The principle of least action in the space of Kähler potentials
aa r X i v : . [ m a t h . C V ] S e p The principle of least action in the space ofK¨ahler potentials
L´aszl´o Lempert ∗ Department of MathematicsPurdue UniversityWest Lafayette, IN 47907-2067, USA
Abstract
Given a compact K¨ahler manifold, the space H of its (relative) K¨ahler potentialsis an infinite dimensional Fr´echet manifold, on which Mabuchi and Semmes have in-troduced a natural connection ∇ . We study certain Lagrangians on T H , in particu-lar Finsler metrics, that are parallel with respect to the connection. We show thatgeodesics of ∇ are paths of least action, and prove a certain convexity property of theleast action. This generalizes earlier results of Calabi, Chen, and Darvas. Let (
X, ω ) be an n dimensional, connected, compact K¨ahler manifold and H = H ω = { u ∈ C ∞ ( X ) : ω + dd c u = ω u > } its space of relative K¨ahler potentials. Here C ∞ ( X ) refers to the Fr´echet space ofreal valued smooth functions on X , and d c = i ( ∂ − ∂ ) /
2, so that dd c = i∂∂ . Thespace H , as an open subset of a Fr´echet space, inherits a F´echet manifold structure,whose tangent bundle has a canonical trivialization T H ≈ H × C ∞ ( X ). Mabuchi andSemmes [M, S] independently and with different motivations have introduced a torsionfree connection ∇ on T H . Mabuchi, as a tool to study special K¨ahler metrics, defined aRiemannian metric on H and obtained ∇ as the Levi–Civita connection of the metric.Somewhat later Semmes found the connection in search for a geometric interpretation ofinterpolation of Banach spaces and of a certain homogeneous complex Monge–Amp`ereequation associated with interpolation. He also determined all Riemannian metricscompatible with the connection: they are linear combinations of Mabuchi’s metric andthe square of a linear form. ∗ Research partially supported by NSF grant DMS 17641672020 Mathematics Subject classification 32Q15, 32U15, 53C35, 58B20, 58E30, 70H99 ne way to explain ∇ is through its parallel transport. We will use dot ˙ to denotederivative of a function of one real variable, and grad v to refer to gradient of a function X → R with respect to the K¨ahler metric of ω v . Let u : [ a, b ] → H be a smooth path.By integrating the time dependent vector field ( − /
2) grad u ( t ) ˙ u ( t ) on X we obtain asmooth family of diffeomorphisms ϕ ( t ) : X → X . In fact ϕ ( t ) : ( X, ω u (0) ) → ( X, ω u ( t ) )is symplectomorphic. The parallel translate of ξ ∈ T u ( t ) H ≈ C ∞ ( X ) to u (0) along thepath u is then(1.1) ξ ◦ ϕ ( t ) ∈ C ∞ ( X ) ≈ T u (0) H . Understanding the geodesics of this connection was already marked in [M, S] as aninteresting and potentially important problem, and Donaldson’s subsequent work [Do]gave further impetus to study them. By now the boundary value problem for geodesicsis well understood. On the one hand Darvas, Hu, Vivas, and myself [D1, DL, Hu, LV]proved that points in H cannot always be connected by a geodesic, not even if theyare close to each other. On the other hand work by Berman–Demailly, Berndtsson,B locki, Chen, Chen–Feldman–Hu, Chu–Tosatti–Weinkove, Darvas, and He [BD, Be1,Bl1, Bl2, C, CFH, CTW, D2, D3, He] gave that the geodesic equation extends tovarious enlargements of H , and in these enlargements any pair of points, or at leastnearby points, can be connected by solutions of the extended geodesic equation, weakgeodesics. It follows from Chen’s work that in those enlargements to which Mabuchi’smetric extends, weak geodesics minimize length. In [D2] Darvas generalized Mabuchi’smetric to certain Orlicz type Finsler metrics on H , determined the metric completionsof H under these metrics, and again found that weak geodesics in these completionsminimize length. In a slight overstatement the length minimizing paths are independentof which of Darvas’s metric we use to compute length. This was surprising at first sight.But in fact in geometry one encounters other similar phenomena. In a normedvector space straight line segments minimize length no matter what norm is chosen.There is also the analogy between H and the space Q of positive definite quadraticforms on R k . Q has a natural torsion free connection that turns it in a symmetric space ≈ GL + k ( R ) / SO k ( R ); and for all parallel Finsler metrics—i.e. those that are invariantunder GL + k ( R )—the shortest paths are the same: subarcs of left translates of certainone parameter subgroups in GL + k ( R ), projected to GL + k ( R ) / SO k ( R ).Now H with Mabuchi’s connection is itself a symmetric space [Do, M, S], at leastaccording some definitions of a symmetric space (while it is not according to someothers, [L3]). Although there is no group acting transitively on ∗ ( H , ∇ ), the holonomygroupoid Γ of ( H , ∇ ) acts on T H . Thus Γ = S u,v ∈H Γ uv , where Γ uv consists of linearisomorphisms T u H → T v H that arise as parallel transport along piecewise smooth pathsfrom u to v . Concatenation of parallel transports defines an operation Γ uv × Γ vw → Γ uw that turns Γ in a grupoid. That a Finsler metric or a function L : T H → R is parallelmeans it is invariant under Γ. ∗ This follows from [L2]. Even though Theorem 1.2 there is formulated for isometries of Mabuchi’s metric,the proof, verbatim, gives that if ω u is analytic while ω v is not, then no diffeomorphism of H can preserve ∇ and map u to v . arvas’s metrics are parallel. They are defined in terms of integrals(1.2) Z X χ ( ξ ) ω nu , u ∈ H , ξ ∈ T u H ≈ C ∞ ( X ) , with a fixed “Young weight” χ : R → [0 , ∞ ], and are invariant under parallel transportsimply because in the formula (1.1) for parallel transport, ϕ ( t ) satisfies ϕ ( t ) ∗ ω u ( t ) = ω u (0) . But there are many parallel Finsler metrics on T H beyond those considered in[D2]. The simplest is, for given 0 < α < p ( ξ ) = sup n Z E | ξ | ω nu . (cid:16) Z E ω nu (cid:17) α : E ⊂ X is measurable o ,ξ ∈ T u H . This is known as weak L q -norm or L q, ∞ Lorentz norm, q = 1 /α .Our thesis is that the proper generality of Darvas’s results on his metrics is parallel,or holonomy invariant, Finsler metrics and more generally, fiberwise convex functions T H → R , “Lagrangians”. In this paper and in a sequel we will show that many of hisresults generalize to this framework. Most of the time we will consider Lagrangians on T H that extend to the space of bounded ω –plurisubharmonic functions. We denoteby B ( X ) the Banach space of bounded Borel functions ξ : X → R with the norm || ξ || = sup | ξ | ; the Lagrangians of interest extend to ( B ( X ) ∩ PSH( ω )) × B ( X ). (Themore common space L ∞ ( X ) is a quotient of B ( X ), but we have little use for it in thispaper.) A generalization of holonomy invariance can be defined for such functions.Our results pertain to invariant Lagrangians that are convex in the B ( X ) variable andhave a certain continuity property, that we call strong continuity. Theorem 1.1 (=Theorem 8.1, Principle of least action) . If v : [0 , T ] → B ( X ) ∩ PSH ( ω ) is a weak geodesic, and C as a map into the Banach space B ( X ) , then it minimizesaction Z T L ( ˙ u ( t )) dt among all piecewise C paths u : [0 , T ] → B ( X ) ∩ PSH ( ω ) with u (0) = v (0) , u ( T ) = v ( T ) . We do not know whether weak geodesy of v already implies it is C , but fromChen’s work [C] we do know that a weak geodesic with endpoints in H is C . Thetheorem can be proved for weak geodesics rather less regular than C , but we still donot know if all weak geodesics have this relaxed regularity.The second result is about how the least action varies as one moves along weakgeodesics; it is a manifestation of seminegative curvature. Fix T >
0. If w, w ′ ∈ B ( X ) ∩ PSH( ω ), the least action L T ( w, w ′ ) between them is the infimum of the actions R T L ( ˙ u ( t )) dt over all piecewise C paths u : [0 , T ] → B ( X ) ∩ PSH( ω ) connecting w with w ′ . It is not obvious, but by Lemma 9.3, L T ( w, w ′ ) is finite. Theorem 1.2 (=Theorem 9.1) . If u, v : [ a, b ] → B ( X ) ∩ PSH ( ω ) are weak geodesics,then the function L T ( u, v ) : [ a, b ] → R is convex. he tools of this paper are Chen’s work on ε –geodesics, rudiments of Guedj–Zeriahi’s pluripotential theory, and our results on invariant convex functions on C ∞ ( X )[C, GZ1, GZ2, L4]. Even if the details are different, overall we will be able to followthe strategy of Calabi, Chen, and Darvas [C, CC, D2, D3] here, and also in a se-quel devoted to least action in spaces of ω –plurisubharmonic functions larger than B ( X ) ∩ PSH( ω ). Once basic properties of our Lagrangians are established, the greatergenerality occasionally results in less computation in the proofs for the following rea-son. Say, for a holonomy invariant Finsler metric p : T H → [0 , ∞ ), there is a family F ⊂ T H ≈ H × C ∞ ( X ) such that(1.3) p ( ξ ) = sup n Z X f ξω nu : f ∈ F ∩ T u H o , ξ ∈ T u H , and the integrals in (1.3), linear in ξ , can be easier to manipulate than the nonlinearintegrals in (1.2).It appears that the greatest generality in which action can be defined by an integralis the space of bounded ω –plurisubharmonic functions. Nonetheless, action can bedefined for any path in PSH( ω ) as a limit of integrals. Whether this action is finite or ±∞ of course depends on the path and on the Lagrangian. We plan to address thisand related questions in a sequel to this paper.Lagrangians even beyond Finsler metrics are not new to the subject. Chen’s ε –geodesics are trajectories of a Lagrangian L : T H → R (albeit not holonomy invariant),with kinetic energy term the square of Mabuchi’s metric and potential energy a multipleof V ( u ) = − R X uω n . Functions on H that its geometry motivates, and that are used inexistence problems in K¨ahler geometry, are also not new. Aubin’s functional I : H → R [Au, p.146], I ( u ) = Z X u ( ω n − ω nu )is a constant multiple of the total geodesic curvature of the line segment [0 , ∋ t tu ∈ H , measured in Darvas’s L Finsler metric. Monge–Amp`ere energy also arisesfrom the geometry of H . It is a convex function on H , for example in the sense thatits restrictions to geodesics of ∇ are convex; but its negative is also convex and, up toscaling and adding a constant, it is the only continuous function that has this property.We hope that a geometrical approach to functions on H and on related spaces, inthe spirit of this paper, will be of use in analytical problems on K¨ahler manifolds.Contents. Section 2 is about basic properties of holonomy invariant convex La-grangians T H → R . Section 3 is about a subclass of Lagrangians that have an extracontinuity property, which makes it possible to extend them to a larger vector bundle.Most of the results in these sections are direct consequences of results in [L4]. Section4 reviews the notion of weak and ε –geodesics, and ε –Jacobi fields. Section 5 introducesthe action and formulates Theorems 1.1 and 1.2 in precise forms. It also gives a roadmap to their proofs, which occupy the rest of the paper, sections 6–9.In this paper we freely use basic notions of infinite dimensional analysis and geom-etry. There are many sources the reader can consult on these matters, one of them[L1], written with an eye on the space H of K¨ahler potentials.Acknowledgement. During the preparation of this paper I have profited frompluripotential theoretic discussions with Darvas and Guedj. Lagrangians
The central objects of this paper are continuous functions L : T H → R that areconvex on each tangent space T u H and have a certain invariance property; as wellas the associated action functional L ( u ) = R ba L ( ˙ u ( t )) dt (cid:0) = R ba L ◦ ˙ u for brevity). Inthis section and in the next we record basic facts about such functions which followmore or less directly from [L4], that dealt with the action on C ∞ ( X ) of Hamiltoniandiffeomorphisms of ( X, ω ) and with invariant convex functions on C ∞ ( X ). As explainedin the Introduction, for L the invariance property in question is invariance under theholonomy grupoid Γ of ( H , ∇ ). Thus, if ξ ∈ T u (1) H is the parallel translate of ξ ∈ T u (0) H along a piecewise smooth path u : [0 , → H , then L ( ξ ) = L ( ξ ). This propertyin fact implies a much stronger and more primitive notion of invariance. Definition 2.1.
Given two measure spaces ( X, µ ) and ( Y, ν ) , we say that measurablefunctions ξ : X → R and η : Y → R are equidistributed, or are strict rearrangementsof each other, if µ ( ξ − B ) = ν ( η − B ) for every Borel set B ⊂ R . In finite measure spaces this is equivalent to requiring µ ( ξ > t ) = ν ( η > t ) for all t ∈ R .Back to our K¨ahler manifold ( X, ω ), if u ∈ H we let µ u denote the measure inducedby ω nu . Given measurable ξ, η : X → R we will write(2.1) ( ξ, u ) ∼ ( η, v ) if ξ, η are equidistributed as functions on ( X, µ u ) , ( Y, µ v ) . When smooth ξ, η are viewed as tangent vectors in T u H , T v H , we will just write ξ ∼ η . Theorem 2.2.
A function L : T H → R , continuous and convex on each fiber T u H ,is invariant under the holonomy gruppoid Γ if and only if it is invariant under strictrearrangements: L ( ξ ) = L ( η ) when ξ ∼ η . For the proof we need to understand the holonomy groups Γ uu . (1.1) shows thatin general, elements of Γ uv , isomorphisms T u H → T v H , are pullbacks by certain sym-plectomorphisms ϕ : ( X, ω v ) → ( X, ω u ). Let us write G for those symplectomorphismsthat induce elements of Γ . Thus G is a subgroup of the Fr´echet–Lie group Diff X ofdiffeomorphisms of X . Lemma 2.3.
The closure of G in Diff X contains all Hamiltonian diffeomorphisms of ( X, ω ) . Recall that Hamiltonian diffeomorphisms are time–1 maps of time dependent Hamil-tonian vector fields sgrad ζ t , i.e., vector fields that are symplectic gradients with ζ t ∈ C ∞ ( X, ω ) a smooth family, t ∈ [0 , Proof.
Let g (the “Lie algebra” of G ) consist of smooth vector fields V on X forwhich there is a smooth map ϕ : [0 , → G ⊂ Diff X such that ϕ (0) = id X and˙ ϕ (0) = dϕ ( t ) /dt | t =0 = V . This is a vector subspace of the space of all vector fields:for example, if ϕ, ψ realize vector fields V, W ∈ g , then ϕ ( t ) ◦ ψ ( t ) realizes V + W .In [S, pp. 512-513] Semmes essentially proved that g contains all Hamiltonian vectorfields. Essentially only, because the proof of his Lemma 4.1 is given only in Sobolev paces, not in C ∞ ( X ). At any rate, we will need a slightly stronger statement, to wit:If ζ : [ a, b ] → C ∞ ( X ) is smooth, then there is a smooth family(2.2) [ a, b ] × [0 , ∋ ( s, t ) ϕ st ∈ G ⊂ Diff X such that ϕ s = id X and ∂ t ϕ st | t =0 = sgrad ζ ( s ) for all s .To verify this, recall Semmes’ construction in [S, top of p. 512] that, given ξ, η ∈ C ∞ ( X ), shows that the Poisson bracket { ξ, η } ∈ C ∞ ( X ), determined by ω , has sym-plectic gradient in g . The same construction works with a parameter appended. Thus,if ξ, η : [ a, b ] → C ∞ ( X ) are smooth, there is a smooth family ϕ st ∈ G as in (2.2), ϕ s = id X and ∂ t ϕ st | t =0 = sgrad { ξ ( s ) , η ( s ) } . But any smooth ζ : [ a, b ] → C ∞ ( X ) suchthat R X ζ ( s ) ω n = 0 can be written(2.3) ζ ( s ) = m X j =1 { ξ j ( s ) , η j ( s ) } , m = 4 n + 1 , with suitable smooth ξ j , η j : [ a, b ] → C ∞ ( X ). In fact ξ j can be chosen constant,and arbitrary as long as ξ j ( s ) ≡ ξ j embed X into R m . The statement, without s –dependence, corresponds to [S, Lemma 4.1], but was already proved in [AG]. Atkinand Grabowski’s proof is easily modified to provide (2.3). The proof of [AG, (5.2)Theorem] depends on [AG, (2.6) Proposition], the s –dependent version of which saysthat if ξ j ∈ C ∞ ( X ), j = 1 , . . . , m , embed X into R m , then any smooth smooth family ψ s of smooth k -forms on X , s ∈ [ a, b ], can be written ψ s = X i ,i ,... f i ...i k ( s ) dξ i ∧ · · · ∧ dξ i k with f i ...i k : [ a, b ] → C ∞ ( X ) smooth. This is proved by an obvious cohomologyvanishing as in [AG]. Another ingredient of the proof of [AG, (5.2) Theorem], on p.325 there, in s –dependent version says that given a smooth family α s of exact smoothforms on X , there is a smooth family β s of smooth forms such that dβ s = α s . One wayto prove this is by Hodge theory, which gives that the unique solution β s of dβ s = α s that is othogonal to Ker d depends smoothly on s . The rest of the proof in [AG]manipulates identities, and changes not if a parameter s is appended. Thus (2.3) isproved.We can now construct ϕ st ∈ G as in (2.2). First, subtracting from ζ a smoothfunction c : [ a, b ] → R we obtain ζ ′ : [ a, b ] → C ∞ ( X ) with R X ζ ′ ( s ) ω m = 0. We find ξ j , η j as in (2.3), corresponding to ζ ′ rather than ζ , and then smooth maps ( s, t ) ϕ sjt ∈ G such that ϕ sj = id X and ∂ t ϕ sjt = { ξ j ( s ) , η j ( s ) } at t = 0. Since sgrad ζ =sgrad ζ ′ , the diffeomorphisms ϕ st = ϕ s t ◦ ϕ s t ◦ · · · ◦ ϕ smt have t –derivative sgrad ζ ( s ) at t = 0.After these preparations we are ready to prove the lemma. Suppose ϕ : ( X, ω ) → ( X, ω ) is a Hamiltonian diffeomorphism. This means it can be included in the flow ϕ s of Hamiltonian vector fields V s = sgrad ζ ( s ),(2.4) ∂ s ϕ s = V s ( ϕ s ) , ≤ s ≤ , ϕ = id X . ere ζ : [0 , → C ∞ ( X ) is smooth. The ϕ st constructed above for this ζ can be used asintegrators in a 1–step scheme to approximate the solution of the initial value problem(2.4). General theory gives that(2.5) ϕ ( k − /k /k ◦ ϕ ( k − /k /k ◦ · · · ◦ ϕ /k → ϕ in C ∞ ( X )as k → ∞ .(Details are as follows. Smoothly embed X in some R m and with p ∈ N , view ϕ s asan element of the Banach space B = C p ( X ) × · · · × C p ( X ), m copies of C p ( X ). Extend ϕ st : X → X to a smooth family of maps ψ st : R m → R m and extend V s to a vectorfield on R m by V s = ∂ t ψ st | t =0 . The error analysis of e.g. [HNW, p.160, Theorem 3.4],or more directly [An, Theorem 4.1, Corollary 4.2], gives that(2.6) ψ ( k − /k /k ◦ ψ ( k − /k /k ◦ · · · ◦ ψ /k ◦ ϕ → ϕ in B as k → ∞ . Both [HNW, An] work in finite dimensional Banach spaces, the latter in C m , but the same reasoning proves the result in any Banach space.)Since the left hand side of (2.6) is ϕ ( k − /k /k ◦ . . . ϕ /k ∈ G , we proved that ϕ isindeed in the closure of G . Proof of Theorem 2.2.
That invariance under strict rearrangements implies holonomyinvariance follows since parallel transport is realized by composition with a symplecto-morphism, and such compositions send functions to their strict rearrangements. Theconverse implication depends on Lemma 2.3. This implies that L ( ξ ) = L ( ξ ◦ ϕ ) if ξ ∈ T H and ϕ ∈ Diff(
X, ω ) is Hamiltonian. By [L4, Theorem 1.2], L | T H is thereforeinvariant under strict rearrangements. To complete the proof, take ξ ∈ T u H , η ∈ T v H such that ξ ∼ η . Parallel translate ξ, η to ξ ′ , η ′ ∈ T H along arbitrary smooth paths.Then ξ ′ ∼ ξ ∼ η ∼ η ′ , whence L ( ξ ) = L ( ξ ′ ) = L ( η ′ ) = L ( η ).In what follows, a fiberwise continuous and convex function L : T H → R that isinvariant under strict rearrangements will be called an invariant convex Lagrangian.The chief device to analyze their finer properties is the following representation theo-rem. We write B ( X ) or B ( X, µ )—when a Borel measure µ on X plays a role—for theBanach space of bounded Borel functions on X , with the supremum norm k k . Theorem 2.4.
Given an invariant convex Lagrangian L : T H → R , there are families A u ⊂ R × B ( X ) , u ∈ H such that for ξ ∈ T u H ≈ C ∞ ( X )(2.7) L ( ξ ) = sup ( a,f ) ∈A u a + Z X f ξω nu . A u can be chosen in R × C ∞ ( X ) , and have the property that whenever ( a, f ) ∈ A u and ϕ : ( X, ω v ) → ( X, ω u ) is a symplectomorphism, then ( a, f ◦ ϕ ) ∈ A v . Alternatively, A u can be chosen to be strict rearrangement invariant: if f ∈ B ( X, µ u ) and g ∈ B ( X, µ v ) are equidistributed, and ( a, f ) ∈ A u , then ( a, g ) ∈ A v .If L is also positively homogeneous, ( L ( cξ ) = L ( ξ ) whenever c ∈ (0 , ∞ )) , then inaddition A u can be chosen in { } × C ∞ ( X ) , respectively, in { } × B ( X ) . roof. Most of the proof was done in [L4]. Lemma 2.1 there produces A ⊂ R × C ∞ ( X )that satisfies (2.7) when u = 0. If we adjoin to A all pairs ( a, f ◦ ϕ ) with ( a, f ) ∈ A and ϕ : ( X, ω ) → ( X, ω ) a symplectomorphism, because of the invariance of L thesupremum in (2.7) is not going to change (for u = 0). So we can assume that A already is invariant under symplectomorphisms. We then define A u to consist of pairs( a, f ◦ ψ ) with ( a, f ) ∈ A and ψ : ( X, ω v ) → ( X, ω ) a symplectomorphism. This willdo, since if ξ ∈ T u H , with the above ψ sup ( a,g ) ∈A u a + Z X gξω nv = sup ( a,g ) ∈A u a + Z X ( g ◦ ψ − )( ξ ◦ ψ − ) ω n = sup ( a,f ) ∈A a + Z X ( ξ ◦ ψ − ) f ω n = L ( ξ ◦ ψ − ) = L ( ξ ) . Alternatively, we can modify the above A u to A ′ u consisting of all ( a, g ) ∈ R × B ( X )for which there is ( a, f ) ∈ A such that ( f, µ ) ∼ ( g, µ u ).This will not change thesupremum in (2.7), with A ′ u now, either. It suffices to check this for u = 0. By avariant of a lemma of Katok, [L4, Lemma 3.2], if ( f, µ ) ∼ ( g, µ ) then there is asequence ϕ k : ( X, ω ) → ( X, ω ) of symplectomorphisms such that R X | g − f ◦ ϕ k | ω n → Z X gξω n = lim k →∞ Z X ( f ◦ ϕ k ) ξω n = lim k →∞ Z X ( ξ ◦ ϕ − k ) f ω n , and so a + Z X gξω n ≤ lim k →∞ L ( ξ ◦ ϕ − k ) = L ( ξ ) . Thus replacing A u with A ′ u , (2.7) will still hold, and A ′ u is now strict rearrangementinvariant.Finally, if L is positively homogeneous, the statement of the theorem follows in thesame way from the corresponding part of [L4, Lemma 2.1].The Lagrangians in this section were required to be continuous on the fibers of T H .But, coupled with invariance, this implies continuity on T H : Theorem 2.5.
An invariant convex Lagrangian L : T H → R is a continuous functionon the Fr´echet manifold T H .Proof. Suppose u, u k ∈ H , ξ ∈ T u H , ξ k ∈ T u k H , and ξ k → ξ . This simply means thatas elements of C ∞ ( X ), u k → u and ξ k → ξ . Parallel translate ξ k to η k ∈ T u H alongthe straight line segment t tu + (1 − t ) u k . This is done by integrating the timedependent vector field (1/2) grad tu +(1 − t ) u k ( u k − u ) on X , for 0 ≤ t ≤
1. If the time–1map is ψ k : X → X , then η k = ξ k ◦ ψ k . Since ψ k → id X in the C ∞ topology, η k → ξ in C ∞ ( X ) ≈ T u H , as k → ∞ . Hence lim k L ( ξ k ) = lim k L ( η k ) = L ( ξ ), as claimed. As said in the Introduction, weak geodesics tend not to stay in the space H . Therefore,even in order to formulate a principle of least action we need to evaluate the action f a Lagragian along paths in spaces larger than H . In this section we will extendcertain invariant convex Lagrangians T H → R to a larger Banach bundle and describeproperties of the extended Lagrangians.We start by recalling definitions. Let Y be a complex manifold and Ω a smoothreal (1 ,
1) form on it, d Ω = 0. A function u : Y → [ −∞ , ∞ ) is Ω–plurisubharmonicif ρ + u is plurisubharmonic whenever ρ is a local potential of Ω, i.e., Ω = dd c ρ . Weuse the convention that ≡ −∞ is not plurisubharmonic, and write PSH(Ω) for the setof Ω–plurisubharmonic functions. Back to our K¨ahler manifold ( X, ω ), we denote by E ( ω ) the class of u ∈ PSH( ω ) with full Monge–Amp`ere mass, see [GZ1]. This classcontains all bounded ω –plurisubharmonic functions. The Monge–Amp`ere measure on X , corresponding to ω nu , will again be denoted µ u . This is a Borel measure on X , itscrucial property is µ u ( X ) = R X ω n . We endow E ( ω ) with the discrete topology, and let(3.1) T ∞ E ( ω ) = E ( ω ) × B ( X ) , a trivial Banach bundle with fibers the bounded Borel functions on X . Correspondingto usage in the subject we will not distinguish between elements ξ ∈ T ∞ u E ( ω ) and theirrepresentation ξ ∈ B ( X ) in the trivialization (3.1). The embedding C ∞ ( X ) ֒ → B ( X )induces an embedding H ֒ → T ∞ E ( ω ) of vector bundles, continuous if H is consideredwith the discrete topology. Definition 3.1.
Suppose u ∈ E ( ω ) and V ⊂ B ( X, µ u ) is a vector subspace. We saythat a function p : V → R is strongly continuous if p ( ξ k ) converges whenever ξ k ∈ V isa uniformly bounded sequence that converges µ u almost everywhere. In this case lim k p ( ξ k ) depends only on ξ = lim k ξ k , since another sequence η k → ξ can be combined with ξ k into one sequence. Theorem 3.2.
Any invariant convex Lagrangian L : T H → R that is strongly con-tinuous on the fibers T u H has a unique extension to L : T ∞ E ( ω ) → R that is strictrearrangement invariant, and strongly continuous on the fibers T ∞ u E ( ω ) . This exten-sion is fiberwise convex. For example, Darvas’s metrics in [D2], coming from a finite Young weight, cf. (1.2),are strongly continuous on the fibers.—The proof of the theorem will use the notion ofdecreasing rearrangement of measurable functions η : ( Y, ν ) → R on a measure space.This is a decreasing, upper semicontinuous function η ⋆ : [0 , ν ( Y )] → R , equidistributedwith η . Thus ν ( s ≤ η ≤ t ) is equal to the length of the maximal interval on which s ≤ η ⋆ ≤ t . The requirement of upper semicontinuity for the decreasing function η ⋆ translates to left continuity, which differs from the more usual right continuityrequirement, but the difference is of no consequence. In our setting(3.2) ν ( η ≥ η ⋆ ( s )) = s, and more generally,(3.3) ν ( η ≥ t ) ≤ τ implies η ⋆ ( τ ) ≤ t, ν ( η ≥ t ) ≥ τ implies η ⋆ ( τ ) ≥ t, roof of Theorem 3.2. By [L4, Theorem 5.2] L | T H : C ∞ ( X ) → R has a uniquestrongly continuous and strict rearrangement invariant extension q : B ( X ) → R ; thisextension is convex, and if uniformly bounded ξ k ∈ B ( X ) converge to ξ ∈ B ( X ) µ –a.e., then q ( ξ k ) → q ( ξ ). From q we obtain L by taking a ξ ∈ T ∞ u E ( ω ) ≃ B ( X, µ u ),finding an η ∈ B ( X, µ ) equidistributed with it, and letting L ( ξ ) = q ( η ). Such an η canbe found in the form η = ξ ∗ ◦ θ , where θ : ( X, µ ) [0 , µ ( X )] is a measure preservingmap (the target endowed with Lebesgue measure), see e.g. [L4, Lemma 5.5]. Thisis clearly the only strict rearrangement invariant way to extend q : T ∞ E ( ω ) → R to L : T ∞ E ( ω ) → R . It is immediate that L thus constructed has the properties claimedin the theorem.Further down we will not distinguish between an invariant convex Lagrangian T H → R that is strongly continuous on the fibers and its extension T ∞ E ( ω ) → R provided by Theorem 3.2, and will just refer to a strongly continuous, invariant, con-vex Lagrangian L : T ∞ E ( ω ) → R . Lemma 3.3.
A strongly continuous, invariant, convex Lagrangian L : T ∞ E ( ω ) → R is equi–Lipschitz continuous on bounded subsets of the fibers T u E ( ω ) in the sense thatgiven R ∈ (0 , ∞ ) there is an A ∈ (0 , ∞ ) such that for u ∈ E ( ω ) and ξ, η ∈ T ∞ u E ( ω )(3.4) if k ξ k , k η k < R then | L ( ξ ) − L ( η ) | ≤ A k ξ − η k . Proof.
According to [L4, Theorem 5.4] (3.4) holds when u = 0. The same A will workfor any u , for with a measure preserving θ : ( X, µ ) → [0 , µ ( X )] as in the proof ofTheorem 3.2 and ξ, η ∈ T ∞ u E ( ω ) | L ( ξ ) − L ( η ) | = | L ( ξ ∗ ◦ θ ) − L ( η ∗ ◦ θ ) | ≤ A sup | ξ ∗ ◦ θ − η ∗ ◦ θ | ≤ A k ξ − η k . Although we have endowed E ( ω ) with the discrete topology, we will need a conti-nuity property of Lagrangians T ∞ E ( ω ) → R stronger than fiberwise. This will involvethe notion of Monge–Amp`ere capacity cap of subsets of X [BT, K, GZ2]. Recall thata function ξ : X → R is quasicontinuous if for every ε > G ⊂ X of capacity cap ( G ) < ε such that ξ | X \ G is continuous; and a sequence of functions ξ j : X → R converges to ξ : X → R in capacity if lim j →∞ cap( | ξ j − ξ | > δ ) = 0 forevery δ >
0. In particular, a uniformly convergent sequence converges in capacity.
Lemma 3.4.
Let L : T ∞ E ( ω ) → R be strongly continuous, invariant, and convex.Suppose u k ∈ E ( ω ) either decrease, or uniformly converge, to a bounded u ∈ E ( ω ) as k → ∞ , and ξ k ∈ T ∞ u k E ( ω ) ≈ B ( X ) converge in capacity to ξ ∈ T ∞ u E ( ω ) ≈ B ( X ) . If ξ is quasicontinuous, then lim k L ( ξ k ) = L ( ξ ) .Proof. Upon adding a constant to the u k and scaling u, u k , ω , and L , we can arrangethat 0 ≤ u, u k ≤
1. Suppose first the u k decrease. The point of the proof is to show thatthe decreasing rearrangements ξ ⋆k of ξ k ∈ B ( X, µ u k ) converge to the rearrangement ξ ⋆ of ξ ∈ B ( X, µ ), away from a countable subset of [0 , µ ( X )]. Define decreasing functions f, g on [0 , µ ( X )] f = lim inf k ξ ⋆k ≤ lim sup k ξ ⋆k = g. et s ∈ (0 , µ ( X )), S = ξ ⋆ ( s ), and ε >
0. With A k = {| ξ k − ξ | ≥ ε } , k ∈ N , { ξ k ≥ S + ε } ⊂ { ξ ≥ S } ∪ A k and { ξ > S − ε } ⊂ { ξ k ≥ S − ε } ∪ A k . For j ∈ N define continuous functions F j , G j : R → [0 , F j ( t ) = t ≤ S − /j t ≥ S linear in between , G j ( t ) = t ≤ S − ε t ≥ S − ε + 1 /j linear in between.Note that F j decreases, G j decreases with increasing j . We can estimate(3.5) µ u k ( ξ k ≥ S + ε ) ≤ µ u k ( ξ ≥ S ) + µ u k ( A k ) ≤ Z X F j ◦ ξ dµ u k + cap ( A k ) µ u k ( ξ k ≥ S − ε ) ≥ µ u k ( ξ > S − ε ) − µ u k ( A k ) ≥ Z X G j ◦ ξ dµ u k − cap ( A k ) . Since F j ◦ ξ , G j ◦ ξ are quasicontinuous, by [GZ2, Theorem 4.2.6]lim k →∞ Z X F j ◦ ξ dµ u k = Z X F j ◦ ξ dµ u , lim k →∞ Z X G j ◦ ξ dµ u k = Z X G j ◦ ξ dµ u . Therefore, letting first k → ∞ in (3.5), then j → ∞ , and using the monotone conver-gence theorem as well,lim sup k →∞ µ u k ( ξ k ≥ S + ε ) ≤ µ u ( ξ ≥ S )lim inf k →∞ µ u k ( ξ k ≥ S − ε ) ≥ µ u ( ξ > S − ε ) ≥ µ u ( ξ ≥ S ) . Now µ u ( ξ ≥ S ) = s , see (3.2). Hence, given σ < s < ρ , for sufficiently large kµ u k ( ξ k ≥ ξ ⋆ ( s ) + ε ) < ρ = µ u ( ξ ≥ ξ ∗ ( ρ )) ,µ u ( ξ ≥ ξ ⋆ ( σ )) = σ < µ u k ( ξ k ≥ ξ ∗ ( s ) − ε ) . We apply (3.3) with ν = µ u k , µ u , t = ξ ⋆ ( s ) + ε , ξ ⋆ ( s ) − ε , and τ = ρ, σ to conclude ξ ⋆k ( ρ ) ≤ ξ ⋆ ( s ) + ε and ξ ⋆k ( σ ) ≥ ξ ⋆ ( s ) − ε . In the limit k → ∞ g ( ρ ) ≤ ξ ⋆ ( s ) + ε and f ( σ ) ≥ ξ ⋆ ( s ) − ε, for all σ < s < ρ. If f, g are continuous at s —which occurs apart from countably many values—, g ( s ) ≤ ξ ⋆ ( s ) ≤ f ( s ) ≤ g ( s ) follows, i.e., lim k ξ ⋆k ( s ) = ξ ⋆ ( s ) as claimed.It is now easy to finish the proof. With a measure preserving θ : ( X, µ ) → [0 , µ ( X )], as in the proof of Theorem 3.2, ξ k ∈ B ( X, µ u k ) and ξ ⋆k ◦ θ ∈ B ( X, µ )are equidistributed, and ξ ⋆k ◦ θ → ξ ⋆ ◦ θ µ –almost everywhere. Hence by Theorem 3.2lim k L ( ξ k ) = lim k L ( ξ ⋆k ◦ θ ) = L ( ξ ⋆ ◦ θ ) = L ( ξ ) . We are done if u k are known to decrease.Now suppose that u k converge uniformly. It suffices to prove that a subsequenceof L ( ξ k ) converges to L ( ξ ), and for this reason we can assume that k u k − u k − k < − k for k = 2 , , . . . . Then the sequence v k = u k + 2 − k decreases to u , and µ u k = µ v k . Wecan view ξ k ∈ T ∞ u k E ( ω ) ≈ B ( X, µ u k ) as elements ξ ′ k ∈ T ∞ v k E ( ω ) ≈ B ( X, µ v k ), which arestrict rearrangements of ξ k . Hence L ( ξ k ) = L ( ξ ′ k ) → L ( ξ ) by the first part of the proof. Weak geodesics, ε –geodesics, Jacobi fields If a < b are real numbers, we let S ab = { s ∈ C : a < Re s < b } , and denote by π the projection S ab × X → X .Following Berndtsson and Darvas [Be1, D4, section 3.3] we make the followingdefinition. Definition 4.1.
A path u : ( a, b ) → PSH ( ω ) is a subgeodesic if the function U : S ab × X → [ −∞ , ∞ ) given by U ( s, x ) = u ( Re s )( x ) is π ⋆ ω –plurisubharmonic.If u a , u b ∈ PSH ( ω ) , the weak geodesic determined by (or connecting) u a , u b is u :( a, b ) → PSH ( ω ) , (4.1) u = sup { v | v : ( a, b ) → PSH ( ω ) is subgeodesic, lim t → a v ≤ u a , lim t → b v ≤ u b } . The limits are understood pointwise on X ; they exist because π ∗ ω –plurisubhar-monicity implies that for each x ∈ X the function v ( · )( x ) is convex. It is possible that(4.1) gives u ≡ −∞ , not valued in PSH( ω ); but otherwise the weak geodesic is indeeda path in PSH( ω ) and is itself a subgeodesic [D4, section 3.1]. Darvas points out thatin general the term “connecting” weak geodesic is misleading, as lim a u may have littleto do with u a . But, if u a , u b are bounded, Berndtsson proves by a simple argumentthat the weak geodesic indeed connects, lim a u = u a , lim b u = u b , uniformly on X ,[Be1, pp. 156-157]. If c < d , the weak geodesic u ′ : ( c, d ) → PSH( ω ) between u a and u b is u , composed with an affine reparametrization, because affine reparametrizationsof subgeodesics yield subgeodesics.In what follows we will only deal with weak geodesics u determined by bounded u a , u b . Such a u is a Lipschitz map into B ( X ), and we will refer to its continuousextension to the closed interval [ a, b ] as a weak geodesic, too.In (1.1) we defined Mabuchi’s connection on T H through its parallel transport. Amore direct definition takes a smooth path u : [ a, b ] → H and a smooth vector field ξ : [ a, b ] → T H , ξ ( t ) ∈ T u ( t ) H , along it; the covariant derivative of ξ along u is thenthe vector field ∇ t ξ given by(4.2) ∇ t ξ ( t ) = ˙ ξ ( t ) −
12 ( d X ˙ u ( t ) , d X ξ ( t )) u ( t )) ∈ C ∞ ( X ) ≈ T u ( t ) H . Here d X is differential on X , for fixed t , and ( , ) u ( t ) is inner product on T ∗ X inducedby the K¨ahler metric of ω u ( t ) . In (4.2) the left hand side is to be computed for ξ asection of T H along u ; on the right ξ stands for the representation of this section inthe canonical trivialization T H ≈ H × C ∞ ( X ), so for a function [ a, b ] → C ∞ ( X ); andthe equality of the two sides again uses the trivialization of T H .Geodesics u : [ a, b ] → H of ∇ satisfy ∇ t ˙ u ( t ) = 0. Chen, however, had the idea thatthe geometry of H can be better accessed through ε –geodesics. Define a vector field F on H by(4.3) F ( v ) ω nv = ω n , v ∈ H , F ( v ) ∈ C ∞ ( X ) ≈ T v H . f ε >
0, an ε –geodesic is a solution u : [ a, b ] → H of(4.4) ∇ t ˙ u ( t ) = εF ( u ( t )) , t ∈ [ a, b ] . In what follows, we will just write d for d X . Chen proves [C, Bl1, Bl2] Theorem 4.2.
Given a < b and two potentials u a , u b ∈ H , (4.4) has a unique C solution u = u ε : [ a, b ] → H satisfying u ( a ) = u a , u ( b ) = u b . The solution u is smooth,and as an element of C ∞ ([ a, b ] → H ) , it depends smoothly on u a , u b (and a, b, ε ).Finally, if u a , u b are in a fixed compact subset of H , the forms dd c u ε ( t ) , d ˙ u ε ( t ) and ¨ u ε ( t ) are uniformly bounded on X for < ε < ε and a ≤ t ≤ b . It follows by the Arzel`a–Ascoli theorem and a maximum principle for the Monge–Amp`ere operator that for fixed u a , u b the uniform limit u = lim ε → u ε exists. The limit u maps into the space H = { w ∈ C ( X ) ∩ PSH( ω ) : the current dd c w is bounded } , and the currents dd c u ( t ) , d ˙ u ( t ) , ¨ u ( t ) are represented by uniformly bounded forms. Also, u is the weak geodesic in the sense of Definition 4.1 to connect u a , u b .Consider an ε –geodesic u : [ a, b ] → H . Definition 4.3.
A vector field ξ : [ a, b ] → T H along u is an ε –Jacobi field if there arean interval I containing ∈ R and a smooth family I ∋ s u s , each u s : [ a, b ] → H an ε –geodesic such that u = u and ξ = ∂ s u s | s =0 . Lemma 4.4. (a) If I ⊂ R is an interval and v : I → H a smooth path, then thecovariant derivative of F along v satisfies (4.5) ω nv ( s ) ∇ s F ( v ( s )) = − nd (cid:0) F ( v ( s )) d c ˙ v ( s ) (cid:1) ∧ ω n − v ( s ) . (b) If ξ : [ a, b ] → T H is an ε –Jacobi field along an ε –geodesic u : [ a, b ] → H , then (4.6) ω nu ( t ) ∇ t ξ ( t ) = 14 {{ ˙ u ( t ) , ξ ( t ) } , ˙ u ( t ) } ω nu ( t ) − εnd (cid:0) F ( u ( t )) d c ξ ( t ) (cid:1) ∧ ω n − u ( t ) , where { , } = { , } u ( t ) is Poisson bracket on T u ( t ) H ≈ C ∞ ( X ) for the symplectic form ω u ( t ) . Calabi and Chen [CC, Section 2.3] derive an equivalent equation for ε –Jacobi fields. Proof. (a) We will apply (4.2) with ξ = F ◦ v . Differentiating F ( v ( s )) ω nv ( s ) = ω n withrespect to s gives ω nv ( s ) ∂ s F ( v ( s )) = − F ( v ( s )) ∂ s ( ω + dd c v ( s )) n = − nF ( v ( s )) dd c ˙ v ( s ) ∧ ω n − v ( s ) . At the same time (cid:0) d ˙ v ( s ) , dF ( v ( s )) (cid:1) v ( s ) ω nv ( s ) = 2 ndF ( v ( s )) ∧ d c ˙ v ( s ) ∧ ω n − v ( s ) , see e.g. [Bl2, p.103]. Substituting into (4.2) now gives (4.5). b) Let u s : [ a, b ] → H be a smooth family of ε –geodesics such that ξ = ∂ s u s | s =0 ,and set U ( s, t ) = u s ( t ). As Mabuchi’s connection is torsion free, ∇ s ∂ t U = ∇ t ∂ s U .The curvature of ∇ , evaluated on ∂ s U ( s, t ) , ∂ t U ( s, t ) ∈ T U ( s,t ) H is an endomorphismof T U ( s,t ) H that acts on a vector field η ( s, t ) by R ( ∂ s U, ∂ t U ) η = ( ∇ s ∇ t − ∇ t ∇ s ) η = {{ ∂ s U, ∂ t U } , η } / , see [M, Theorem 4.3]. (Mabuchi’s formula does not contain the factor 1 /
4, due todifferent conventions.)We apply ∇ s to the ε –geodesic equation ∇ t ∂ t U ( s, t ) = εF ( U ( s, t )), to obtain at s = 0 ε ∇ s F ( U ) = ∇ s ∇ t ∂ t U = R ( ∂ s U, ∂ t U ) ∂ t U + ∇ t ∇ s ∂ t U = (1 / {{ ∂ s U, ∂ t U } , ∂ t U } + ∇ t ∇ t ∂ s U = (1 / {{ ξ, ˙ u } , ˙ u } + ∇ t ξ. Combining (4.5) with this, (4.6) follows.
Consider an invariant convex Lagrangian T H → R . If u : [ a, b ] → H is a piecewise C path, its action is(5.1) L ( u ) = Z ba L ( ˙ u ( t )) dt. Depending on the nature of L , this can represent length or energy of a path, butin general it is neither. No mather what L , the integral (5.1) is that of a piecewisecontinuous function by Theorem 2.5, so that it exists as a Riemann integral.For the purposes of this paper we must consider action for paths beyond H . Thematerial developed in section 3 allows to define action for paths in the space B ( X ) ∩ PSH( ω ) ⊂ E ( ω ) of continuous ω –plurisubharmonic functions. This is a subset of theBanach space B ( X ) and, viewing maps into it as maps into B ( X ), we can talk aboutvarious regularity classes of such maps. If a < b are real, the following is easy to check. Lemma 5.1.
A map u : [ a, b ] → B ( X ) is continuous if and only if the functions (5.2) u ( · )( x ) , for x ∈ X are equicontinuous, and it is C k for k = 1 , , . . . if and only if the functions in (5.2)are k times differentiable, and the k ’th derivatives are also equicontinuous. According to Theorem 3.2, an invariant convex Lagrangian T H → R that is stronglycontinuous on the fibers determines a strongly continuous, invariant, convex Lagrangian L : T ∞ E ( ω ) → R . Suppose u : [ a, b ] → B ( X ) ∩ PSH( ω ) is a C path. Since ω –plurisubharmonic functions are quasicontinuous [BT, Theorem 3.5], [GZ2, Corollary9.12], the difference quotients ( u ( t ) − u ( s )) / ( t − s ) are quasicontinuous and so are theiruniform limits ˙ u ( t ). Hence by Lemma 3.4 L ◦ ˙ u : [ a, b ] → R is continuous. Clearly if is just piecewise C , the integral in (5.1) still exists as the integral of a piecewisecontinuous function, and defines action L ( u ).If w, w ′ ∈ B ( X ) ∩ PSH( ω ) and T ∈ (0 , ∞ ), we define the least action, or just action, L T ( w, w ′ ) between w, w ′ as(5.3) L T ( w, w ′ ) = inf u L ( u ) , the infimum taken over all piecewise C paths u : [0 , T ] → B ( X ) ∩ PSH( ω ) such that u (0) = w , u ( T ) = w ′ . Note that w, w ′ can be connected by a smooth path, e.g. u ( t ) = (1 − t/T ) w + ( t/T ) w ′ connects. We will see that L T ( w, w ′ ) > −∞ (Lemma 9.3).Instead of [0 , T ] if we minimize over paths [ a, a + T ] → B ( X ) ∩ PSH( ω ), the infimumin (5.3) does not change. However, in general L T ( w, w ′ ) will depend on T ; it will notif L is positively homogeneous. In general L T ( w, w ′ ) + L S ( w ′ , w ′′ ) ≥ L T + S ( w, w ′′ )follows by concatenating paths. Of course, L T ( w, w ′ ) = L T ( w ′ , w ) should be expectedonly if L is even, L ( − ξ ) = L ( ξ ).In our two main results below, L : T ∞ E ( ω ) → R is a strongly continuous, invariant,convex Lagragian. Theorem 5.2 (Principle of least action) . If a C path v : [0 , T ] → B ( X ) ∩ PSH ( ω ) isa weak geodesic, then L ( v ) = L T ( v (0) , v ( T )) . Theorem 5.3. If u, v : [ a, b ] → B ( X ) ∩ PSH ( ω ) are weak geodesics, then for S > the function [ a, b ] ∋ t
7→ L S ( u ( t ) , v ( t )) ∈ R is convex. If L is absolutely homogeneous, L ( cξ ) = | c | L ( ξ ), and vanishes only on zero vectors,then action L S is distance measured in a Finsler metric and is independent of S ; thestatement of Theorem 5.3 is an indication of seminegative curvature.The proofs will take the rest of the paper. First we prove them in approximateversions, with ε –geodesics in H replacing weak geodesics. The approximate versionsdepend on two facts. First, that L is convex along ε –Jacobi fields; second, as a con-sequence, for some Lagrangians a triangle inequality holds for triangles in H with twosides ε –geodesics (Theorem 6.1, Lemma 7.2). It is a technical point but noteworthythat the approximate results contain no error term, no O ( ε ). By letting ε → H (Corollary 7.4). Approximating weak geodesicsin B ( X ) ∩ PSH( ω ) by weak geodesics in H we obtain the same in B ( X ) ∩ PSH( ω ) insection 8. Theorem 5.3 is proved by the same approximation scheme in section 9. ε –Jacobi fields In this section we stay in H , and consider invariant convex Lagrangians L : T H → R ,strongly continuous or not. Recall that given ε >
0, an ε –geodesic u : [ a, b ] → H satisfies the equation(6.1) ∇ t ˙ u ( t ) = εF ( u ( t )) , here the vector field F : H → T H is defined by F ( v ) ω nv = ω n . Infinitesimal variationsof ε –geodesics are ε –Jacobi fields. If ξ : [ a, b ] → T H is an ε –Jacobi field along an ε –geodesic u : [ a, b ] → T H , by Lemma 4.4(b)(6.2) ω nu ( t ) ∇ t ξ ( t ) = 14 {{ ˙ u ( t ) , ξ ( t ) } , ˙ u ( t ) } ω nu ( t ) − εnd (cid:0) F ( u ( t )) d c ξ ( t ) (cid:1) ∧ ω n − u ( t ) . All our subsequent results rest on the following theorem.
Theorem 6.1. If ξ : [ a, b ] → T H is an ε –Jacobi field along an ε –geodesic u : [ a, b ] → H ,then L ◦ ξ is a convex function on [ a, b ] . This will be derived from a special case.
Lemma 6.2.
Given u ∈ H and f ∈ B ( X ) , Theorem 6.1 holds for the Lagrangian (6.3) L ( η ) = sup ( f,v ) ∼ ( f ,u ) Z X f ηdµ v , η ∈ T v H , cf. (2.1). To prove Lemma 6.2 we need some preparation. If Y is any set, we say thatfunctions g, h : Y → R are similarly ordered if ( g ( x ) − g ( y ))( h ( x ) − h ( y )) ≥ x, y ∈ Y . Equivalently, g ( x ) < g ( y ) should imply h ( x ) ≤ h ( y ). The relation is nottransitive, any function is similarly ordered as a constant. Lemma 6.3.
Let Y be a smooth manifold and a ij smooth functions, V i smooth vectorfields on it, i, j = 1 , . . . , k . Assume the matrix ( a ij ) is symmetric and positive semidef-inite everywhere. If g ∈ C ∞ ( Y ) and a locally integrable h : Y → R are similarlyordered, then the current Q = P i,j a ij ( V i g )( V j h ) ≥ .Proof. Assume first that there is a smooth increasing H : R → R such that h = H ◦ g .Then Q = H ′ ( g ) P a ij ( V i g )( V j g ) ≥
0. The same follows if H is any increasing function,by writing it as lim p lim q H pq (pointwise limit), with locally uniformly bounded smoothincreasing H pq .Now consider general g, h . Let I denote the range of g , and for t ∈ I define m ( t ) = inf { h ( x ) : g ( x ) = t } , M ( t ) = sup { h ( x ) : g ( x ) = t } . If g ( x ) = t < g ( y ) = τ , then h ( x ) ≤ h ( y ), which means that m ( t ) ≤ M ( t ) ≤ m ( τ ) ≤ M ( τ ) when t ≤ τ. In particular, m and M are increasing functions, and coincide on int T wherever oneof them is continuous, that is, apart from a countable set T ⊂ I . On g − ( I \ T ) wehave h = m ◦ g . If t is a regular value of g , then g − ( t ) has measure 0. Hence on theregular set of g the functions h and m ◦ g agree a.e., and the induced currents simplyagree there. By what we already proved, Q ≥ dg = 0. We still needto understand what happens on the critical set C = ( dg = 0).Let χ : [0 , ∞ ) → [0 ,
1] be a smooth function, χ ( t ) = 0 if t ≤ χ ( t ) = 1 if t ≥ Y with a Riemannian metric and denote by dist( · , C ) distance to C ; this is a ipschitz function with Lipschitz constant 1. For s >
0, the function χ ( s dist( · , C ))has Lipschitz constant O ( s ); it vanishes in the 1 /s neighborhood of C and equals 1outside the 2 /s neighborhood. Let ρ s ∈ C ∞ ( Y ) have the same properties. To provethe lemma we need to show that if θ ≥ Y , then 0 ≤ Z Y Qθ = − Z Y h X i,j £ j ( θa ij V i g ) , where £ j stands for Lie derivative along V j .The inequality holds if θ is replaced by θρ s , because Q ≥ θρ s . The point is that the functions £ j ( θρ s a ij V i g ) are uniformly bounded andtend to £ j ( θa ij V i g ) pointwise as s → ∞ . The latter is obvious; the former is verifiedby applying Leibniz rule to the products, and checking each term. The only term thatneeds speaking for is θa ij ( V i g )( V j ρ s ). But since | V i g | | V j ρ s | attains its maximum on { y ∈ Y : 1 /s ≤ dist( y, C ) ≤ /s } , this maximum is O (1 /s ) O ( s ) = O (1). Therefore bydominated convergence Z Y Qθ = − lim s →∞ Z Y h X i,j £ j ( θρ s a ij V i g ) = lim s →∞ Z Y Qθρ s ≥ . Proof of Lemma 6.2.
The plan is to construct for every t ∈ ( a, b ) a family f ( t ) ∈ B ( X )such that ( f ( t ) , u ( t )) ∼ ( f , u ) and A ( t ) = R X f ( t ) ξ ( t ) dµ u ( t ) ≤ L ( ξ ( t )) satisfies A ( t ) = L ( ξ ( t )) , ¨ A ( t ) ≥ . To simplify notation we can assume t = 0. At the price of replacing f by f suchthat ( f , u ) ∼ ( f , u (0)), we can assume u (0) = u . Further to simplify we can arrangethat f = f realizes sup ( f,u (0)) ∼ ( f ,u (0)) Z X f ξ (0) dµ u (0) ;this is possible simply because the supremum is attained, see e.g., [L4, Lemma 6.2].The same lemma says that there is a maximizing f that is similarly ordered as ξ (0),and accordingly we will work with f similarly ordered as ξ (0).For a moment suppose u : [ a, b ] → H is an arbitrary smooth path, and paralleltransport T u (0) H → T u ( t ) H along u is given by pull back by a symplectomorphism ϕ ( t ) : ( X, ω u ( t ) ) → ( X, ω u (0) ). Suppose η : [ a, b ] → T u (0) H is smooth; then t η ( t ) ◦ ϕ ( t ) defines a vector field along u . Parallel transport intertwines differentiationand covariant differentiation: ∇ t ( η ( t ) ◦ ϕ ( t )) = ˙ η ( t ) ◦ ϕ ( t ) and ∇ t ( η ( t ) ◦ ϕ ( t )) = ¨ η ( t ) ◦ ϕ ( t ) . When u is an ε –geodesic and ξ an ε –Jacobi field along it, as in the lemma, choose η so that η ( t ) ◦ ϕ ( t ) = ξ ( t ). By (6.2), at t = 0,(6.4) ¨ η (0) ω nu (0) = (1 / {{ ˙ u (0) , η (0) } , ˙ u (0) } ω nu (0) − εnd (cid:0) F ( u (0)) d c η (0) (cid:1) ∧ ω n − u (0) . With f ( t ) = f ◦ ϕ ( t ) we let A ( t ) = Z X f ( t ) ξ ( t ) ω nu ( t ) = Z X f η ( t ) ω nu (0) , hen ¨ A ( t ) = R X f ¨ η ( t ) ω nu (0) . In view of (6.4)¨ A (0) = 14 Z X f {{ ˙ u (0) , η (0) } , ˙ u (0) } ω nu (0) − εn Z X f d (cid:0) F ( u (0)) d c η (0) (cid:1) ∧ ω n − u (0) = 14 Z X { ˙ u (0) , f }{ ˙ u (0) , η (0) } ω nu (0) + εn Z X F ( u (0)) df ∧ d c η (0) ∧ ω n − u (0) . In the last line { ˙ u (0) , f } and df are currents. By Lemma 6.3 the first integrand inthis last line is ≥
0, since f and η (0) = ξ (0) are similarly ordered; and so is, for thesame reason, 2 ndf ∧ d c η (0) ∧ ω n − u (0) = ( df , dη (0)) u (0) ω nu (0) , cf. [Bl2, p.103].To summarize, we have shown that for every t ∈ ( a, b ) there is a function A ∈ C ∞ [ a, b ] such that A ( t ) ≤ L ( ξ ( t )) , with equality when t = t , and ¨ A ( t ) ≥ . By a standard argument this implies that L ◦ ξ is convex. First one notes that if p > q ∈ R , the function L ( ξ ( t )) + pt + qt cannot have a local maximum atany t ∈ ( a, b ), because with the A we have constructed A ( t ) + pt + qt has no localmaximum at t . It follows that on any subinterval [ α, β ] ⊂ [ a, b ], L ( ξ ( t )) + pt + qt attains its maximum at the endpoints, whence L ( ξ ( t )) + pt is convex. Letting p → L ◦ ξ itself is also convex. Proof of Theorem 6.1.
Clearly, Lemma 6.2 implies that if a ∈ R , g ∈ B ( X ), and L a,g ( η ) = a + sup ( f,v ) ∼ ( g,u ) Z X f ηdµ v , η ∈ T v H , then L a,g ◦ ξ is convex for any ε –Jacobi field. Since by Theorem 2.4 a general invariantconvex Lagrangian is the supremum of a family of such L a,g , the theorem follows. H and H In this section unless otherwise indicated the Lagrangian L is strongly continuous,invariant, and convex on T ∞ E ( ω ). We will compare the actions along weak geodesicsin H and along general paths in H . Theorem 7.1.
Consider a piecewise C path u : [0 , T ] → H and a weak geodesic v : [0 , T ] → H . If u (0) = v (0) and u ( T ) = v ( T ) , then (7.1) 1 T Z T L ◦ ˙ u ≥ L ( ˙ v (0)) . First we prove a variant.
Lemma 7.2.
Suppose an invariant convex Lagrangian L : T H → R is positivelyhomogeneous, L ( cξ ) = cL ( ξ ) if c > . Consider a triangle in H formed by a piecewise C path u : [ a, b ] → H and ε –geodesics v a , v b : [0 , T ] → H ; so that v a (0) = v b (0) and v a ( T ) = u ( a ) , v b ( T ) = u ( b ) . Then (7.2) 1 T Z ba L ◦ ˙ u ≥ L ( ˙ v b (0)) − L ( ˙ v a (0)) . ote that positive homogeneity implies the triangle inequality L ( ξ + η ) ≤ L ( ξ )+ L ( η )for w ∈ H and ξ, η ∈ T w H . Proof.
Because of the additive nature of (7.2), we can assume u is C , not only piece-wise, and then by simple approximation that it is even C ∞ . For each s ∈ [ a, b ] let U ( s, · ) : [0 , T ] → H denote the ε –geodesic connecting v a (0) = v b (0) with u ( s ). Ac-cording to Theorem 4.2, that is, by Chen’s work, there is a unique such geodesic, and U ∈ C ∞ ([ a, b ] × [0 , T ]). Thus ξ s = ∂ s U ( s, · ) is an ε –Jacobi field and ξ s (0) = 0. By The-orem 6.1 L ◦ ξ s is convex on [0 , T ]. Using ∂ t (and later, dot) to denote right derivative,therefore L ( ξ s ( T )) ≥ L ( ξ s (0)) + T ∂ t | t =0 L ( ξ s ( t )) . By homogeneity, the first term on the right is 0. To compute the second, let η ( t ) ∈ T U (0 , H denote the parallel translate of ξ s ( t ) ∈ T U ( s,t ) H along U ( s, · ). Thuslim t → L ( ξ s ( t )) /t = lim t → L ( η ( t )) /t = lim t → L ( η ( t ) /t )= L ( ∇ t | t =0 ξ s ( t )) = L ( ∇ t | t =0 ∂ s U ( s, t )) = L ( ∂ s ∂ t | t =0 U ( s, t )) . The last equality is because ∇ has no torsion, and U ( s,
0) is constant. Hence, usingJensen’s inequality as well,1 T Z ba L ( ∂ s u ( s )) ds = 1 T Z ba L ( ξ s ( T )) ds ≥ Z ba L ( ∂ s ∂ t | t =0 U ( s, t )) ds ≥ L (cid:16) Z ba ∂ s ∂ t | t =0 U ( s, t ) ds (cid:17) = L ( ˙ v b (0) − ˙ v a (0)) ≥ L ( ˙ v b (0)) − L ( ˙ v a (0)) , as claimed. Proof of Theorem 7.1.
For ε > v ε : [0 , T ] → H denote the ε –geodesic connecting u (0) and u ( T ). Again by Chen [C], see also B locki [Bl1, Bl2], v ε → v in such a way that˙ v ε (0) → ˙ v (0) in T ∞ v (0) E ( ω ) as ε →
0. Suppose first that L is positively homogeneous,and apply Lemma 7.2 with [ a, b ] = [0 , T ], v a ≡ u (0), v b = v ε . We obtain1 T Z T L ◦ ˙ u ≥ L ( ˙ v ε (0)) . Hence (7.1) follows by letting ε → L is not positively homogeneous but L plusa constant is. Since a general L is the supremum of Lagrangians of form positivelyhomogeneous plus constant, see Theorem 2.4, (7.1) holds in complete generality. Lemma 7.3. If v : [0 , T ] → H is a weak geodesic, then L ◦ ˙ v is constant. Hence L ( ˙ v (0)) = 1 T Z T L ◦ ˙ v. This can be seen as an instance of Noether’s theorem on conserved quantities, albeitin an unusual setting. roof. Berndtsson [Be2, Proposition 2.2] discovered that ˙ v ( t ) ∈ B ( X, µ v ( t ) ) are equidis-tributed for all t , although he worked with integral K¨ahler classes [ ω ] only. At any rate,[D2, Lemma 4.6] implies the general result. Since L is invariant, the lemma follows.Together with Theorem 7.1 this almost proves the principle of least action in H : Corollary 7.4. If u : [0 , T ] → H is a piecewise C path and v : [0 , T ] → H is a weakgeodesic between the same endpoints, then R T L ◦ ˙ u ≥ R T L ◦ ˙ v . B ( X ) ∩ PSH( ω ) Here we will extend Corollary 7.4 to u, v taking values in B ( X ) ∩ PSH( ω ) (Theorem5.2). In this section L : T ∞ E ( ω ) → R is assumed to be strongly continuous, invariant,and convex. Theorem 8.1.
Suppose u, v : [0 , T ] → B ( X ) ∩ PSH ( ω ) have the same endpoints: u (0) = v (0) , u ( T ) = v ( T ) . If u is piecewise C and v is a C weak geodesic, then R T L ◦ ˙ u ≥ R T L ◦ ˙ v. This will be derived from Corollary 7.4 by approximation.
Lemma 8.2.
Suppose u : [0 , T ] → B ( X ) ∩ PSH ( ω ) is a piecewise C path, and w j , w ′ j ∈H decrease to u (0) , respectively, u ( T ) , as j → ∞ . Then there are a sequence J ⊂ N and for j ∈ J piecewise linear u j : [0 , T ] → H such that u j (0) = w j , u j ( T ) = w ′ j , and R T L ◦ ˙ u j → R T L ◦ ˙ u as J ∋ j → ∞ . As said, at points where u, u j are not differentiable, ˙ u, ˙ u j mean right derivatives. Proof.
Choose t = 0 < t < · · · < t p = T so that u is C on each [ t i − , t i ]. Supposefirst that u is even linear on [ t i − , t i ]. In this case J will be all of N . A simple specialcase of regularization, see [De, DP] and especially [BK], provides z ij ∈ H such that z ij decreases to u ( t i ) as j → ∞ for i = 0 , . . . , p . We take z j = w j , and z pj = w ′ j , andarrange that the z ij are uniformly bounded. Linearly interpolating on [ t i − , t i ] between z i − ,j and z ij we obtain the functions u j sought. Indeed, u j ( t ) decreases to u ( t ), and˙ u j ( t ) = z ij − z i − ,j t i − t i − ∈ T u j ( t ) H , when t ∈ [ t i − , t i ] , are uniformly bounded and tend to ˙ u ( t ) as j → ∞ . Since ω –plurisubharmonic func-tions are quasicontinuous [GZ2, Corollary 9.12], so are the difference quotients ˙ u ( t ).According to [GZ2, Proposition 9.11] this implies convergence in capacity, and solim j L ( ˙ u j ( t )) = L ( ˙ u ( t )) by Lemma 3.4. Since ˙ u j ( t ) are uniformly bounded, so are L ( ˙ u j ( t )) by equi–Lipschitz continuity, Lemma 3.3. The dominated convergence theo-rem gives therefore lim j R T L ◦ ˙ u j = R T L ◦ ˙ u .For general u , partition each [ t i − , t i ] into k equal parts. Construct v k : [0 , T ] → B ( X ) ∩ PSH( ω ) that agrees with u at each partition point, and is linear in between.Then v k → u and ˙ v k → ˙ u uniformly, because ˙ u is uniformly continuous on [ t i − , t i ).Hence L ◦ ˙ v k → L ◦ ˙ u by Lemma 3.4 and, again by dominated convergence, R T L ◦ ˙ v k → T L ◦ ˙ u . By what we have already proved, for each k we can find j = j k > j k − andpiecewise linear u j : [0 , T ] → H such that u j (0) = w j , u j ( T ) = w ′ j , and (cid:12)(cid:12)(cid:12) Z T L ◦ ˙ u j − Z T L ◦ ˙ v k (cid:12)(cid:12)(cid:12) < k . Thus J = { j , j , . . . } will do. Lemma 8.3.
Let v, v j : [ a, b ] → PSH ( ω ) be weak geodesics. If v j ( t ) decreases to v ( t ) when t = a, b , then v j ( t ) decreases to v ( t ) for all t ∈ [ a, b ] . This is [D4, Proposition 3.15].—There is one more ingredient that goes into theproof of Theorem 8.1.
Lemma 8.4.
Consider a weak geodesic v : [0 , T ] → B ( X ) ∩ PSH ( ω ) . If it is rightdifferentiable at t ∈ [0 , T ) , then the right derivative ˙ v ( t ) is quasicontinuous. Moreover, L ◦ ˙ v is constant on the subset D ⊂ (0 , T ) where v is differentiable. Finally, if v j :[0 , T ] → H are weak geodesics that decrease to v , then L ◦ ˙ v j → L ◦ ˙ v on D .Proof. As said, plurisubharmonic functions are quasicontinuous, hence so are the dif-ference quotients ( v ( t + s ) − v ( t )) /s , and their uniform limit, ˙ v ( t ). Next we turn tothe last statement, that we reduce to Lemma 3.4. First we show that ˙ v j ( t ) → ˙ v ( t ) in B ( X ) if t ∈ D . Let t ∈ D and ε >
0. There is an s > (cid:13)(cid:13)(cid:13) ˙ v ( t ) − v ( t ± s ) − v ( t ) ± s (cid:13)(cid:13)(cid:13) < ε, and so there is a j such that for j > j (cid:13)(cid:13)(cid:13) ˙ v ( t ) − v j ( t ± s ) − v j ( t ) ± s (cid:13)(cid:13)(cid:13) < ε. Convexity implies v j ( t − s ) − v j ( t ) − s ≤ ˙ v j ( t ) ≤ v j ( t + s ) − v j ( t ) s , whence k ˙ v j ( t ) − ˙ v ( t ) k < ε .Given that v j ( t ) decreases to v ( t ), that ˙ v j ( t ) → ˙ v ( t ) in B ( X ), and that ˙ v ( t ) isquasicontinuous, t ∈ D , Lemma 3.4 implies L ◦ ˙ v j → L ◦ ˙ v on D .To prove the second statement, construct w j , w ′ j ∈ H that decrease to v (0), v ( T ),and let v j : [0 , T ] → H be the weak geodesic that joins them. By Lemma 8.3 v j decreases to v and by Lemma 7.3 L ◦ ˙ v j is constant. According to what we just proved, L ◦ ˙ v j → L ◦ ˙ v on D , and L ◦ ˙ v must be constant there.In particular, if v : [ a, b ] → B ( X ) ∩ PSH( ω ) is a weak geodesic of class C , then L ◦ ˙ v is constant on ( a, b ). Using this with different Lagrangians one can show thatin fact ˙ v ( t ) ∈ B ( X, µ v ( t ) ) are equidistributed for a < t < b . Darvas points out in [D3,p. 1305] that for general weak geodesics even in C ( X ) ∩ PSH( ω ) this is no longer truefor t = a, b . The big question is whether it is true for a < t < b and a general weak eodesic in B ( X ) ∩ PSH( ω ) that the left and right derivatives lim s → ± ( v ( t + s )( x ) − v ( t )( x )) /s , computed pointwise on X , are equidistributed. If so, at least for almostevery t , the Principle of least action could be extended to include all weak geodesicsin B ( X ) ∩ PSH( ω ). Proof of Theorem 8.1.
Construct w j , w ′ j ∈ H decreasing to u (0) , u ( T ), and let u j :[0 , T ] → H , j ∈ J , be as in Lemma 8.2. Let v j : [0 , T ] → H be the weak geodesicconnecting w j and w ′ j , j ∈ J . By Corollary 7.4(8.1) Z T L ◦ ˙ u j ≥ Z T L ◦ ˙ v j . The integral on the left tends to R T L ◦ ˙ u as j → ∞ . The integrand on the right isconstant for each j , and on (0 , T ) converges unformly to L ◦ ˙ v by Lemma 8.4. Hencelim J ∋ j →∞ R T L ◦ ˙ v j = R T L ◦ ˙ v and letting j → ∞ in (8.1) we obtain the theorem. In this section the Lagrangian L : T ∞ E ( ω ) → R is strongly continuous, invariant, andconvex. We first investigate the least action, cf. (5.3), between two ε –geodesics, andthen by letting ε → Theorem 9.1. If u, v : [ a, b ] → B ( X ) ∩ PSH ( ω ) are weak geodesics, then for any S ∈ (0 , ∞ ) the function L S ( u, v ) : [ a, b ] → R is convex. The ε –variant is as follows: Lemma 9.2. If u, v : [ a, b ] → H are ε –geodesics, then for any S ∈ (0 , ∞ ) the function L S ( u, v ) : [ a, b ] → R is convex.Proof. Let a ≤ α < β ≤ b . Suppose U : [0 , S ] × [ α, β ] → H is a smooth map such that U ( s, · ) is an ε –geodesic for all s , and U (0 , · ) = u , U ( S, · ) = v . Hence ξ s = ∂ s U ( s, · ) isan ε –Jacobi field, 0 ≤ s ≤ S , and by Theorem 6.1 L ◦ ξ s is convex. Therefore, with0 ≤ λ ≤ t λ = (1 − λ ) α + λβ (9.1) L ( U ( · , t λ )) = Z S L ( ξ s ( t λ )) ds ≤ (1 − λ ) Z S L ( ξ s ( α )) ds + λ Z S L ( ξ s ( β )) ds. Fix δ >
0. Given u, v , we can choose U (uniquely) so that both w δα = U ( · , α ) and w δβ = U ( · , β ) are δ –geodesics. From (9.1)(9.2) L ( u ( t λ ) , v ( t λ )) ≤ L ( U ( · , t λ )) ≤ (1 − λ ) Z S L ◦ ˙ w δα + λ Z S L ◦ ˙ w δβ . Now lim δ → w δα = w α and lim δ → w δβ = w β are the weak geodesics in H connecting u ( α ) , v ( α ), respectively, u ( β ) , v ( β ); and, as explained in section 4, w δα → w α , w δβ → w β , ˙ w δα → ˙ w α , ˙ w δβ → ˙ w β niformly as δ →
0. Thus by Lemma 3.4 and Theorem 8.1lim δ → Z S L ◦ ˙ w δα = Z S L ◦ ˙ w α = L ( u ( α ) , v ( α )) , and similarly for the other integral in (9.2). Hence letting δ → L ( u ( t λ ) , v ( t λ )) ≤ (1 − λ ) L ( u ( α ) , v ( α )) + λ L ( u ( β ) , v ( β )) , what was to be proved. Lemma 9.3. If w, w ′ ∈ B ( X ) ∩ PSH ( ω ) and T > , then L T ( w, w ′ ) is finite. If w j , w ′ j ∈ C ( X ) ∩ PSH ( ω ) decrease, or converge uniformly, to w , resp. w ′ , then (9.3) L T ( w j , w ′ j ) → L T ( w, w ′ ) as j → ∞ . We do not know if (9.3) holds when w j , w ′ j ∈ B ( X ) ∩ PSH( ω ). Proof.
We will prove for decreasing sequences w j , w ′ j ; the case of uniformly convergentsequences can be reduced to decreasing sequences in a standard way, as in Lemma 3.4.Invariance implies that L is constant on the zero section of T ∞ E ( ω ). Since addinga constant to L will not affect the validity of the lemma, we will assume L vanisheson the zero section. Let us start with (9.3). It suffices to prove it along a subsequence j = j k .Assume first that w j , w ′ j ∈ H . Let u : [0 , T ] → B ( X ) ∩ PSH( ω ) be piecewise C connecting w and w ′ . At the price of passing to a subsequence, by Lemma 8.2there are u j : [0 , T ] → H piecewise C such that u j (0) = w j , u j ( T ) = w ′ j , and R T L ◦ ˙ u j → R T L ◦ ˙ u . Therefore L ( u ) = lim j →∞ L ( u j ) ≥ lim sup j →∞ L T ( w j , w ′ j ) . Passing to the infimum over all paths u connecting w, w ′ ,(9.4) L T ( w, w ′ ) ≥ lim sup j →∞ L T ( w j , w ′ j ) . Let v j : [0 , T ] → H be the weak geodesics connecting w j and w ′ j .We take a pause in the proof of (9.3) and show how (9.4) implies L T ( w, w ′ ) > −∞ .Fix numbers m, M so that for all jm ≤ inf X w j , inf X w ′ j , M ≥ sup X w j , sup X w ′ j . By convexity ˙ v j (0) ≤ ( v j ( T ) − v j (0)) /T ≤ ( M − m ) /T . Furthermore, u ( t ) = w j + ( m − M ) t/T is a subgeodesic, u (0) = w j , u ( T ) ≤ w ′ j . Hence u ( t ) ≤ v j ( t ) for all t , and˙ v j (0) ≥ lim t → u ( t ) − v (0) t ≥ m − MT . hus k ˙ v j (0) k ≤ ( M − m ) /T . Since L is equi–Lipschitz on bounded subsets of the fibers(Lemma 3.3), using Lemma 7.3 as well, L T ( w j , w ′ j ) = L ( v j ) = T L ( ˙ v j (0)) is a boundedsequence, and (9.4) implies L T ( w, w ′ ) > −∞ .We return to the proof of (9.3); we need to estimate L T ( w, w ′ ) from above. Forfixed δ > k with(9.5) lim inf j →∞ L T ( w j , w ′ j ) ≥ L T ( w k , w ′ k ) − δ = T L ( ˙ v k (0)) − δ. If 0 < ε < T /
2, define v εj ( t ) = tw j /ε + ( ε − t ) w/ε if 0 ≤ t < εv j (cid:16) t − ε T − tT − ε (cid:17) if ε ≤ t < T − ε ( T − t ) w ′ j /ε + ( t + ε − T ) w ′ /ε if T − ε ≤ t ≤ T .The piecewise C paths v εj : [0 , T ] → B ( X ) ∩ PSH( ω ) connect w and w ′ , hence(9.6) L T ( w, w ′ ) ≤ L ( v εj ) = (cid:16) Z ε + Z T − εε + Z TT − ε (cid:17) L ◦ ˙ v εj . The middle integral on the right is Z T − εε L ◦ ˙ v εj = ( T − ε ) L (cid:16) T ˙ v j (0) T − ε (cid:17) . As we saw, the ˙ v j (0) are uniformly bounded. By the equi–Lipschitz property of L an ε ∈ (0 ,
1) can be chosen so that for all j (9.7) Z T − εε L ◦ ˙ v εj ≤ T L ( ˙ v j (0)) + δ. When 0 ≤ t ≤ ε , we have ˙ v εj ( t ) = ( w j − w ) /ε ∈ T ∞ v εj ( t ) E ( ω ). Again by the equi–Lipschitz property, if j is sufficiently large, | L ( ˙ v εj ( t )) | < δ ; and similarly for T − ε ≤ t < T . Putting this and (9.6), (9.7) together, L T ( w, w ′ ) ≤ δ + T L ( ˙ v j (0))if j is sufficiently large. Choosing j from among the k in (9.5) therefore yields L T ( w, w ′ ) ≤ δ + lim inf j →∞ L T ( w j , w ′ j ) . This being true for all δ >
0, (9.3) follows in view of (9.4).So far we dealt with w j , w ′ j ∈ H . If w j , w ′ j ∈ C ( X ) ∩ PSH( ω ) only, upon addingconstants to them we can arrange that w j < w j − and w ′ j < w ′ j − everywhere. We willexpress this by saying that w j , w ′ j strictly decrease. We construct recursively z j > w j , z ′ j > w ′ j in H that strictly decrease to w, w ′ and satisfy |L T ( z j , z ′ j ) − L T ( w j , w ′ j ) | < /j as follows. Suppose we already have z j − , z ′ j − . Construct sequences y i < z j − , y ′ i < z ′ j − ( i ∈ N ) in H that decrease to w j , w ′ j . By what we have already proved, |L T ( y i , y ′ i ) − L T ( w j , w ′ j ) | < /j for some i , and we let z j = y i , z ′ i = y ′ i with that i . Thus L T ( w, w ′ ) = lim j L T ( z j , z ′ j ) = lim j L T ( w j , w ′ j ) , as claimed. roof of Theorem 9.1. Assume first that u, v are weak geodesics in H with endpointsin H , and connect u ( a ) , u ( b ), respectively, v ( a ) , v ( b ) by ε –geodesics u ε , v ε . By Chen’stheorem u ε → u and v ε → v uniformly as ε →