v ) (cid:55)→ | v | y ∈ R is Borel.A section of T G Y is a map s : Y → T G Y such that s y ∈ T y Y for every Y. A section issaid Borel if it is measurable w.r.t. B (Y) and B (T G Y). Among the various sections, simple onesplay a special role, similar to the one played by finite-ranged functions in the theory of Bochnerintegration: s is a simple section provided there are ( y n ) ⊂ Y, ( α n ) ⊂ R + and ( E n ) Borel partitionof Y such that y n ∈ B r y ( y ) for every y ∈ E n and s | E n = α n ( G y n · ) (cid:48) . If this is the case we write s = (cid:80) n χ E n α n ( G y n · ) (cid:48) , although the ‘sum’ here is purely formal. The following basic result -obtained in [11] - will be useful, we report the proof for completeness: Proposition 2.12.
Let Y be separable and locally CAT ( κ ) . Then, simple sections of T G Y asdefined above are Borel.Proof. It is sufficient to prove that for any given ¯ y ∈ Y , z ∈ B r ¯ y (¯ y ) and α ∈ R + the assignment B r ¯ y (¯ y ) (cid:51) y (cid:55)→ s y := α ( G zy ) (cid:48) is Borel and to this aim, by the very definition of B (T G Y), it issufficient to check that π Y ◦ s : Y → Y is Borel - which it is, being this map the identity on Y -and, for any w ∈ B r ¯ y (¯ y ), the map B r ¯ y (¯ y ) (cid:51) y (cid:55)→ (cid:10) s y , ( G wy ) (cid:48) (cid:11) y is Borel. Thus fix w and notice thatthanks to (2.10) and to the definition of scalar product on T y Y to conclude it is sufficient to checkthat y (cid:55)→ d y ( s y , ( G wy ) (cid:48) ) is Borel. We have d y ( s y , ( G wy ) (cid:48) ) = d y ( α ( G zy ) (cid:48) , ( G wy ) (cid:48) ) = lim t ↓ d Y (cid:0) ( G zy ) αt , ( G wy ) t (cid:1) t . From the continuous dependence of geodesics on their endpoints we deduce that y (cid:55)→ d Y (cid:0) ( G zy ) αt , ( G wy ) t (cid:1) is a continuous function for every t ∈ (0 , ∧ α − ). The conclusion then follows from the fact thata pointwise limit of continuous functions is Borel. (cid:3) DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 9 It has been proved in [11] that simple sections are dense among Borel ones (see also Lemma 4.6below in the case X = Y and u = Identity). Moreover, the operations on a single tangent spaceT y Y induce in a natural way operations on the space of Borel sections of T G Y: these are Borelregular, as recalled in the next statement (see [11, Proposition 3.6] for the proof).
Proposition 2.13.
Let Y be separable and locally CAT ( κ ) , s , t Borel sections of T G Y and f : Y → R + Borel. Then, the maps from Y to R sending y to | s y | y , d y ( s y , t y ) , (cid:10) s y , t y (cid:11) y are Borel and thesections y (cid:55)→ f ( y ) s y , s y ⊕ t y are Borel as well. Gradient flows on
CAT ( κ ) -spaces Metric approach.
We recall here the basic definitions and properties of gradient flows onlocally
CAT ( κ )-spaces. We begin with the definition of (descending) slope | ∂ − E | of the functional E : for y ∈ D ( E ) we put(3.1) | ∂ − E | ( y ) := lim z → y ( E ( y ) − E ( z )) + d Y ( y, z ) , and we denote the points where the slope is finite by D ( | ∂ − E | ) ⊂ D ( E ). It is easy to prove that for λ -convex functionals, the slope admits the following ‘global’ formulation (see [2, Theorem 2.4.9]for the proof): Lemma 3.1.
Let Y be a geodesic space and E : Y → R ∪ { + ∞} be λ -convex, λ ∈ R , and lowersemicontinuous. Then, for every y ∈ D ( E ) , | ∂ − E | ( y ) = sup z (cid:54) = y (cid:18) E ( y ) − E ( z ) d Y ( y, z ) + λ d Y ( y, z ) (cid:19) + . Moreover, y (cid:55)→ | ∂ − E | ( y ) is a lower semicontinuous function. We now come to various equivalent definitions of gradient flows on locally
CAT ( κ )-spaces. Theequivalence between the first two notions below is due to the convexity assumption, while theequivalence of these with the EVI is due to the geometric properties of CAT ( κ )-spaces, and inparticular their Hilbert-like structure at small scales. Theorem 3.2 (Gradient flows on locally
CAT ( κ )-spaces: equivalent definitions) . Let Y be a locally CAT ( κ ) -space, E : Y → R ∪ { + ∞} a λ -convex and lower semicontinuous functional, λ ∈ R , y ∈ Y and (0 , ∞ ) (cid:51) t (cid:55)→ y t ∈ Y a locally absolutely continuous curve such that y t → y as t ↓ . Then,the following are equivalent: ( i ) Energy Dissipation Inequality
We have − ∂ t E ( y t ) ≥ | ˙ y t | + 12 | ∂ − E | ( y t ) where the derivative in the left hand side is intended in the sense of distributions. ( ii ) Sharp dissipation rate t (cid:55)→ E ( y t ) is locally absolutely continuous and (3.2) lim h ↓ E ( y t ) − E ( y t + h ) h = | ˙ y + t | = | ∂ − E | ( y t ) for every t > , where | ˙ y + t | := lim h ↓ d Y ( y t + h ,y t ) h is the right metric speed, which in this case exists for every t > . ( iii ) Evolution Variational Inequality
For every z ∈ Y we have (3.3) dd t d ( y t , z )2 + E ( y t ) + λ d ( y t , z ) ≤ E ( z ) a.e. t > . Proof.
The fact that ( ii ) implies ( i ) is obvious. The converse implication has been proved in [2]as a consequence of the so called strong upper gradient property of the slope. The implication( iii ) → ( ii ) is proved in [29] (the argument in [29] has been also reported in [15]). The fact that onlocally CAT ( κ )-spaces ( ii ) implies ( iii ) has also been proved in [29] (see in particular Theorems 4.2and 3.14 there). More precisely, in [29] only the ‘global’ case of CAT ( κ )-spaces has been considered,but the arguments there can be quickly adapted to cover our case by noticing that: - arguing as for the proof of (3.7) below, we see that (3.3) holds at some t if and only if itholds at t for z varying only in a neighbourhood of y t ,- property (3.2) is local by nature,- if B ⊂ Y is closed, convex and
CAT ( κ ), then a curve I (cid:51) t (cid:55)→ y t ∈ B satisfies ( ii ) (resp.( iii )) in B if and only if it satisfies ( ii ) (resp. ( iii )) in Y. (cid:3) A curve satisfying any of the equivalent conditions in this last theorem will be called gradientflow trajectory . Moreover, we define the gradient flow map GF E : (0 , ∞ ) × Y → Y via GF E t ( y ) := y t for every t ∈ (0 , ∞ ) , y ∈ Y, where, evidently, y t is the gradient flow trajectory starting at y andassociated to the functional E evaluated at time t . Some of their main properties are collected inthe following statement: Theorem 3.3 (Gradient flows on locally
CAT ( κ )-spaces: some basic properties) . Let Y be alocally CAT ( κ ) -space, E : Y → R ∪ { + ∞} a λ -convex and lower semicontinuous functional. Then,the following holds: ◦ Existence
For every y ∈ D ( E ) there exists a gradient flow trajectory for E starting from y . ◦ Uniqueness and λ -contraction For any two gradient flow trajectories ( y t ) , ( z t ) starting from y, z respectively we have (3.4) d Y ( y t , z t ) ≤ e − λ ( t − s ) d Y ( y s , z s ) ∀ t ≥ s > ◦ Monotonicity properties
For ( y t ) gradient flow trajectory for E starting from y wehave that t (cid:55)→ y t is locally Lipschitz in (0 , + ∞ ) with values in D ( | ∂ − E | ) ⊂ D ( E ) , t (cid:55)→ E ( y t ) is nonincreasing in [0 , + ∞ ) ,t (cid:55)→ e λt | ∂ − E | ( y t ) is nonincreasing in [0 , + ∞ ) . (3.5) Proof.
In the
CAT (0) case, existence of a limit of the so-called minimizing movements scheme inthis setting has been proved in [28] and [22]. The fact that the limit curve obtained in this waysatisfies the EVI condition has been proved in [2]. The contractivity property, also at the levelof the discrete scheme, has been proved in [28] and [22] (at least in the case λ = 0, the generalcase can be found e.g. in [2] as a consequence of the EVI condition). Then, uniqueness is directlyimplied by (3.4) and the last claims are a consequence of (3.2) and the contraction property.The CAT ( κ ) case has been treated in [30], at least under some compactness assumptions on thesublevels of the functional. Such compactness assumption has been removed in [29]. Finally, thecase of locally CAT ( κ ) spaces can be dealt with as in the proof of Theorem 3.2 above. (cid:3) Finally, we conclude the section with an a priori estimate, a variant of the ones investigated in[29], concerning contraction properties along the gradient flow trajectories at different times. Theproof is inspired by the one of [31, Lemma 2.1.4] in the context of CBB-spaces.
Lemma 3.4 (A priori estimates) . Let Y be locally CAT ( κ ) and E : Y → [0 , ∞ ] be a λ -convex andlower semicontinuous functional, λ ∈ R . Let y, z ∈ Y and consider the gradient flow trajectories ( y t ) , ( z t ) associated with E .Then, for any t ≥ s > , it holds d ( y t , z s ) ≤ e − λs (cid:16) d ( y, z )+2( t − s )( E ( z ) − E ( y ))+ 2 | ∂ − E | ( y ) ˆ t − s θ λ ( r ) d r − λ ˆ t − s d ( y r , z ) d r (cid:17) , (3.6) where θ λ ( t ) := ´ t e − λr d r . DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 11 Proof.
We start fixing t >
0. First, we notice that, in light of ( ii ) of Theorem 3.2 and the basicproperties in Theorem 3.3, we have for any r > r ), − E ( y r ) + E ( y ) = ˆ r e − λq e +2 λq | ∂ − E | ( y q ) d q ≤ | ∂ − E | ( y ) θ λ ( r ) . Thus, we can integrate from 0 to t the EVI condition (3.3) to get12 ( d ( y t , z ) − d ( y, z )) ≤ ˆ t E ( z ) − E ( y r ) − λ d ( y r , z ) d r ≤ t ( E ( z ) − E ( y )) + | ∂ − E | ( y ) ˆ t θ λ ( r ) d r − λ ˆ t d ( y r , z ) d r. Finally, for general t ≥ s >
0, we can reduce to above case by appealing to property (3.4). (cid:3)
The object − ∂ − E ( y ) . In this section we introduce the key object − ∂ − E ( y ) of this manuscriptassociated to a semiconvex and lower semicontinuous functional E over a local CAT ( κ ) space. Asthe notation suggests, and as will be clear from Definition 3.6, for functionals on Hilbert spacesthis corresponds to {− v : v ∈ ∂ − E ( y ) } .We start recalling the following well known fact: Proposition 3.5 (Metric projection) . Let Y be a CAT (0) -space and C ⊂ Y be a closed convexsubset. Then, for every y ∈ Y , there is a unique Pr C ( y ) ∈ C , called metric projection of y onto C , such that d Y ( y, Pr C ( y )) = inf C d Y ( y, · ) .Proof. Since the function to be minimized is continuous and C closed, it is sufficient to prove thatany minimizing sequence ( c n ) for I := inf c ∈ C d ( c, y ) (which is equivalent to be minimizing forinf C d Y ( y, · )) is Cauchy. Fix such sequence and, for every n, m ∈ N , let c m,n be the mid-pointbetween c n and c m . Observe that since C is convex, c n,m belongs to C and thus is a competitorfor the minimization problem. Condition (2.3) therefore implies I ≤ d ( c n,m , y ) ≤ d ( c n , y ) + 12 d ( c m , y ) − d ( c n , c m ) , for every n, m ∈ N . Rearranging terms, and taking the limsup as n, m go to infinity we observelim n,m → + ∞ d ( c n , c m ) ≤ lim n,m → + ∞ d ( c n , y ) + 12 d ( c m , y ) − I = 0 , i.e. ( c n ) is Cauchy, as desired. (cid:3) We remark that the metric projection can be also shown to be 1-Lipschitz and to satisfy a‘Pythagoras’ inequality’ (see [5, Theorem 2.1.12]), but we will not make use of this fact.bFinally,we are ready to give an effective definition of (opposite of the) subdifferential of E as a subset ofthe tangent cone. Definition 3.6 (Minus-subdifferential) . Let Y be locally CAT ( κ ) , E : Y → R ∪{ + ∞} be a λ -convexand lower semicontinuous functional, λ ∈ R , and y ∈ D ( E ) . We define the minus-subdifferential of E at y , denoted by − ∂ − E ( y ) , as the collection of v ∈ T y Y satisfying the subdifferential inequality E ( y ) − (cid:10) v, γ (cid:48) (cid:11) y + λ d ( y, z ) ≤ E ( z ) , for every z ∈ Y , and some geodesic γ from y to z . Moreover, by D ( − ∂ − E ) , we denote the collectionof y ∈ Y for which − ∂ − E ( y ) (cid:54) = ∅ . Notice that v ∈ − ∂ − E ( y ) if and only if(3.7) − (cid:10) v, γ (cid:48) (cid:11) y ≤ lim t ↓ E ( γ t ) − E ( y ) t ∀ z ∈ Y , for some geodesic γ from y to z. so that in particular the definition of − ∂ − E ( y ) does not depend on λ . Indeed the ‘if’ is obviousby λ -convexity while for the ‘only if’ we apply the defining inequality with z t := γ t in place of z and, for t small enough, rearrange to get − (cid:10) v, ( G z t y ) (cid:48) (cid:11) y + λ d ( y, z t ) ≤ E ( z t ) − E ( y )so that the conclusion follows noticing that d ( y, z t ) = t d ( y, z ), ( G z t y ) (cid:48) = tγ (cid:48) (because for t (cid:28) y to z t in unique), then dividing by t and letting t ↓
0. The same argumentsalso show that both in Definition 3.6 and in (3.7) we can take γ to be any geodesic from y to z .It is also worth to point out thatFor E convex and lower semicontinuous we have that: x is a minimum point for E if and only if 0 ∈ − ∂ − E ( x ).(3.8)The proof of this fact being obvious. Remark 3.7.
It would certainly be possible to define the analogous notion of subdifferential ∂ − E by replacing − (cid:10) v, γ (cid:48) (cid:11) y with (cid:10) v, γ (cid:48) (cid:11) y in the defining formula, however, since the tangent cone isonly a cone and not a space, there is no obvious relation between the two definitions.For our purposes, − ∂ − E is the correct object to work with because, as discussed in the intro-duction, we aim at showing the existence of the Laplacian of a CAT (0)-valued Sobolev map bylooking at the gradient flow of the Korevaar-Schoen energy E KS , thus we notice on one hand that,by definition and imitating what happens in the smooth category, the Laplacian of u has to beintroduced as (the element of minimal norm in) − ∂ − E KS ( u ), and on the other one that in thegradient flow equation (1.1) it is − ∂ − E who appears.In this direction, it is interesting to point out that the classical procedure of minimizing y (cid:55)→ E ( y ) + d ( y, ¯ y )2 τ , which is the cornerstone of most existence results about gradient flows in the metric setting (see e.g.[2]), produces a (unique, if τ > y τ for which we have τ ( G ¯ yy τ ) (cid:48) ∈ ∂ − E ( y τ ).In particular it gives no informations about whether − ∂ − E ( y τ ) is not empty. In our approachthis latter fact, and the related one that the slope at y coincides with the norm of the element ofleast norm in − ∂ − E ( y ), will be a consequence of the fact that gradient flow trajectories satisfy ananalogue of (1.1), see Theorem 3.10. (cid:4) It will be important to know that in − ∂ − E ( y ) there is always an element of minimal norm: Proposition 3.8.
Let Y be a locally CAT ( κ ) -space, E : Y → R ∪ { + ∞} be a λ -convex and lowersemicontinuous functional, λ ∈ R , and y ∈ Y . Then, − ∂ − E ( y ) is a closed and convex subset of T y Y . In particular, if this set is not empty, the optimization problem inf v ∈− ∂ − E ( y ) | v | y admits a unique minimiser.Proof. Recalling that T y Y is
CAT (0), by Proposition 3.5 the existence of a unique minimizer in − ∂ − E ( y ) for the norm, i.e. of a unique metric projection of 0 y onto − ∂ − E ( y ), will follow once weshow that − ∂ − E ( y ) is closed and convex.The fact that it is closed follows from the definition and the consideration already stated inProposition 2.6 that the scalar product (cid:10) · , · (cid:11) y is continuous on T y Y. The convexity follows fromthe inequality − (cid:10) ( G v v ) t , w (cid:11) y ≤ − (1 − t ) (cid:10) v , w (cid:11) y − t (cid:10) v , w (cid:11) y ∀ v , v , w ∈ T y Y , t ∈ [0 , , which is a direct consequence of (2.7b) and (2.7f). (cid:3) DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 13 Subdifferential formulation.
Here we prove the main results of this note, namely Theorem3.10 and Corollary 3.11 below. We shall use the following preliminary result (notice that the factthat equality holds in (3.9) will be obtained in (3.10)):
Proposition 3.9.
Let Y be locally CAT ( κ ) and E : Y → R ∪ { + ∞} be a λ -convex and lowersemicontinuous functional, λ ∈ R . Then, for every y ∈ D ( − ∂ − E ) , we have (3.9) | ∂ − E | ( y ) ≤ inf v ∈− ∂E ( y ) | v | y . In particular, D ( − ∂ − E ) ⊂ D ( | ∂ − E | ) .Proof. Let v ∈ − ∂ − E ( y ) and notice that E ( y ) − E ( z ) + λ d ( y, z ) ≤ | (cid:10) v, ( G zy ) (cid:48) (cid:11) y | (2.7c) ≤ | v | y d Y ( y, z ) , ∀ z ∈ Ywhich in turns implies (cid:18) E ( y ) − E ( z ) d Y ( y, z ) + λ d Y ( y, z ) (cid:19) + ≤ | v | y ∀ z ∈ Y , z (cid:54) = y. Taking the supremum over z (cid:54) = y and recalling Lemma 3.1 we conclude. (cid:3) We now come to the main result of this manuscript, namely the existence of right incrementalratios of the flow for all time.
Theorem 3.10 (Right derivatives of the flow) . Let Y be locally CAT ( κ ) and E : Y → R ∪ { + ∞} be a λ -convex and lower semicontinuous functional, λ ∈ R . Let y ∈ D ( E ) , and ( y t ) be the gradientflow trajectory starting from y (recall Theorem 3.3).Then, for every t > , the right ‘difference quotient’ h ( G y t + h y t ) (cid:48) strongly converges to the elementof minimal norm in − ∂ − E ( y t ) ⊂ T y t Y (i.e. to Pr − ∂ − E ( y t ) (0 y t ) ) as h goes to + . The same holdsfor t = 0 if (and only if ) we have y ∈ D ( | ∂ − E | ) .Moreover, D ( − ∂ − E ) = D ( | ∂ − E | ) and the identity (3.10) | ∂ − E | ( y ) = min v ∈− ∂ − E ( y ) | v | y ∀ y ∈ Y holds, where as customary the minimum of the empty set is declared to be + ∞ . In particular, D ( − ∂ − E ) is dense in D ( E ) .Proof. By the semigroup property ensured by the uniqueness of gradient flow trajectories andtaking into account that y t ∈ D ( | ∂ − E | ) for every t > t = 0 under the assumption y ∈ D ( | ∂ − E | ). Suppose y is not a minimum point for E , otherwisethere is nothing to prove. In particular, ( ii ) of Theorem 3.2 ensures that | ˙ y | exists and it ispositive. Also, notice that the continuity at time t = 0 of the gradient flow trajectory ensuresthat for (cid:15) > y h ∈ B r y ( y ) for every h ∈ (0 , (cid:15) ). In particular for such h the tangent vector v h := h ( G y h y ) (cid:48) ∈ T y Y is well defined and the statement makes sense. Fix such (cid:15) > Step 1
For every h ∈ (0 , (cid:15) ) we have(3.11) | v h | y = d Y ( y h , y ) h ≤ h | ˙ y t | d t (3.2) = h | ∂ − E | ( y t ) d t (3.5) ≤ | ∂ − E | ( y ) h e − λt d t. Hence sup h ∈ (0 ,(cid:15) ) | v h | y < ∞ , therefore point ( iii ) of Proposition 2.11 gives that for every sequence h n ↓ v h n (cid:42) v for some v ∈ T y Y.Fix such sequence and such weak limit v . To conclude it is sufficient to prove that the conver-gence is strong and that v is the element of minimal norm in − ∂ − E ( y ), as this in particular grantsthat the limit is independent on the particular subsequence chosen. Step 2
We claim that v ∈ − ∂ − E ( y ). To see this, integrate (3.3) from 0 to h and divide by h toobtain d ( y h , z ) − d ( y, z )2 h + h E ( y t ) + λ d ( y t , z ) d t ≤ E ( z ) ∀ z ∈ Y , h ∈ (0 , (cid:15) ) . Letting h = h n ↓ E is lower semicontinuous we deduce that(3.12) lim n →∞ d ( y h n , z ) − d ( y, z )2 h n + E ( y ) + λ d ( y, z ) ≤ E ( z ) ∀ z ∈ Y . Next, fix z ∈ Y, let γ ∈ Geo y Y with γ = z , denote z s := γ s and notice that, for s sufficientlysmall, z s , y h n ∈ B r y ( y ). Now (2.9) yieldslim n →∞ − (cid:10) v h n , ( G z s y ) (cid:48) (cid:11) y ≤ d Y ( y, z ) lim n →∞ d ( y h n , y ) h n d Y ( y h n , z s ) − d Y ( y, z s ) | ˙ y | h n = lim n →∞ d ( y h n , z s ) − d ( y, z s )2 h n , having used the fact that lim n a n b n = a lim n b n if lim n a n = a > a n ) , ( b n ) ⊂ R are bounded,and a chain rule argument in the last equality. Thus, recalling the weak upper semicontinuity ofthe scalar product proved in point ( vi ) of Proposition 2.11 we getlim n →∞ d ( y h n , z s ) − d ( y, z s )2 h n ≥ − (cid:10) v, ( G z s y ) (cid:48) (cid:11) y . Now, combine with (3.12) to get E ( y ) − (cid:10) v, ( G z s y ) (cid:48) (cid:11) y + λ d ( z s , y ) ≤ E ( z s ) ≤ (1 − s ) E ( y ) + s E ( z ) − λ s (1 − s ) d ( z, y ) . Finally, using that ( G z s y ) (cid:48) = sγ (cid:48) , d ( z s , y ) = s d ( y, z ) and (2.7b), we can rearrange terms andtake the limit as s ↓ E ( y ) − (cid:10) v, γ (cid:48) (cid:11) y + λ d ( z, y ) ≤ E ( z ) for every γ geodesic from y to z. Given that z was arbitrary, we conclude. Step 3
Since | · | y : T y Y → R is convex and continuous, by point ( v ) of Proposition 2.11 we get | v | y ≤ lim n →∞ | v h n | y ≤ lim n →∞ | v h n | y (3.11) ≤ | ∂ − E | ( y ) (3.9) ≤ inf w ∈− ∂ − E ( y ) | w | y ≤ | v | y , and thus all the inequalities must be equalities. This proves at once the strong convergence of( v h n ) to v (by the convergence of norms and point ( ii ) of Proposition 2.11) and that v is theelement of minimal norm in − ∂ − E ( y ).The argument also proves that if y ∈ D ( | ∂ − E | ), then y ∈ D ( − ∂ − E ) and in this case the equalityin (3.10) holds. Taking into account Proposition 3.9 we conclude that D ( | ∂ − E | ) = D ( − ∂ − E ) that(3.10) holds for every y ∈ Y, as desired.The last claim then follows from the existence of gradient flow trajectories starting from pointsin D ( E ) (Theorem 3.3) and (3.2). (cid:3) As a direct consequence of the above result, we see that we can characterize gradient flowtrajectories by means of the classical differential inclusion x (cid:48) t ∈ − ∂ − E ( x t ) which can be used todefine such evolution in the Hilbert setting: Corollary 3.11.
Let Y be locally CAT ( κ ) and E : Y → R ∪ { + ∞} be a λ -convex and lowersemicontinuous functional, λ ∈ R . Let y ∈ D ( E ) , and (0 , ∞ ) (cid:51) t (cid:55)→ y t ∈ D ( E ) be a locallyabsolutely continuous curve. Then, the following are equivalent: (i) ( y t ) is a gradient flow trajectory for E starting from y , i.e. satisfies any of the threeequivalent conditions in Theorem 3.2. (ii) The right derivative y (cid:48) + t exists for every t > and (cid:40) y (cid:48) + t ∈ − ∂ − E ( y t ) ∀ t > and is the element of minimal norm , lim t ↓ y t = y. If y ∈ D ( | ∂ − E | ) = D ( − ∂ − E ) then the above holds also at t = 0 . DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 15 (iii) It holds (cid:40) y (cid:48) + t ∈ − ∂ − E ( y t ) a.e. t > , lim t ↓ y t = y. Proof.
The implication ( i ) ⇒ ( ii ) is proved in Theorem 3.10 above and the one ( ii ) ⇒ ( iii ) isobvious. The fact that ( iii ) implies ( i ) (in the form of the Evolution Variation Inequality) is adirect consequence of Proposition 2.8 (applied in a CAT ( κ ) neighbourhood of y t in combinationwith arguments similar to those outlined in the proof of Theorem 3.2 to cover the case of a local CAT ( κ ) space) and the definition of − ∂ − E . (cid:3) Remark 3.12.
In the setting of Alexandrov geometry it is more customary to study the gradientflow of semi concave functions F , thus studying (a properly interpreted version of) y (cid:48) t ∈ ∂ + F .Let E be semiconvex on a CAT ( κ )-space Y and put F := − E . Then it is clear that the slope | ∂ − E | as we defined it coincides with the absolute gradient |∇ F | as defined in [26, Definition 4.1],therefore, taking into account the characterization (3.2), we see that up to a different choice ofparametrization, our notion of gradient flow trajectory coincides with the one of gradient-likecurve studied in [26, Definition 6.1].The property dd t + F ( y t ) = − dd t + E ( y t ) = | ∂ − E | ( y t ) = | v t | y t , where v t ∈ − ∂ − E ( y t ) is the elementof minimal norm, together with the existence of the right derivative of y t and the characterization(3.7) show that the element of minimal norm in − ∂ − E ( y ) coincides with ∇ F ( y ) as defined in [1,Definition 11.4.1] on spaces with curvature bounded from below . This shows that our ‘differential’perspective on gradient flows is compatible with the one studied in [1] on CBB spaces. (cid:4) Laplacian of
CAT (0)-valued maps
Pullback geometric tangent bundle.
The general non-separable case.
For the purpose of this manuscript, a metric measure space (X , d , m ) is always intended to be given by: a complete and separable metric space (X , d ) equippedwith a non-negative and non-zero Borel measure giving finite mass to bounded sets. In somecircumstances we shall add further assumptions on X, typically in the form a RCD( K, N ) condition.Thus let us fix a pointed
CAT (0) space (Y , d Y , ¯ y ), a metric measure space (X , d , m ) and an opensubset Ω ⊂ X. We recall that the space L (Ω , Y) is the collection of all Borel maps u : Ω → Y whichare essentially separably valued (i.e. for some separable subset ˜Y ⊂ Y we have m ( u − (Y \ ˜Y)) = 0),where two maps agreeing m -a.e. are identified. Then L (Ω , Y ¯ y ) ⊂ L (Ω , Y) is collection of those(equivalence classes of maps) u such that ´ Ω d ( u ( x ) , ¯ y ) d m ( x ) < ∞ . The space L (Ω , Y ¯ y ) comesnaturally with the distance d L ( u, v ) := ˆ Ω d ( u ( x ) , v ( x )) d m ( x )and by standard means one sees that with such distance the space is complete and that finite-ranged maps are dense. Moreover, for u, v ∈ L (Ω , Y ¯ y ) a direct computation shows that t (cid:55)→ ( G vu ) t ∈ L (Ω , Y ¯ y ), where ( G vu ) t ( x ) := ( G v ( x ) u ( x ) ) t , is a geodesic from u to v (the fact that ( G vu ) t : Ω → Y is Borel follows from the continuous dependence of the - unique - geodesics on Y w.r.t. theendpoints). Also, by appealing to the equivalent characterization (2.3) of
CAT (0)-spaces, thecomputation d L (( G vu ) t , w ) = ˆ d (( G v ( x ) u ( x ) ) t , w ( x )) d m ( x ) (2.3) ≤ ˆ (1 − t ) d ( u ( x ) , w ( x )) + t d ( v ( x ) , w ( x )) − t (1 − t ) d ( u ( x ) , v ( x )) d m ( x )= (1 − t ) d L ( u, w ) + t d L ( v, w ) − t (1 − t ) d L ( u, v )valid for any w ∈ L (Ω , Y ¯ y ) and every t ∈ [0 , L (Ω , Y ¯ y ) is a CAT (0)-space as welland thus G vu is the only geodesic from u to v . In particular, given u ∈ L (Ω , Y ¯ y ) we have a well defined tangent cone T u L (Ω , Y ¯ y ) containingwhat we may think of as the set of ‘infinitesimal variations’ of u . Intuitively, these variationsshould correspond to a collection, for m -a.e. x ∈ Ω, of a variation of u ( x ) ∈ Y, i.e. to a collectionof elements of T u ( x ) Y.We now want to make this intuition rigorous and, due to the fact that
CAT (0)-spaces aretypically studied in non separable environments, we first discuss this case, postponing to the nextsections the separable case and its relations with the Borel structure on T G Y seen in Section 2.4.Fix u ∈ L (Ω , Y ¯ y ) and a Borel representative of it, which by abuse of notation we shall continueto denote by u . By u ∗ T G Y we intend the set u ∗ T G Y := (cid:8) ( x, y, v ) : x ∈ Ω , y = u ( x ) , v ∈ T y Y (cid:9) ⊂ X × T G YA section of u ∗ T G Y is a map S : Ω → u ∗ T G Y such that π X ( S ( x )) = x , where π X : u ∗ T G Y → X is u ( x ) Y u X T G Y u ∗ T G Y x Figure 1.
Pullback geometric tangent bundle u ∗ T G Y via u : X → Y. the canonical projection. Given such a section S we write S ( x ) = ( x, u ( x ) , S x ) for any x ∈ Ω. Weshall denote by the zero section defined by x := 0 u ( x ) ∈ T u ( x ) Y.Then given another v ∈ L (Ω , Y ¯ y ) and a Borel representative of it, still denoted by v , and α ≥
0, we can consider the section S of u ∗ T G Y given by x (cid:55)→ ( x, u ( x ) , α ( G v ( x ) u ( x ) ) (cid:48) ). We then havethe following simple and useful lemma. Lemma 4.1.
Let (Y , d Y , ¯ y ) be a pointed CAT (0)-space, (X , d , m ) a metric measure space, Ω ⊂ X an open subset, u, v , v : Ω → Y Borel representatives of maps in L (Ω , Y ¯ y ) . Also, let α , α ∈ R + and consider the sections S i of u ∗ T G Y given by S ix := α i ( G v i ( x ) u ( x ) ) (cid:48) , i = 1 , .Then the maps Ω (cid:51) x (cid:55)→ | S x | u ( x ) , d u ( x ) ( S x , S x ) , (cid:10) S x , S x (cid:11) u ( x ) are Borel.Proof. It is sufficient to prove that Ω (cid:51) x (cid:55)→ d u ( x ) ( S x , S x ) ∈ R is Borel, as then the other Borelregularities will follow. We have already noticed that the maps x (cid:55)→ ( G v i u ) α i t ( x ) ∈ Y, i = 1 ,
2, areBorel, hence so is the map x (cid:55)→ d Y (cid:0) ( G v u ) α t ( x ) , ( G v u ) α t ( x ) (cid:1) t for any 0 < t (cid:28)
1. Since these maps pointwise converge to x (cid:55)→ d u ( x ) ( S x , S x ) as t ↓
0, the claimfollows. (cid:3)
In particular, for S , S as in the above statement, the quantity(4.1) d L (cid:0) S , S (cid:1) := (cid:115) ˆ Ω d u ( x ) ( S x , S x ) d m ( x ) , is well defined. Standard arguments then show that d L is symmetric, satisfies the triangle in-equality and d L ( S , S ) = 0 (but it might happen that d L ( S , S ) = 0 for S (cid:54) = S and that d L ( S , S ) = + ∞ ).We then give the following definitions: Definition 4.2 ( L sections of u ∗ T G Y) . Let (Y , d Y , ¯ y ) be a pointed CAT (0)-space, (X , d , m ) ametric measure space, Ω ⊂ X an open subset and u a Borel representative of a map in L (Ω , Y ¯ y ) .Then, L ( u ∗ T G Y , m | Ω ) is the collection of sections S of u ∗ T G Y such that: DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 17 i) For any α ∈ R + and v : Ω → Y Borel and essentially separably valued we have that x (cid:55)→ d u ( x ) ( S x , α ( G v ( x ) u ( x ) ) (cid:48) ) is a Borel function. ii) There is a sequence ( α n ) ⊂ R + and maps v n : Ω → Y Borel and essentially separablyvalued such that for the sections S n given by S nx := α n ( G v n ( x ) u ( x ) ) (cid:48) we have sup n ∈ N d L ( S n , ) < ∞ , lim n →∞ d u ( x ) ( S nx , S x ) = 0 ∀ x ∈ Ω . (4.2)It is clear from the definitions that for S , S ∈ L ( u ∗ T G Y , m | Ω ) the map x (cid:55)→ d u ( x ) ( S x , S x ) isBorel and L ( m | Ω )-integrable, therefore d L ( S , S ) is well defined by (4.1) and finite. Definition 4.3 ( L sections of u ∗ T G Y) . Let (Y , d Y , ¯ y ) be a pointed CAT (0)-space, (X , d , m ) ametric measure space, Ω ⊂ X an open subset and u a Borel representative of a map in L (Ω , Y ¯ y ) .We define L ( u ∗ T G Y , m | Ω ) as the quotient of L ( u ∗ T G Y , m | Ω ) with respect to the relation S ∼ S if d L ( S , S ) = 0 . It is obvious that the relation indicated in the previous definition is an equivalence relation, sothat L ( u ∗ T G Y , m | Ω ) is well defined. Also, the quantity d L passes to the quotient and definesa distance, still denoted by d L , on L ( u ∗ T G Y , m | Ω ) and standard considerations show that theresulting object is a complete metric space.Now let ˜ u : Ω → Y be Borel and m -a.e. equal to u and consider the identification I : L ( u ∗ T G Y , m | Ω ) → L (˜ u ∗ T G Y , m | Ω ) sending S to the section I ( S ) defined by I ( S ) x := (cid:26) S x , if u ( x ) = ˜ u ( x ) , ˜ u ( x ) , if u ( x ) (cid:54) = ˜ u ( x ) . It is clear that this map passes to the quotients and thus induces a map, still denoted by I , from L ( u ∗ T G Y , m | Ω ) to L (˜ u ∗ T G Y , m | Ω ). Also, the fact that u = ˜ u m -a.e. trivially implies that such I is an isometry.Thanks to these considerations, it makes sense to consider the space L ( u ∗ T G Y , m | Ω ) for u ∈ L (Ω , Y ¯ y ), i.e. even when u is only given up to m -a.e. equality: it is just sufficient to pick anyBorel representative of u , consider the corresponding space of L -sections up to m -a.e. equalityand notice that such space does not depend on the representative of u chosen.The basic properties of the space L ( u ∗ T G Y , m | Ω ) are collected in the following statement. Proposition 4.4 (Properties of L ( u ∗ T G Y , m | Ω )) . Let (Y , d Y , ¯ y ) be a pointed CAT (0)-space, (X , d , m ) a metric measure space, Ω ⊂ X an open subset and u ∈ L (Ω , Y ¯ y ) .Then: ( i ) For every S , S ∈ L ( u ∗ T G Y , m | Ω ) the functions Ω (cid:51) x (cid:55)→ d u ( x ) ( S x , S x ) , | S x | u ( x ) , (cid:10) S x , S x (cid:11) u ( x ) are (equivalence classes up to m -a.e. equality of ) Borel functions. ( ii ) For every S , S ∈ L ( u ∗ T G Y , m | Ω ) the section S ⊕ S given by the (equivalence class ofthe) map x (cid:55)→ ( x, u ( x ) , S x ⊕ S x ) belongs to L ( u ∗ T G Y , m | Ω ) . ( iii ) For every S ∈ L ( u ∗ T G Y , m | Ω ) and f ∈ L ∞ (Ω) the section f S given by the (equivalenceclass of the) map x (cid:55)→ ( x, u ( x ) , f ( x ) S x ) belongs to L ( u ∗ T G Y , m | Ω ) .Proof. The Borel regularity of x (cid:55)→ d u ( x ) ( S x , S x ) has already been noticed. Then the one of | S x | u ( x ) follows from the fact that the zero section belongs to L ( u ∗ T G Y , m | Ω ) and thus the oneof (cid:10) S x , S x (cid:11) u ( x ) follows by the definition of scalar product.We pass to ( ii ) and first consider the case of S , S ∈ L ( u ∗ T G Y , m | Ω ) of the form S x = α ( G v ( x ) u ( x ) ) (cid:48) and S x = β ( G w ( x ) u ( x ) ) (cid:48) for v, w : Ω → Y Borel and essentially separably valued and α, β ≥ T := min { α − , β − } ∈ (0 , ∞ ] and for t ∈ (0 , T ) put v t := ( G vu ) αt , w t := ( G wu ) βt and let m t ( x )be the midpoint of v t ( x ) , w t ( x ) for every x ∈ Ω. From the continuity of the ‘midpoint’ operation and the triangle inequality it easily follows that x (cid:55)→ m t ( x ) is Borel, essentially separably valuedand in L (Ω , Y ¯ y ). Then, define the section M t as M t,x := t ( G m t ( x ) u ( x ) ) (cid:48) and recall (2.8) to see that M t,x → ( S ⊕ S ) x in T u ( x ) Y as t ↓ x ∈ Ω: this proves that ( S ⊕ S ) satisfies therequirement ( i ) in Definition 4.2. The same convergence together with the bound | M t, · | u ( · ) ≤ t d Y ( u, m t ) ≤ t (cid:0) d Y ( u, v t ) + d Y ( u, w t ) (cid:1) ≤ (cid:0) α d Y ( u, v ) + β d Y ( u, w ) (cid:1) on Ωvalid for every t ∈ (0 , T ) shows that ( S ⊕ S ) satisfies also the requirement ( ii ) in Definition 4.2.Now the fact that ( S ⊕ S ) ∈ L ( u ∗ T G Y , m | Ω ) for generic S , S ∈ L ( u ∗ T G Y , m | Ω ) followsby approximation (recall point ( ii ) in Definition 4.2) and the continuity of the ‘sum’ operationnoticed in Proposition 2.6, then the analogous properties for elements of L ( u ∗ T G Y , m | Ω ) triviallyfollow.Finally, the fact that ( S ⊕ S ) ∈ L ( u ∗ T G Y , m | Ω ) implies S ⊕ S ∈ L ( u ∗ T G Y , m | Ω ) is trivialfrom the definitions (see also the arguments below).For ( iii ) we notice that it is sufficient to prove that f S is in L ( u ∗ T G Y , m | Ω ) whenever f : Ω → R is Borel and bounded and S ∈ L ( u ∗ T G Y , m | Ω ). In this case the fact that f S satisfies therequirement ( i ) in Definition 4.2 is obvious. For ( ii ) we consider sections S nx = α n ( G v n ( x ) u ( x ) ) (cid:48) forwhich (4.2) hold and put ˜ S nx := α n (cid:107) f (cid:107) L ∞ ( G w n ( x ) u ( x ) ) (cid:48) , where w n ( x ) := ( G v n ( x ) u ( x ) ) f ( x ) / (cid:107) f (cid:107) L ∞ . Thefact that the w n ’s are Borel representatives of maps in L (Ω , Y ¯ y ) can be easily checked from thedefinition while the fact that (4.2) holds for f S and (˜ S n ) is obvious. (cid:3) Let us now come back to the initial discussion and, for given u ∈ L (Ω , Y ¯ y ), let us define themap ι : Geo u L (Ω , Y ¯ y ) → L ( u ∗ T G Y , m | Ω ) as follows. For v ∈ L (Ω , Y ¯ y ) and α ≥ t (cid:55)→ ( G vu ) αt to the (equivalence class of the) section given by x (cid:55)→ α ( G v ( x ) u ( x ) ) (cid:48) . The relationbetween T u L (Ω , Y ¯ y ) and L ( u ∗ T G Y , m | Ω ) is then described by the following result: Proposition 4.5 ( L ( u ∗ T G Y , m | Ω ) and T u L (Ω , Y ¯ y )) . Let (Y , d Y , ¯ y ) be a pointed CAT (0)-space, (X , d , m ) a metric measure space, Ω ⊂ X an open subset and u ∈ L (Ω , Y ¯ y ) .Then, the map ι : Geo u L (Ω , Y ¯ y ) → L ( u ∗ T G Y , m | Ω ) passes to the quotient and induces amap, still denoted ι , from Geo u L (Ω , Y ¯ y ) / ∼ to L ( u ∗ T G Y , m | Ω ) that can be uniquely extended bycontinuity to a bijective isometry, again denoted ι , from T u L (Ω , Y ¯ y ) to L ( u ∗ T G Y , m | Ω ) .Moreover, the so defined isometry ι respects the operations on the tangent cones, i.e. | v | u = ˆ Ω | ι ( v ) x | u ( x ) d m ( x ) , (4.3a) (cid:10) v , v (cid:11) u = ˆ Ω (cid:10) ι ( v ) x , ι ( v ) x (cid:11) u ( x ) d m ( x ) , (4.3b) d u ( v , v ) = ˆ Ω d u ( x ) ( ι ( v ) x , ι ( v ) x ) d m ( x ) , (4.3c) ι ( λ v ) = λι ( v ) , (4.3d) ι ( v ⊕ v ) = ι ( v ) ⊕ ι ( v ) , (4.3e) for any v , v , v ∈ T u L (Ω , Y ¯ y ) and λ ∈ R + . DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON
CAT ( κ )-SPACES AND APPLICATIONS 19 Proof.
Let v , v ∈ L (Ω , Y ¯ y ), α , α ≥
0, consider the sections S , S ∈ L ( u ∗ T G Y , m | Ω ) given by S i := ι ( α i ( G v i u ) (cid:48) ). Notice that d u (cid:0) α ( G v u ) (cid:48) , α ( G v u ) (cid:48) (cid:1) = lim t ↓ d L (cid:0) ( G v u ) α t , ( G v u ) α t (cid:1) t = lim t ↓ ˆ Ω d (cid:0) ( G v ( x ) u ( x ) ) α t , ( G v ( x ) u ( x ) ) α t (cid:1) t d m ( x )= ˆ Ω lim t ↓ d (cid:0) ( G v ( x ) u ( x ) ) α t , ( G v ( x ) u ( x ) ) α t (cid:1) t d m ( x )= ˆ Ω d u ( x ) (cid:0) S x , S x (cid:1) d m ( x ) , where, in passing the limit inside the integral, we used the dominate convergence theorem andthe fact that the integrand is non-negative and non-decreasing in t (recall (2.5)). This proves atonce that ι passes to the quotient to a map on Geo u L (Ω , Y ¯ y ) / ∼ and that the so induced mapis an isometry which therefore can be extended to a map from T u L (Ω , Y ¯ y ) to L ( u ∗ T G Y , m | Ω ).The fact that such extension is surjective follows from an approximation argument based on therequirement ( ii ) in Definition 4.2.Now observe that (4.3c) has already been proved by the fact that ι is an isometry. Then(4.3a) and (4.3b) follow as well. Also, (4.3d) is obvious by definition and then (4.3e) follows from(4.3c), (4.3d) and the metric characterization of the midpoints of x, y as the point m such that d ( x, m ) + d ( m, y ) = d ( x, y ) / (cid:3) The separable setting.
In this section we assume instead that Y is a separable and locally
CAT ( κ )-space and we study the Borel structure of the pullback u ∗ T G Y of the geometric tangentbundle of Y. We shall then see in the space case of Y being separable and
CAT (0) how such Borelstructure relates to the space L ( u ∗ T G Y , m | Ω ) studied in the previous section.Thus let Y be separable and locally CAT ( κ ), (X , d ) be a complete and separable metric spaceand Ω ⊂ X be open.As before, for a given Borel map u : Ω → Y, the pullback geometric tangent bundle u ∗ T G Y isdefined as u ∗ T G Y := (cid:8) ( x, y, v ) : x ∈ Ω , y = u ( x ) , v ∈ T y Y (cid:9) ⊂ X × T G Yand a section of u ∗ T G Y is a map S : Ω → u ∗ T G Y such that S x ∈ T u ( x ) Y for every x ∈ Ω.Now equip u ∗ T G Y ⊂ X × T G Y with the restriction of the product σ -algebra B (X) ⊗ B (T G Y),which, with abuse of terminology, we shall call Borel σ -algebra on u ∗ T G Y and denote B ( u ∗ T G Y).In particular, we shall say that a section is Borel if it is measurable w.r.t. B (X) and B ( u ∗ T G Y).A section is simple provided there are a Borel partition ( E n ) of Ω, ( α n ) ⊂ R + and points( y n ) ⊂ Y s.t. y n ∈ B r u ( x ) ( u ( x )), for every x ∈ E n and S | E n = α n ( G y n u ( · ) ) (cid:48) . We shall formallydenote such section by (cid:80) n χ E n α n u ∗ ( G y n · ) (cid:48) . Notice that the restriction of such section to E n coincides with the (graph of) the composition of u with the simple section of T G Y given by y (cid:55)→ ( y, α n ( G y n y ) (cid:48) ). In particular, recalling Proposition 2.12 we see that simple sections of u ∗ T G Yare Borel.Moreover, they are dense in the space of Borel sections:
Lemma 4.6 (Density of simple sections) . Let (X , d ) be a metric space, (Y , d Y ) be separable andlocally CAT ( κ ) -space, Ω ⊂ X an open subset and u : Ω → Y Borel. Let S : Ω → u ∗ T G Y be a Borelsection of u ∗ T G Y and ε > .Then, there is a simple section T such that d u ( x ) ( S x , T x ) < ε for every x ∈ Ω .Proof. We can reduce the proof to the case of Y being
CAT ( κ ) by using the Lindel¨of property ofY and the coverings made by B r y / ( y ). Doing so, we achieve uniqueness of geodesics between anycouple of points. Let D ⊂ Y be countable and dense ( y n , α n ) be an enumeration of D × Q + . Then for every n ∈ N consider the function F n : T G Y → R given by F n ( y, v ) := d y ( v, α n ( G y n y ) (cid:48) ) = (cid:113) | v | y + | α n | d ( y, y n ) − (cid:10) v, α n ( G y n y ) (cid:48) (cid:11) y . The defining requirements of B (T G Y) and the property (2.10) ensure that F n is Borel. Hence so isthe map ˜ F n : u ∗ T G Y → R defined as ˜ F n := F n ◦ π T G Y , where π T G Y : u ∗ T G Y ⊂ X × T G Y → T G Yis the canonical projection.Hence given a Borel section S of u ∗ T G Y the map ˜ F n ◦ S : X → R is Borel and thus, for given ε >
0, so is the set ˜ E n := ( ˜ F n ◦ S ) − ([0 , ε )). We then put E n := ˜ E n \ ∪ i Let (X , d ) be a metric space, (Y , d Y ) be separable and locally CAT ( κ ) , Ω ⊂ X an open subset and u : Ω → Y a Borel map. Let S , S be Borel sections of u ∗ T G Y and f : X → R + be a Borel map.Then the functions sending x ∈ X to | S x | u ( x ) , (cid:10) S x , S x (cid:11) u ( x ) , d u ( x ) ( S x , S x ) are Borel and the sec-tions x (cid:55)→ f ( x ) S x , S x ⊕ S x are Borel as well.Proof. Let S , S be simple of the form: S = χ E α u ∗ ( G y · ) (cid:48) and S = χ E β u ∗ ( G y · ) (cid:48) , with E i := u − ( A i ) and A i ∈ B (Y) such that y i ∈ B r y ( y ) for every y ∈ A i , i = 1 , 2. Then they arethe (graph of the) composition of u with the simple sections of T G Y given by χ A α ( G y · ) (cid:48) and χ A β ( G y · ) (cid:48) respectively, hence in this case the conclusion comes from Proposition 2.13.Then the conclusion comes from the ‘fiberwise’ continuity of all the expressions considered(granted by Proposition 2.6) and the density of simple sections established in Lemma 4.6 above. (cid:3) We now come to the relation between the space of (equivalence classes up to m -a.e. equalityof) Borel sections of u ∗ T G Y and the space L ( u ∗ T G Y , m | Ω ) in the case where Y is separable and CAT (0). As expected, these spaces coincide when the right integrability of the first ones is in place: Proposition 4.8. Let (X , d , m ) be a metric measure space, (Y , d Y , ¯ y ) be a pointed separable CAT (0)-space, Ω ⊂ X an open subset and u : Ω → Y be a Borel map.Then, S ∈ L ( u ∗ T G Y , m | Ω ) if and only if it is the equivalence class up to m -a.e. equality of aBorel section T of u ∗ T G Y with ´ Ω | T | u ( x ) d m ( x ) < ∞ .Proof. Assume at first that S ∈ L ( u ∗ T G Y , m | Ω ). Then the fact that ´ Ω | S x | u ( x ) d m ( x ) < ∞ is adirect consequence of the definition and of Proposition 4.5 above, thus we only need to prove that S is the equivalence class up to m -a.e. equality of a Borel section of u ∗ T G Y. To see this we need toprove that, letting π X , π T G Y be the projections of u ∗ T G Y ⊂ X × T G Y to X , T G Y respectively, themaps π X ◦ S and π T G Y ◦ S are equivalence classes up to m -a.e. equality of Borel maps. For the firstone this is obvious, because it is the identity on X. For the second one we recall the definition of B (T G Y) to see that we need to prove that π Y ◦ π T G Y ◦ S is Borel (which it is, because it coincideswith u ) and that x (cid:55)→ (cid:10) S x , ( G zu ( x ) ) (cid:48) (cid:11) is Borel for every z ∈ Y (which is easily seen to be the casefrom the requirement ( i ) in Definition 4.2).We pass to the converse implication and start observing that Lemma 4.1 and the definitionof B (T G Y) just recalled ensure that for any v ∈ L (Ω , Y ¯ y ) the section given by ( G v ( x ) u ( x ) ) (cid:48) is theequivalence class up to m -a.e. equality of a Borel section. It follows that if T is a Borel section as inthe statement, then it satisfies the requirement ( i ) in Definition 4.2. We now claim that if T is alsosimple, then it also satisfies the requirement ( ii ). To see this write T = (cid:80) n χ E n α n u ∗ ( G y n · ) (cid:48) andput T i := (cid:80) n ≤ i χ E n α n u ∗ ( G y n · ) (cid:48) where it is intended that for x / ∈ ∪ n ≤ i E n we have T ix = 0 u ( x ) ∈ T u ( x ) Y. Then putting β i := max n ≤ i α n , y n,i := ( G y n u ( x ) ) α n /β i and defining v i ∈ L (Ω , Y ¯ y ) as v i | E n := y n,i for n ≤ i and v i | Ω \∪ n ≤ i E n ≡ u we see that T i = ι ( β i ( G v i u ) (cid:48) ), so that (the equivalence DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON CAT ( κ )-SPACES AND APPLICATIONS 21 class up to m -a.e. equality of) T i belongs to L ( u ∗ T G Y , m | Ω ). It is then clear that d L ( T i , T ) → T belongs to L ( u ∗ T G Y , m | Ω ).Then the conclusion for a generic section T as in the statement can be easily obtained by anapproximation argument starting from the density result in Lemma 4.6. (cid:3) The Korevaar-Schoen energy. We recall here the key definitions and results of [21], wherethe original analysis done in [24] has been generalized to the setting of RCD( K, N ) spaces ([3],[17]).For the definitions of all the objects appearing below we refer to [21] (but see also [18] for thedefinition of the differential d u appearing in the statement below). Theorem 4.9 (The Korevaar-Schoen energy) . Let (X , d , m ) be a RCD( K, N ) space, K ∈ R , N ∈ [1 , ∞ ) , (Y , d Y , ¯ y ) a pointed CAT (0)-space, Ω ⊂ X open and u ∈ L (Ω , Y ¯ y ) . Then thefollowing are equivalent: i) Letting ks ,r [ u, Ω] : Ω → R + be defined by ks ,r [ u, Ω]( x ) := (cid:12)(cid:12)(cid:12) ffl B r ( x ) d ( u ( x ) ,u (˜ x )) r d m (˜ x ) (cid:12)(cid:12)(cid:12) / if B r ( x ) ⊂ Ω , otherwise.and the energy E KS ( u ) be given by (4.4) E KS ( u ) := lim r ↓ ˆ Ω ks ,r [ u, Ω] d m , we have E KS ( u ) < ∞ . ii) There is G ∈ L (Ω) such that for every ϕ : Y → R ϕ (¯ y ) = 0 we have ϕ ◦ u ∈ W , (Ω) with | d( ϕ ◦ u ) | ≤ G m -a.e..If any of these hold, the ‘energies at scale r ’ ks ,r [ u, Ω] converge to ( d + 2) − | d u | HS in L (Ω) as r ↓ . In particular, the lim in (4.4) is actually a limit and the energy admits the representation E KS ( u ) = 12( d + 2) ˆ Ω | d u | HS d m . Finally, the functional E KS : L (Ω , Y ¯ y ) → [0 , + ∞ ] is convex and lower semicontinuous. Remark 4.10. It should be noticed that the smallest function G for which ( ii ) holds is notthe Hilbert-Schmidt norm | d u | HS of the differential d u of u , but rather the (pointwise) operatornorm of d u . The two quantities are nevertheless comparable, i.e. one controls the other up tomultiplication with a dimensional constant. (cid:4) We shall denote by KS , (Ω , Y ¯ y ) ⊂ L (Ω , Y ¯ y ) the collection of maps with finite energy andrecall from [21] that for u, v ∈ KS , (Ω , Y ¯ y ) we always have d Y ( u, v ) ∈ W , (Ω). Therefore itmakes sense to ask whether u, v attain the same boundary value by checking whether or not wehave d Y ( u, v ) ∈ W , (Ω).Then given ¯ u ∈ KS , (Ω , Y ¯ y ) the ‘energy E KS ¯ u : L (Ω , Y) → [0 , ∞ ] with ¯ u as prescribed boundaryvalue’ can be defined as E KS ¯ u ( u ) := (cid:26) E KS ( u ) if u ∈ KS , (Ω , Y ¯ y ) and d Y ( u, ¯ u ) ∈ W , (Ω) , + ∞ otherwise . We shall denote the domain of E KS ¯ u by KS , u (Ω , Y ¯ y ) ⊂ L (Ω , Y ¯ y ) and recall that(4.5) E KS ¯ u : L (Ω , Y ¯ y ) → [0 , + ∞ ] is convex and lower semicontinuous,moreover it admits a unique minimizer, called harmonic map with ¯ u as boundary value. Remark 4.11. Even if the definition of E KS ¯ u can be given in high generality, it should be notedthat it may happen that E KS ¯ u = E KS . This happens when W , (Ω) = W , (Ω) which in turn occursif X \ Ω has null capacity. Thus in practical situations if one wants to enforce some boundarycondition, it should be checked that X \ Ω has positive capacity. (cid:4) For later use we recall that the convexity of both E KS and E KS ¯ u can be improved to the followinginequality:(4.6) E KS (( G vu ) t ) + t (1 − t ) E KS ( d ) ≤ (1 − t ) E KS ( u ) + t E KS ( v ) ∀ t ∈ [0 , , where d ( x ) := d ( u ( x ) , v ( x )). Such inequality has been proved for the case t = in [21] (imitatingthe arguments in [24]), the general case follows along the same arguments. It is worth to underlinethat in the above the maps u, v, ( G vu ) t are Y-valued, while d is real valued. In this sense the energyof E KS ( d ) of d has a different meaning w.r.t. the energy of the other maps. Still, we recall (see [21]and [24]) that for a constant c ( d ) depending only on the essential dimension d ≤ N of X we have E KS ( f ) = c ( d ) Ch ( f ) for any f ∈ L (Ω), where Ch is the standard Cheeger/Dirichlet energy on X.4.3. The Laplacian of a CAT (0)-valued map. Let us start by giving the general definition ofLaplacian of a CAT (0)-valued Sobolev map: Definition 4.12 (Tension field/Laplacian) . Let (X , d , m ) be a RCD( K, N ) space, Ω ⊂ X an opensubset, (Y , d Y , ¯ y ) a pointed CAT (0)-space and ¯ u ∈ KS , (Ω , Y ¯ y ) .Then the domain of the Laplacian D (∆ ¯ u ) ⊂ KS , u (Ω , Y ¯ y ) is defined as D (∆ ¯ u ) := D ( | ∂ − E KS ¯ u | ) and for u ∈ D (∆ ¯ u ) we put ∆ ¯ u u := ι ( v ) ∈ L ( u ∗ T G Y , m | Ω ) , where v is the element of minimal norm in − ∂ − E KS ¯ u ( u ) .Similarly, for maps u from X to Y we say that u is in the domain of the Laplacian D (∆) if | ∂ − E KS | ( u ) < ∞ and in this case ∆ u := ι ( v ) , where ι ( v ) is the element of minimal norm in − ∂ − E KS ( u ) . Proposition 4.13 (Laplacian and variation of the energy) . Let (X , d , m ) be a RCD( K, N ) space, Ω ⊂ X an open subset, (Y , d Y , ¯ y ) a pointed CAT (0)-space and ¯ u ∈ KS , (Ω , Y ¯ y ) . Also, let u ∈ D (∆ ¯ u ) . Then, for every v ∈ L (Ω , Y ¯ y ) , we have (4.7) − ˆ X (cid:10) ∆ ¯ u u ( x ) , (cid:0) G v ( x ) u ( x ) (cid:1) (cid:48) (cid:11) u ( x ) d m ( x ) ≤ lim t ↓ E KS ¯ u (( G vu ) t ) − E KS ¯ u ( u ) t . Moreover, u is harmonic with ¯ u as boundary value if and only if u ∈ D (∆ ¯ u ) with ∆ ¯ u u = 0 .Proof. Inequality (4.7) follows applying (3.7), the definition of ∆ ¯ u u and recalling Proposition 4.5.The second claim is a restatement of (3.8) in this setting. (cid:3) Remark 4.14. This last proposition shows that our definition is compatible with the classical onevalid in the smooth category. Indeed, if X , Y are smooth Riemannian manifold, ¯ u, u : ¯Ω ⊂ X → Yare smooth maps with the same boundary values, v is a smooth section of u ∗ TY (in the smoothsetting T G Y is canonically equivalent to the standard tangent bundle TY) which is 0 on ∂ Ω, thenwe can produce a smooth perturbation of u by putting u t ( x ) := exp u ( x ) ( t v x ). A direct computationthen shows that dd t | t =0 E KS ¯ u ( u t ) = − ˆ Ω (cid:10) τ ( u ) x , v x (cid:11) u ( x ) d m ( x ) , where τ ( u ) is the tension field of u , see for instance [23, Section 9.2]. This formula is the smoothversion of (4.7). Notice indeed that u t = ( G u u ) t for t ∈ [0 , 1] (and similarly u t = ( G u − u ) − t for t ∈ [ − , t (cid:55)→ E KS ¯ u ( u t ) is C , hence differentiable in 0, sothat the one-sided bound in (4.7) becomes an equality in the smooth case.It is worth to underline that in our framework the lack of equality in (4.7) is not only relatedto the lack of smoothness of t (cid:55)→ E KS ¯ u ( u t ), which a priori could produce different left and rightderivatives in 0, but also to the fact that tangent cones are not really tangent spaces : the oppositeof a vector field does not necessarily exist and thus we are forced to take one-sided perturbationsonly. (cid:4) A direct consequence of Proposition 4.13 above is the following: DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON CAT ( κ )-SPACES AND APPLICATIONS 23 Corollary 4.15. With the same assumptions and notation as in Proposition 4.13 we have E KS ¯ u ( u ) − ˆ X (cid:10) ∆ ¯ u u ( x ) , (cid:0) G v ( x ) u ( x ) (cid:1) (cid:48) (cid:11) u ( x ) d m ( x ) + E KS ( d ) ≤ E KS ¯ u ( v ) , where d := d ( u, v ) .Proof. Couple (4.7) with (4.6). (cid:3) In the next discussion, we are interested in properties of the composition f ◦ u , whenever u is aharmonic map and f is λ -convex functional. Observe that, in a smooth framework, the chain rule∆( f ◦ u ) = Hess f ( ∇ u, ∇ u ) + d f (∆ u ) immediately implies that(4.8) ∆( f ◦ u ) ≥ λ | d u | HS if f is λ -convex and u is harmonic.A nonsmooth version of (4.8) has already been addressed in [27] (see Theorem 1.2 there) formaps with euclidean source domain and CAT (0)-target. Nevertheless, as we are going to show inTheorem 4.18, the discussion generalizes to our framework: the main stumbling block to overcomebeing the absence of Lipschitz vector field on a RCD-space. In the next, we shall need the followingproperty of Sobolev functions and, specifically, of their directional derivatives (for the definitionof test vector field see [16] and for the concept of Regular Lagrangian Flow see [4]): Proposition 4.16. Let (X , d , m ) be a RCD( K, N ) space, (Y , d Y , ¯ y ) a pointed complete metricspace, Ω ⊂ X open, v ∈ L m ( T X) a test vector field and ( FI vs ) the associated Regular LagrangianFlow. Also, let u ∈ KS , (Ω , Y ¯ y ) .Then, for every K ⊂ Ω compact, we have that (4.9) lim s → d Y ( u ◦ FI vs , u ) s = | d u ( v ) | in L ( K ) . (notice that for | s | small the map u ◦ FI vs is well defined from K to Y ).Similarly, for a real valued Sobolev function g ∈ W , (Ω) we have (4.10) lim s → g ◦ FI vs − gs = d g ( v ) in L ( K ) . Proof. Property (4.10) is (an equivalent version of) the definition of Regular Lagrangian Flow, seefor instance [19, Proposition 2.7]. For (4.9) recall first [21, Remark 4.15] to get that functions in KS , (Ω , Y ¯ y ) also belong to the ‘direction’ Korevaar-Schoen space as defined in [20], then recall[20, Theorem 4.5]. (cid:3) The next Lemma deals with variations of a map u , suitably obtained through gradient flowstrajectories in the target space, and the rate of change at the level of Korevaar-Schoen energy (see(4.11)-(4.12) below). In the following statement, notice that f ◦ u belongs to W , (Ω) - and thusd( f ◦ u ) is well defined - because f is Lipschitz, Ω has finite measure and by ( ii ) in Theorem 4.9.Also, for the very same reason, we shall drop the subscript ¯ y from Y when Ω is bounded as the L -integrability depends no more on the particular chosen point ¯ y ∈ Y. Compare the proof with[27, Lemma 3.1]. Lemma 4.17. Let (X , d , m ) be a RCD( K, N ) space, Y CAT (0)-space and Ω ⊂ X open and bounded.Also, let f ∈ Lip(Y) be λ -convex, λ ∈ R , and u ∈ KS , (Ω , Y) . For g ∈ Lip bs (X) + , define the(equivalence class of the) variation map u t ( x ) = GF ftg ( x ) ( u ( x )) ∀ t > , x ∈ Ω . Then, u t ∈ KS , (Ω , Y) for every t > and there is a constant C > depending on f, g suchthat (4.11) | d u t | HS ≤ e − λtg (cid:0) | d u | HS − t (cid:10) d g, d( f ◦ u ) (cid:11) + Ct (cid:1) m - a.e. in Ω , holds for every t ∈ [0 , . In particular (4.12) lim t ↓ E KS ( u t ) − E KS ( u ) t ≤ − ˆ Ω λd + 2 g | d u | HS + (cid:10) d( f ◦ u ) , d g (cid:11) d m . Proof. The map x (cid:55)→ ( tg ( x ) , u ( x )) is Borel and essentially separably valued and the map ( t, y ) (cid:55)→ GF ft ( y ) is continuous, hence x (cid:55)→ u t ( x ) is Borel and essentially separably valued. Also, the identity(3.2) and the trivial estimate | ∂ − f | ≤ Lip( f ) show that t (cid:55)→ GF ft ( y ) is Lip( f )-Lipschitz for every y ∈ Y, thus d Y ( u t ( x ) , ¯ y ) ≤ t sup( g )Lip( f )+ d Y ( u ( x ) , ¯ y ), for every ¯ y ∈ Y, from which it easy followsthat u t ∈ L (Ω , Y). Taking also into account the contraction property (3.4) we obtain that d Y ( u t ( x ) , u t ( y )) ≤ e λ − t ( g ( x )+ g ( y )) d Y (cid:0) u ( x ) , GF ft | g ( y ) − g ( x ) | ( u ( y )) (cid:1) ≤ e λ − t sup g (cid:0) d Y ( u ( x ) , u ( y )) + t Lip( g )Lip( f ) d ( x, y ) (cid:1) and thus ks ,r [ u t , Ω]( x ) ≤ e λ − t sup g (cid:0) ks ,r [ u, Ω]( x ) + t Lip ( g )Lip ( f ) (cid:1) . Integrating and using the fact that m (Ω) < ∞ we conclude that u t ∈ KS , (Ω , Y).In order to obtain (4.11) we need to be more careful in our estimates and to this aim we shalluse Lemma 3.4 and Proposition 4.16 above. Let γ : [0 , S ] → Ω be a Lipschitz curve: for any s ∈ [0 , S ] the bound (3.6) gives (here we are fixing a Borel representative of u and thus of u t , butnotice that the estimate (4.15) does not depend on such choice): d ( u t ( γ ) , u t ( γ s )) ≤ e − λt ( g ( γ ) ±| g ( γ ) − g ( γ s ) | ) (cid:16) d ( u ( γ ) , u ( γ s )) + 2 t ( g ( γ s ) − g ( γ ))( f ( u ( γ )) − f ( u ( γ s )))+ ˆ | t ( g ( γ ) − g ( γ s )) | ( f ) θ λ ( r ) + λ − (cid:0) d ( GF fr ( u ( γ )) , u ( γ s )) + d ( GF fr ( u ( γ s )) , u ( γ )) (cid:1) d r (cid:17) , where the sign in ±| g ( γ ) − g ( γ s ) | depends on the sign of λ . Now use again the fact that r (cid:55)→ GF fr ( u ( γ )) is Lip( f )-Lipschitz to get that d ( GF fr ( u ( γ )) , u ( γ s )) ≤ r Lip ( f ) + 2 d ( u ( γ ) , u ( γ s )) , notice that the same bounds holds for d ( GF fr ( u ( γ s )) , u ( γ )), that | t ( g ( γ ) − g ( γ s )) | ≤ ts Lip( g )Lip( γ )and that θ λ ( t ) ≤ te λ − t to conclude that, for some constant C depending only on f, g, Lip( γ ) , T and every t ∈ [0 , T ], we have d ( u t ( γ ) , u t ( γ s )) ≤ e − λtg ( γ )+ Cs (cid:16) d ( u ( γ ) , u ( γ s ))+ 2 t ( g ( γ s ) − g ( γ ))( f ( u ( γ )) − f ( u ( γ s ))) + Ct s + Cts d ( u ( γ ) , u ( γ s )) (cid:17) . (4.13)Now let v be a test vector field on X and FI vs its Regular Lagrangian Flow and recall that since g, f ◦ u ∈ W , (Ω), by (4.10) we know that for any K ⊂ Ω compact we have(4.14) g ◦ FI vs − gs → d g ( v ) and f ◦ u ◦ FI vs − f ◦ us → d( f ◦ u )( v )in L ( K ) as s ↓ 0. Thus writing (4.13) for γ s := FI vs ( x ) for m -a.e. x ∈ Ω, dividing by s , letting s ↓ | d u t ( v ) | ≤ e − λtg (cid:16) | d u ( v ) | − t d g ( v ) d( f ◦ u )( v ) + Ct (cid:17) m - a.e. in Ω , having also used the arbitrariness of K ⊂ Ω compact and the fact that the Lipschitz constantof t (cid:55)→ FI vs ( x ) is bounded by (cid:107) v (cid:107) L ∞ . We have established (4.15) for v regular, but both sides ofthe inequality are continuous w.r.t. L -convergence of uniformly bounded vectors v with values in L m ( T X), thus by density we deduce that (4.15) is valid for any v ∈ L ∞ m ( T X). Hence writing (4.15)for v varying in a local Hilbert base of L m ( T X) and adding up we deduce (4.11). Then (4.12) alsofollows. (cid:3) DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON CAT ( κ )-SPACES AND APPLICATIONS 25 In order to state the analogue of (4.8) in the non-smooth setting we need to recall the notion ofmeasure-valued Laplacian as introduced in [17] (the presentation that we make here is simplifiedby the fact that RCD spaces are infinitesimally Hilbertian).Thus let X be a RCD( K, N ) space, Ω ⊂ X open and bounded and f ∈ W , (Ω). We say that f has a measure valued Laplacian in Ω, and write f ∈ D ( ∆ , Ω), provided there is a (signed) Radonmeasure µ on Ω such that ˆ g d µ = − ˆ (cid:10) d f, d g (cid:11) d m ∀ g ∈ Lip c (Ω) . It is clear that this measure is unique and, denoting it by ∆ f | Ω , that the assignment f (cid:55)→ ∆ f | Ω is linear.We shall need the following criterium for checking whether f ∈ D ( ∆ , Ω): for f ∈ W , (Ω) and h ∈ L ( m | Ω ) we have(4.16) − ˆ X (cid:10) d f, d g (cid:11) d m ≥ ˆ X gh d m ∀ g ∈ Lip c (Ω) + ⇒ f ∈ D ( ∆ , Ω) and ∆ f | Ω ≥ h m . We are now ready to state and prove the next theorem. Theorem 4.18. Let (X , d , m ) be a RCD( K, N ) space, Y be CAT (0) and Ω ⊂ X open and bounded.Also, let f ∈ Lip(Y) be λ -convex, λ ∈ R and u ∈ KS , (Ω , Y) be harmonic.Then, f ◦ u ∈ D ( ∆ , Ω) and ∆ ( f ◦ u ) | Ω is a (signed) locally finite Radon measure satisfying (4.17) ∆ ( f ◦ u ) | Ω ≥ λd + 2 | d u | HS m . Proof. As noticed before Lemma 4.17, under the stated assumptions we have f ◦ u ∈ W , (Ω).Now let g ∈ Lip c (Ω) + be arbitrary and apply Lemma 4.17 with these functions f, g, u and define u t ∈ KS , u (Ω , Y) accordingly. Notice that since supp( g ) ⊂ Ω, we have that u t and u agree on aneighbourhood of ∂ Ω and thus have the same boundary value.Therefore from the fact that u is harmonic and (4.12) we deduce − ˆ Ω (cid:10) d( f ◦ u ) , d g (cid:11) d m ≥ λd + 2 ˆ Ω g | d u | HS d m ∀ g ∈ Lip c (Ω) + and the conclusion comes from (4.16). (cid:3) Corollary 4.19. Let Ω ⊂ X be open, Y be CAT (0), ¯ u ∈ KS , (Ω , Y) , u harmonic map with ¯ u asboundary values and f ∈ Lip(Y) be -convex. If f ◦ u is constant then u itself is constant map.Proof. Apply Theorem 4.18, then | d u | HS vanishes and conclude. (cid:3) Let us now discuss a simple and explicit example of Laplacian of a map. Example 4.20. Let Y := R , X := R / Z equipped with the standard distances and measure, andΩ = X. Then a direct application of the definitions in Theorem 4.9 show that u = ( u , u ) : X → Yis in KS , (X , Y) if and only if u ◦ p , u ◦ p : R → R are in W , loc ( R ), where p : R → R / Z = X isthe natural projection, with E KS ( u ) = c (cid:16) ˆ X | u (cid:48) | ( θ ) + | u (cid:48) | ( θ ) d θ (cid:17) , for some universal constant c > 0. Then it is clear that u ∈ D (∆) if and only if ( u ◦ p) (cid:48)(cid:48) , ( u ◦ p) (cid:48)(cid:48) ∈ L loc ( R ) and that in this case ∆ u = c ( u (cid:48)(cid:48) , u (cid:48)(cid:48) ) . Now let u ( θ ) := (cos(2 πθ ) , sin(2 πθ )) be the canonical embedding of X in Y. Then ∆ u = − u andin particular for any θ ∈ X we have that ∆ u ( θ ) ∈ T u ( θ ) R ∼ R is orthogonal to the tangent spaceof X seen as a subset of R = Y.This is interesting because one can define the differential d u of u , even in very abstract situations[18], by means related to Sobolev calculus on the metric measure space (Y , d Y , µ := u (cid:93) ( | d u | HS m ))and tangent vector fields in this metric measure space only see directions which are tangent to the graph of u (this is rather obvious in this example, but see for instance [12] for a discussion of thisphenomenon in more general cases). This means that, curiously, ∆ u cannot be computed startingfrom d u and using Sobolev calculus in the spirit developed in [16], [14], simply because ∆ u doesnot belong to the tangent module L µ ( T Y) (cid:4) We conclude pointing out that while in the Definition 4.12 of Laplacian of a map we called intoplay the space L ( u ∗ T G Y , m | Ω ) as introduced in Definition 4.3, in some circumstances it mightbe useful to deal with a notion of Laplacian related to the Borel σ -algebra B ( u ∗ T G Y) - and thusto the characterization given in Proposition 4.8 -, which however is only available for separablespaces Y.In this direction it is worth to underline that one can always reduce to such case thanks to thefollowing two simple results: the first says that given u ∈ L (Ω , Y ¯ y ) we can always find a separable CAT (0) subspace ˜Y of Y containing the gradient flow trajectory of E KS ¯ u starting from u , the secondensures that this restriction does not affect the notion of minus-subdifferential. Proposition 4.21. Let (X , d , m ) be a RCD( K, N ) space, (Y , d Y , ¯ y ) a pointed CAT (0)-space, Ω ⊂ X open, ¯ u ∈ KS , (Ω , Y ¯ y ) and u ∈ L (Ω , Y ¯ y ) . Also, let ( u t ) be the gradient flow trajectory for E KS ¯ u starting from u .Then, there exists a separable CAT (0) subspace ˜Y ⊂ Y such that m ( u − t (Y \ ˜Y)) = 0 for every t ≥ . Similarly for the functional E KS .Proof. From the fact that geodesics on Y are unique and vary continuously with the endpoint itis easy to see that the closed convex hull of a separable set (i.e. the smallest closed and convexset containing the given set) is also separable. Use this and the fact that maps in L (Ω , Y ¯ y ) areby definition essentially separably valued to find ˜Y ⊂ Y which is CAT (0) with the induced metricand such that m ( u − t (Y \ ˜Y)) = 0 for every t ∈ Q + . We claim that ˜Y satisfies the conclusion.To see this, pick t ≥ 0, let ( t n ) ⊂ Q + be converging to t and up to pass to a non-relabeledsubsequence assume that (cid:80) n d L ( u t n +1 , u t n ) < ∞ . Then from the triangle inequality in L (Ω)and the monotone convergence we see that (cid:107) (cid:80) n d Y ( u t n +1 , u t n ) (cid:107) L ≤ (cid:80) n d L ( u t n +1 , u t n ) < ∞ sothat in particular for m -a.e. x ∈ Ω we have (cid:80) n d Y ( u t n +1 , u t n )( x ) < ∞ which in turn implies that( u t n ( x )) ⊂ ˜Y is a Cauchy sequence, so that its limit v ( x ) also belongs to ˜Y. The same kind ofargument also shows that ( u t n ) converges to v in L (Ω , Y ¯ y ) and since we know, by the continuityof ( u t ) as L (Ω , Y ¯ y )-valued curve, that u t n → u t in L (Ω , Y ¯ y ) we conclude that u t = v , whichproves our claim. (cid:3) To present our final result we need a bit of notation. Let Y be a CAT (0)-space and ˜Y a subspacewhich is also CAT (0) with the induced metric. Call I Y˜Y : ˜Y → Y the inclusion map. Then for every y ∈ ˜Y the tangent space T y ˜Y embeds isometrically into T y Y via the continuous extension of themap which sends α ( G zy ) (cid:48) ∈ T y ˜Y to α ( I Y˜Y ( G zy )) (cid:48) ∈ T y Y. In other words, we can regard a geodesicin ˜Y also as a geodesic in Y and this provides a canonical immersion of T y ˜Y in T y Y which fortrivial reasons is an isometry. Abusing a bit the notation we shall denote such isometry by I Y˜Y . Proposition 4.22. Let Y be a CAT (0)-space, E : Y → R ∪ { + ∞} a λ -convex and lower semicon-tinuous functional, ( y t ) a gradient flow trajectory for E starting from y ∈ Y and ˜Y ⊂ Y a subsetwhich is also a CAT (0) -space with the induced metric and such that ( y t ) ⊂ ˜Y . Denote by ˜ E therestriction of E to ˜Y Then, − ∂ − E ( y ) (cid:54) = ∅ if and only if − ∂ − ˜ E ( y ) (cid:54) = ∅ and letting v, ˜ v be the respective elements ofminimal norm we have I Y˜Y (˜ v ) = v . Moreover, ( y t ) is also a gradient flow trajectory for ˜ E .Proof. Assume that − ∂ − ˜ E ( y ) (cid:54) = ∅ . Then we know from Theorem 3.10 that h ( G y h y ) (cid:48) → ˜ v as h ↓ I Y˜Y ( h ( G y h y ) (cid:48) ) → I Y˜Y (˜ v ) and thus by Theorem 3.10 to conclude it is sufficient to provethat | ∂ − E | ( y ) < ∞ , because in that case we have that I Y˜Y ( h ( G y h y ) (cid:48) ) converges to the element ofminimal norm in − ∂ − E ( y ) (cid:54) = ∅ (which in particular is not empty) as h ↓ DIFFERENTIAL PERSPECTIVE ON GRADIENT FLOWS ON CAT ( κ )-SPACES AND APPLICATIONS 27 Since h ( G y h y ) (cid:48) → ˜ v we have in particular that d Y ( y ,y h ) h = | h ( G y h y ) (cid:48) | y → | v | y and thus S :=sup h ∈ (0 , d Y ( y ,y h ) h < ∞ . By the contractivity property (3.4) we deduce thatsup t,h ∈ (0 , d Y ( y t , y t + h ) h < ( e λ ∨ S =: S (cid:48) and thus letting h ↓ | ˙ y + t | ≤ S (cid:48) for every t ∈ (0 , − ∂ − E ( y ) (cid:54) = ∅ . 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