aa r X i v : . [ m a t h . DG ] F e b The isometries of the space of K¨ahler metrics
Tam´as Darvas
To Anita.
Abstract
Given a compact K¨ahler manifold, we prove that all global isometries of the space ofK¨ahler metrics are induced by biholomorphisms and anti-biholomorphisms of the man-ifold. In particular, there exist no global symmetries for Mabuchi’s metric. Moreover,we show that the Mabuchi completion does not even admit local symmetries. Closelyrelated to these findings, we provide a large class of metric geodesic segments that cannot be extended at one end, pointing out the first such examples in the literature.
Let (
X, ω ) be a compact connected K¨ahler manifold. Given a K¨ahler metric ω ′ cohomologuosto ω , by the ∂ ¯ ∂ -lemma of Hodge theory there exists u ∈ C ∞ ( X ) such that ω ′ := ω + i∂ ¯ ∂u. Such a metric ω ′ is said to belong to the space of K¨ahler metrics H . By the above, up to aconstant, one can identify H with the space of K¨ahler potentials : H ω := { u ∈ C ∞ ( X ) s.t. ω + i∂ ¯ ∂u > } . This space can be endowed with a natural infinite dimensional L type Riemannian metric[24, 26, 17]: h ξ, ζ i v := 1 V Z X ξζ ω nv , v ∈ H ω , ξ, ζ ∈ T v H ω ≃ C ∞ ( X ) , (1)where V = R X ω n . Additionally, Donaldson and Semmes pointed out that ( H ω , h· , ·i ) can bethought of as a formal symmetric space [27, 17]: H ω ≃ Ham C ω Ham ω , (2)where Ham ω is the group of Hamiltonian symplectomorphisms of ω , and Ham C ω is its formalcomplexification. Though not quite precise, the underlying heuristic of (2) led to manyadvances in the understanding of the geometry of H ω , as well as the formulation of stabilityconditions aiming to characterize existence of canonical metrics (for an exposition see [28]).1 lobal L isometries and symmetries of H ω . For finite dimensional Riemannian man-ifolds, the existence of a symmetric structure arising as a quotient of Lie groups, as in (2), isequivalent with existence of global symmetries at all points of the manifold [19]. Such mapsare global involutive isometries reversing geodesics at a specific point. If such symmetriesexisted for ( H ω , h· , ·i ) it would perhaps allow to make a precise sense of (2).Recently a large class of local symmetries of H ω were constructed in [2], via complexLegendre transforms, that also found applications to interpolation of norms [3]. Moreover,it was shown in [21] that all local symmetries of H ω arise from the construction of [2]. Belowwe show that global symmetries actually do not exist, in particular these local symmetriescan not be extended to H ω . This will follow from our characterization of the isometry groupof ( H ω , h· , ·i ).First we recall some terminology. Let U , V ⊂ H ω be open sets. We say that a map F : U → V is C , or (with slight abuse of terminology) differentiable, if ( F, F ∗ ) : U × C ∞ ( X ) → V × C ∞ ( X ) is continuous as a map of Fr´echet spaces. Here F ∗ is the differentialof F (see [22, p. 3] and references therein for more details). Moreover, F : U → U is a differentiable L symmetry at φ ∈ U if F = Id , F ( φ ) = φ , F ∗ | φ = − Id and Z X | ξ | ω nv = Z X | F ∗ ξ | ω nG ( v ) , v ∈ H ω , ξ ∈ T v H ω . (3)If F : U → V is C , satisfies (3) and it is bijective, then it is called a differentiable L isometry . Due to infinite dimensionality, it is not yet known if differentiable L isometriesare automatically smooth [20], hence the isometries we consider in this work are possiblymore general than the ones in [2, 21].A small class of global L isometries has been previously known in the literature [20, p.16]. One of them is the so called Monge–Amp`ere flip I : H ω → H ω , and is defined by theformula I ( u ) = u − I ( u ), where I : H ω → R is the Monge–Amp`ere energy: I ( u ) = 1 V ( n + 1) n X j =0 Z X uω j ∧ ω n − ju . The map I is involutive and its name is inspired by the fact that it flips the sign of I . Indeed, I ( I ( u )) = − I ( u ).We say that a biholomorphism f : X → X preserves the K¨ahler class [ ω ] if [ f ∗ ω ] = [ ω ].Similarly, an anti-biholomorphism g : X → X flips the the K¨ahler class [ ω ] if [ g ∗ ω ] = − [ ω ].Such maps also induce a class of global L isometries, and we refer to Section 2.3 for thedetailed construction.In our first main result we point out that these maps and their compositions are the onlyglobal differentiable L isometries: Theorem 1.1.
Let F : H ω → H ω be a differentiable L isometry. Then exactly one of thefollowing holds:(i) F is induced by a biholomorphism or anti-biholomorphism f : X → X that preserves orflips [ ω ] , respectively.(ii) F ◦ I is induced by a biholomorphism or anti-biholomorphism f : X → X that preservesor flips [ ω ] , respectively. H ω admits a Riemannian splitting H ω = H ⊕ R , via the Monge–Amp`ere energy I . As the fixed point set of I is exactly H = I − (0), we obtain the followingcorollary regarding isometries of H : Corollary 1.2.
Let F : H → H be a differentiable L isometry. Then F is induced by abiholomorphism or anti-biholomorphism f : X → X that preserves or flips [ ω ] , respectively. The above results answer explicitly questions raised by Lempert regarding the extensionproperty of local isometries [20, p. 3], though questions surrouding the isometry group of( H ω , h· , ·i ) go back to early work of Semmes [26, 27].Lastly, via the classification theorem of Lempert (recalled in Theorem 2.1), we will seethat neither of the maps in the statement of Theorem 1.1 are symmetries, immediately givingthe following non-existence result for differentiable L symmetries: Corollary 1.3.
There exists no differentiable L symmetry F : H ω → H ω at any φ ∈ H ω . Non-existence of local L symmetries on the completions. It was shown in [7]that (1) induces a path length metric space ( H ω , d ). By ( E ω , d ) we denote the d -metriccompletion of this space, that can identified with a class of finite energy potentials [11].Using density, any differentiable L isometry F : H ω → H ω extends to a unique metric d -isometry F : E ω → E ω . The proof of Theorem 1.1 consists of showing that contradictionsarise in this extension process, unless F is very special. With this and the above resultsin mind, one may hope that the isometry group of the metric space ( E ω , d ) could possiblyadmit elements beyond the ones that arise from the global differentiable L isometries of H ω . Though this may be true, we point out below that even local symmetries fail to exist inthe context of the completion, further elaborating on phenomenon related to Corollary 1.3.Before stating our result, we recall some facts about the d -geodesics of E . For moredetails we refer to Section 2.2 and the recent survey [13]. Let V ⊂ E ω be d -open with φ ∈ V ∩ H ω . Given a d -geodesic [0 , ∋ t → φ t ∈ V with φ = φ , since t → φ t ( x ) is t -convexfor almost every x ∈ X , it is possible to introduce ˙ φ = ddt | t =0 φ t . Moreover, due to [12,Theorem 2], it follows that ˙ φ ∈ L ( ω nφ ) . Let G : V → G ( V ) ⊂ E ω be an L isometry, i.e, a bijective map satisfying d ( v , v ) = d ( G ( v ) , G ( v )) , v , v ∈ V . It is clear that in this case t → G ( φ t ) is also a d -geodesic.Furthermore, we say that G is a metric L symmetry at φ if G = Id , G ( φ ) = φ and˙ G ( φ ) = − ˙ φ , i.e., G “reverses” d -geodesics at φ .Unfortunately, metric L symmetries actually do not exist, implying that the analog of[2, Theorem 1.2] does not hold in the context of the metric completion, answering questionsof Berndtsson and Rubinstein [25]: Theorem 1.4.
Let
V ⊂ E ω be a d -open set and φ ∈ V ∩ H ω . There exists no metric L symmetry F : V → V at φ . Given that ( E ω , d ) is CAT(0), the group of isometries of this metric space has specialstructure [6], as pointed by B. McReynolds during the Ph.D. thesis defense of the author. Inlight of the above result, we expect that the group of metric isometries can be characterizedas in Theorem 1.1, though this remains an open question.3 he extension property of geodesic segments. As an intermediate step in the proofof Theorem 1.4 we show that a large class of d -geodesic segments inside E ω can not beextended at one of the endpoints. Previously no such examples were known. Theorem 1.5.
Let φ ∈ H ω and φ ∈ E ω \ L ∞ . Then the d -geodesic t → ψ t connectingthese potentials can not be extended to a d -geodesic ( − ε, ∋ t → φ t ∈ E ω for any ε > . For finite dimensional manifolds, topological and geodesical completeness are equivalentdue to the classical Hopf–Rinow theorem. According to the above result, this is not the casefor the completion ( E ω , d ), despite the fact that this space it is non-positively curved [8, 11].It will be interesting to see if a similar property holds for the C , -geodesics of Chen andChu–Tosatti–Weinkove, joining the potentials of H ω [7, 9]. Relation to the L p geometry of H ω . In [12] the author introduced a family of L p Finslermetrics on H ω for any p ≥
1, generalizing (1): k ξ k p,v = (cid:18) V Z X | ξ | p ω nv (cid:19) p , v ∈ H ω , ξ ∈ T v H ω . These induce path length metric spaces ( H ω , d p ), and in [12] the author computed the corre-sponding metric completions, that later found applications to existence of canonical metrics(for a survey see [13]). Though this more general context lacks the symmetric space inter-pretation, all of our above results can be considered in the L p setting as well.As the reader will be able to deduce from our arguments below, the L p version of Theorem1.4 holds for any p >
1. Our proof does not work when p = 1, since the class of finite energygeodesics may not be stable under isometries in this case (see [14, Theorem 1.2]). On theother hand, the L p version of Theorem 1.5 does hold for all p ≥
1. Lastly, our argument forTheorem 1.1 would most likely go through in the L p context in case one could obtain theanalog of Theorem 2.1 for differentiable L p isometries. Acknowledgements.
We thank L. Lempert for extensive feedback on our manuscript, andfor generously explaining to us details about his paper [20]. We also thank B. Berndtssonand Y.A. Rubinstein for suggestions on how to improve the presentation. This work waspartially supported by NSF grant DMS 1610202.
For simplicity we assume throughout the paper the the K¨ahler metric ω satisfies the followingvolume normalization: V = Z X ω n = 1 . Using a dilation of ω this can always be achieved and does not represent loss of generality.4 .1 The classification theorem of Lempert In this short section we recall the particulars of a result due to Lempert on the classificationof local C isometries on H ω ([20, Theorem 1.1]), tailored to our global setting: Theorem 2.1.
Suppose that F : H ω → H ω is a differentiable L isometry. Then for u ∈ H ω there exists a unique C ∞ diffeomorphism G u : X → X such that G ∗ u ω u = ± ω F ( u ) and F ∗ ( u ) ξ = aξ ◦ G u − b Z X ξω nu , ξ ∈ T u H ω ≃ C ∞ ( X ) , (4) where a = 1 , or a = − , or b = 0 , or b = 2 a . In the particular case of the (local) L symmetries constructed in [2], formula (4) is aconsequence of [2, Theorem 5.1, Theorem 6.1, Proposition 7.1] with a = − b = 0. Remark 2.2.
It follows from the proof of [20, Theorem 1.1] that the integers a and b inthe statement depend continuously on u ∈ H ω (as does G u ), hence in our case they areindependent of u , as H ω is connected. This was pointed out to us by L. Lempert [23]. From the classification theorem we obtain the following simple monotonicity result:
Proposition 2.3.
Suppose that F : H ω → H ω is a differentiable L isometry with b = 0 .Let c ∈ R and u, v ∈ H ω with u ≤ v . Then the following hold:(i) if a = 1 then F ( u ) ≤ F ( v ) and F ( u + c ) = F ( u ) + c .(ii) if a = − then F ( u ) ≥ F ( v ) and F ( u + c ) = F ( u ) − c .Proof. We only address (ii), as the proof of (i) is analogous. Let [0 , ∋ t → γ t := v + t ( u − v ) ∈ H ω . Then t → F ( γ t ) is a C curve connecting F ( v ) and F ( u ). Moreover, Theorem 2.1implies that F ( u ) − F ( v ) = Z ddt F ( γ t ) dt = Z − ( u − v ) ◦ G γ t dt ≥ . The fact that F ( u + c ) = F ( u ) − c , follows after another application of Theorem 2.1 to thecurve [0 , ∋ t → η t := u + tc ∈ H ω . Corollary 2.4.
Suppose that F : H ω → H ω is a differentiable L isometry with b = 0 . Then,in the language of Theorem 2.1 applied to F , we have that G u + c = G u for all u ∈ H ω and c ∈ R .Proof. We only address the case a = 1, as the argument for a = − ξ ∈ C ∞ ( X ). By Proposition 2.3(i) and Theorem 2.1 we have that ξ ◦ G u + c = F ∗ ( u + c ) ξ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 F ( u + tξ + c ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 F ( u + tξ ) = F ∗ ( u ) ξ = ξ ◦ G u . Since ξ ∈ C ∞ ( X ) is arbitrary, we obtain that G u + c = G u .5 .2 The complete metric space ( E ω , d ) In this short subsection we recall aspects from the work of the author related to the metriccompletion of ( H ω , d ). For details we refer to the survey [13].As conjectured by V. Guedj [18], ( H ω , d ) can be identified with ( E ω , d ), where E ω ⊂ PSH(
X, ω ) is an appropriate subset of ω -plurisubharmonic potentials [11, Theorem 1]. More-over, ( E ω , d ) is a non-positively curved complete metric space, whose points can be joinedby unique d -geodesics.Given u , u ∈ E ω , the unique d -geodesic [0 , ∋ t → u t ∈ E ω connecting these pointshas special properties. To start, we recall that this curve arises as the following envelope: u t := sup { v t | where t → v t is a subgeodesic } , t ∈ (0 , . (5)Here a subgeodesic (0 , ∋ t → v t ∈ PSH(
X, ω ) is a curve satisfying lim sup t → , v t ≤ u , and u ( s, x ) := u Re s ( x ) ∈ PSH( S × X, ω ), where S = { < Re s < } ⊂ C .It follows from (5) that t → u t ( x ) , t ∈ (0 ,
1) is convex for all x ∈ X away from a set ofmeasure zero. On the complement we have that u t ( x ) = −∞ , t ∈ (0 , t → u t ( x ) = u ( x ) and lim t → u t ( x ) = u ( x ) (6)for all x ∈ X away from a set of measure zero. In the particular case when u , u ∈ H ω , thecurve t → u t is C , on [0 , × X [7, 4, 9].By C ω we denote the set of continuous potentials in PSH( X, ω ). As pointed out previously,a differentiable L isometry F : H ω → H ω induces a unique d -isometry F : E ω → E ω ,extending the original map (using density). Going forward, we do not distinguish F fromits unique extension. Moreover, if F is an isometry with b = 0 (see Theorem 2.1), we pointout that C ω is stable under the extension: Proposition 2.5.
Suppose that F : H ω → H ω is a differentiable L isometry with b = 0 .Then F ( C ω ) ⊂ C ω . More importantly, sup X k u j − u k → implies sup X k F ( u j ) − F ( u ) k → for any u j , u ∈ C ω .Proof. We only argue the case when a = 1, as the proof is analogous in case a = −
1. Since d -convergence implies pointwise a.e. convergence (see [12, Theorem 5]), Proposition 2.3(i)holds for the extension F : E ω → E ω and u, v ∈ E ω satisfying u ≤ v .Let u ∈ C ω . Then [5] implies existence of u k ∈ H ω such that u k ց u . In fact, due to Dini’slemma, the convergence is uniform. From Proposition 2.3 it follows that { F ( u k ) } k ⊂ H ω is monotone decreasing. Due to uniform convergence, we have that for any ε > k such that u ≤ u k ≤ u + ε for k ≥ k . Then Proposition 2.3 implies that F ( u ) ≤ F ( u k ) ≤ F ( u ) + ε, k ≥ k . This gives that F ( u k ) converges to F ( u ) uniformly, in particular F ( u ) ∈ C ω .Lastly, we can essentially repeat the above argument for continuous potentials u j con-verging uniformly to u , concluding the last statement of the proposition.6 .3 Examples of differentiable L isometries on H ω In this short subsection we describe three examples of global differentiable L isometries on H ω . Later we will argue that in fact all isometries arise as compositions of these examples. • First we take a closer look at the Monge–Amp`ere flip I : H ω → H ω , defined in Section1, perhaps first introduced in [20]. Let [0 , ∋ t → γ t ∈ H ω be a smooth curve. Since ddt I ( γ t ) = R X ˙ γ t ω nγ t , we obtain that Z X (cid:16) ddt I ( γ t ) (cid:17) ω nγ t = Z X (cid:16) ˙ γ t − Z X ˙ γ t ω nγ t (cid:17) = Z X ˙ γ t ω nγ t , hence I is indeed an involutive L isometry, with a = 1 and b = 2 (see Theorem 2.1). Thissimple map has the following intriguing property, that will help to adjust the b parameterof arbitrary isometries without changing the a parameter: Lemma 2.6.
Suppose that F : H ω → H ω is a differentiable L isometry. The a parameterof F and F ◦ I is always the same. Regarding the b parameter the following hold:(i) If b = 0 for F , then b = 2 a for F ◦ I .(ii) If b = 2 a for F , then b = 0 for F ◦ I .Proof. Let [0 , ∋ t → γ t ∈ H ω be a smooth curve. Then we have that ddt F ( I ( γ t )) = F ∗ ( I ∗ ˙ γ t ) = F ∗ (cid:16) ˙ γ t − Z X ˙ γ t ω nγ t (cid:17) . If a = 1 and b = 0 for F , then we get that ddt F ( I ( γ t )) = ˙ γ t ◦ G u − R X ˙ γ t ω nγ t . If a = − b = 0 for F , then ddt F ( I ( γ t )) = − ˙ γ t ◦ G u + 2 R X ˙ γ t ω nγ t , addressing (i).In case a = 1 and b = 2 a for F , then ddt F ( I ( γ t )) = ˙ γ t ◦ G u . Similarly, if a = − b = 2 a for F , then ddt F ( I ( γ t )) = − ˙ γ t ◦ G u , addressing (ii). • Now let f : X → X be a biholomorphism preserving the K¨ahler class [ ω ]. Then f induces a map L f : H → H via pullbacks: ω L f ( u ) := f ∗ ω u , where we made the identification H ≃ I − (0). Using this identification it is possible to describe the action of F on the levelof potentials in the following manner [15, Lemma 5.8]: L f ( u ) = L f (0) + u ◦ F, u ∈ I − (0) , (7)where 0 ∈ I − (0) is simply the zero K¨ahler potential. More importantly, L f further extendsto a map L f : H ω → H ω in the following manner: L f ( v ) = L f ( v − I ( v )) + I ( v ) , v ∈ H ω . It is well known that L f thus described gives a differentiable L isometry of H ω with a = 1and b = 0. Actually, using the language of Theorem 2.1 applied to L f , we obtain that G u = f for all u ∈ H ω . We leave the related simple computation to the reader. • Now let g : X → X be an anti-biholomorphism that flips the K¨ahler class [ ω ]. Bydefinition, such a map is a diffeomorphism satisfying ∂g j ∂z k = 0 for all j, k ∈ { , . . . , n } in any7hoice of local coordinates. For example, the map g ( z ) = ¯ z is an anti-biholomorphism of theunit torus C / Z [ i ] that flips that class of the flat K¨ahler metric.Such a map g induces another map N g : H → H via pullbacks: ω N g ( u ) := − g ∗ ω u . Herewe used again the identification H ≃ I − (0). Similar to (7), it is possible to describe theaction of N g on the level of potentials in the following manner: N g ( u ) = N g (0) + u ◦ g, u ∈ I − (0) . (8)To show this, we have to go through the proof of [15, Lemma 5.8] in the anti-holomorphiccontext. As a beginning remark, we notice that g ∗ ∂ ¯ ∂v = − ∂ ¯ ∂v ◦ g for all smooth functions v . With this in mind, we have that ω + i∂ ¯ ∂ ( N g (0) + u ◦ g ) = − g ∗ ω − g ∗ i∂ ¯ ∂u = − g ∗ ω u = ω N g ( u ) = ω + i∂ ¯ ∂N g ( u ) . In particular, N g (0) + u ◦ g − N g ( u ) is a constant. To show that this constant is equal tozero, we only need to argue that I ( N g (0) + u ◦ g ) = 0 = I ( N g ( u )). But this holds because ofthe following computation: I ( N g (0) + u ◦ g ) = I ( N g (0) + u ◦ g ) − I ( N g (0)) = 1 n + 1 n X j =0 Z X ( u ◦ g ) ω jN g (0)+ u ◦ g ∧ ω n − jN g (0) = ± n + 1 n X j =0 Z X ( u ◦ g ) g ∗ ( ω ju ∧ ω n − j )= ± n + 1 n X j =0 Z X uω ju ∧ ω n − j = ± I ( u ) = 0 . As above, N g extends to a map N g : H ω → H ω in the following manner: N g ( v ) = N g ( v − I ( v )) + I ( v ) , v ∈ H ω . We point out that N g thus described gives a differentiable L isometry of H ω with a = 1and b = 0. To see this, let [0 , ∋→ γ t ∈ H ω be a smooth curve. Using (8) we can write thefollowing ddt N g ( γ t ) = ddt ( γ t ◦ g − I ( γ t )) + ddt I ( γ ( t )) = ˙ γ t ◦ g. In the language of Theorem 2.1 applied to N g , we actually obtained that G u = g for all u ∈ H ω . The argument of Theorem 1.1 is split into two parts. First we show that there exist noglobal differentiable isometries with a = −
1. Later we will classify all global differentiableisometries with a = 1.Before we go into specific details, we recall the following simple lemma that will be usednumerous times in our arguments: 8 emma 3.1. [10, Lemma 3.1] Suppose that u , u ∈ C ω and [0 , ∋ t → u t ∈ E ω is the d -geodesic connecting these potentials. Then we have that inf X ˙ u = inf X ( u − u ) , sup X ˙ u = sup X ( u − u ) . Proof.
First we argue that inf X ˙ u = inf X ( u − u ). From (5) we obtain the estimate u t ≥ u + t inf X ( u − u ) , t ∈ [0 , u ≥ inf X ( u − u ). Using t -convexity it followsthat u t ( y ) = u ( y ) + t inf X ( u − u ) for y ∈ X such that u ( y ) − u ( y ) = inf X ( u − u ). Thisimplies that t → u t ( y ) is linear, implying that inf X ˙ u = inf X ( u − u ).For the second identity, we notice that t -convexity implies sup X ˙ u ≤ sup X ( u − u ).In addition, (5) implies that u − (1 − t ) sup X ( u − u ) ≤ u t , t ∈ [0 , t -convexity again, we obtain that ˙ u ( z ) = u ( z ) − u ( z ) = sup X ( u − u ), for z ∈ X with u ( z ) − u ( z ) = sup X ( u − u ). Summarizing, we obtain that sup X ˙ u = sup X ( u − u ), asdesired. a = − We start with a lemma:
Lemma 3.2.
Suppose that F : H ω → H ω is a differentiable L isometry with a = − and b = 0 . Let φ ∈ H ω and u ∈ H ω with u ≤ φ . Then we have that F ( u ) ≥ F ( φ ) and sup X ( F ( u ) − F ( φ )) = − inf X ( u − φ ) . (9) Proof.
That F ( u ) ≥ F ( φ ) follows from Proposition 2.3(ii). As it is pointed out on [20, p.2],Theorem 2.1 implies that F is a d p -isometry for any p ≥
1. This implies that d p ( φ, u ) = d p ( F ( φ ) , F ( u )) for any p ≥ , ∋ t → u t , v t ∈ H , ω be the C , geodesic connecting u := φ, u := u , respectively v := F ( φ ) and v := F ( u ). By the comparison principle for weak geodesics (see for example[4, Proposition 2.2]) it follows that v t ≥ F ( φ ) and u t ≤ φ for any t ∈ [0 , v ≥ u ≤ Z X | ˙ u | p ω nφ = d p ( φ, u ) p = d p ( F ( φ ) , F ( u )) p = Z X | ˙ v | p ω nF ( φ ) , p ≥ . Raising to the p -power, and letting p → ∞ gives thatsup X ˙ v = − inf X ˙ u . (10)From Lemma 3.1 we get that inf X ˙ u = inf X ( u − φ ) and sup X ˙ v = sup X ( F ( u ) − F ( φ )).Putting this together with (10), we obtain (9), as desired. Theorem 3.3.
There exists no differentiable L isometry F : H ω → H ω with a = − . We note that this result already implies Corollary 1.3.9 roof.
Due to Lemma 2.6, after possibly composing F with I , we only need to worry aboutthe case a = − b = 0.Since F : H ω → H ω is a differentiable L -isometry, it is also a d -isometry, hence itextends to a unique d -isometry F : E ω → E ω .Let φ ∈ H ω . Let u ∈ E ω \ L ∞ with u ≤ φ −
1, and we choose u k ∈ H ω such that u k ց u and u k ≤ φ . Such a sequence can always be found [5].Due to our choice of u we have that inf X ( u k − φ ) ց −∞ . From Lemma 3.2 it followsthat sup X F ( u k ) = sup X ( F ( u k ) − F ( φ )) ր + ∞ . Since F is a d -isometry, we have that d ( F ( u k ) , F ( u )) = d ( u, u k ) →
0. However [12, Theorem 5(i)] gives that sup X F ( u k ) → sup X F ( u ) < + ∞ , which is a contradiction. a = 1 To start, we point out an important relationship between d -geodesics and differentiable L isometries with a = 1 and b = 0: Proposition 3.4.
Suppose that F : H ω → H ω is a differentiable L isometry with a = 1 and b = 0 . Let [0 , ∋ t → u t ∈ E ω be the d -geodesic connecting u ∈ H ω and u ∈ C ω . Then ˙ u ◦ G u = ˙ F ( u ) . (11)Here and below ˙ u := ddt (cid:12)(cid:12) t =0 F ( u t ) and ˙ F ( u ) := ddt (cid:12)(cid:12) t =0 F ( u t ) are the initial tangent vectorsof the d -geodesics t → u t and t → F ( u t ), interpreted according to the discussion precedingTheorem 1.4. Proof.
There exists a constant c ∈ R such that u > u + c . Since F ( u t + tc ) = F ( u t ) + tc (Proposition 2.3(i)), we can assume without loss of generality that u > u .First, we show (11) in case u ∈ H ω . Let [0 , ∋ u εt ∈ H ω be the smooth ε -geodesics ofX.X. Chen, connecting u and u [7]. It is well known that u εt ր u t as ε →
0, where t → u t is the C , -geodesic joining u and u . Due to Proposition 2.3 and Proposition 2.5, for thecurves t → F ( u εt ) , F ( u t ) we obtain that F ( u εt ) ր F ( u t ). Since t → F ( u εt ) is a C curve, viaTheorem 2.1, we obtain that˙ u ε ◦ G u = ˙ F ( u ε ) ≤ ˙ F ( u ) ≤ , ε > . Taking the limit ε →
0, since u ε → C ,α u , we arrive at ˙ u ◦ G u ≤ ˙ F ( u ) ≤
0. By Theorem2.1 we have that G ∗ u ω nu = ± ω nF ( u ) . Using this and [7] (see also [12, Theorem 1]) we obtainthat Z X ( ˙ u ◦ G u ) ω nF ( u ) = Z X ˙ u ω nu = d ( u , u ) = d ( F ( u ) , F ( u )) = Z X ˙ F ( u ) ω nF ( u ) . Due to continuity we conclude that ˙ u ◦ G u = ˙ F ( u ), as desired.Now we treat the general case. Let u k ∈ H ω , k ∈ N such that u > u k and u k ց u ∈ C ω .Also, by [0 , ∋ t → u t , u kt ∈ E ω we denote the d -geodesics connecting u and u , respectively u and u k . Since F is a d -isometry, we obtain that [0 , ∋ t → F ( u t ) , F ( u kt ) ∈ E ω are the d -geodesics connecting F ( u ) and F ( u ), respectively F ( u ) and F ( u k ). Due to t -convexity,10 -monotonicity and Proposition 2.3, we obtain that ˙ u k ց ˙ u and ˙ F ( u k ) ց ˙ F ( u ). Letting k → ∞ we arrive at the desired conclusion: ˙ u ◦ G u = lim k ( ˙ u k ◦ G u ) = lim k ˙ F ( u k ) =˙ F ( u ).This result together with Lemma 3.1 gives the following corollary, paralleling Lemma 3.2: Corollary 3.5.
Suppose that F : H ω → H ω is a differentiable L isometry with a = 1 and b = 0 . Suppose that u, v ∈ C ω . Then we have that F ( u ) , F ( v ) ∈ C ω and inf X ( F ( u ) − F ( v )) = inf X ( u − v ) . (12)By the switching the role of u and v , we obtain that the above identity holds for thesuprema as well. Proof.
That F ( u ) , F ( v ) ∈ C ω , follows from Proposition 2.5. First we deal with the case when u, v ∈ H ω . If [0 , ∋ t → h t ∈ H ω is the C , -geodesic connecting h := u and h := v , thenLemma 3.1 gives thatinf X ( v − u ) = inf X ˙ h and inf X ( F ( v ) − F ( u )) = inf X ˙ F ( h ) . Putting this together with (11), we obtain that inf X ( v − u ) = inf X ( F ( v ) − F ( u )), as desired.When u, v ∈ C ω , by [5] one can find u k , v k ∈ H ω such that sup X | u k − u | → X | v k − v | →
0. Then Proposition 2.5 implies that sup X | F ( u k ) − F ( u ) | → X | F ( v k ) − F ( v ) | → X ( u k − v k ) → inf X ( u − v ) and inf X ( F ( u k ) − F ( v k )) → inf X ( F ( u ) − F ( v )). The conclusion follows after taking the k -limit of inf X ( u k − v k ) =inf X ( F ( u k ) − F ( v k )).To continue, we need an an auxiliary construction. Fixing x ∈ X and a small enoughcoordinate neighborhood O x ⊂ X , we can find a function ρ x ∈ C ∞ ( X ) such that ρ x ( y ) = e − k y − x k for all y ∈ O x , and there exists β > β ≤ ρ x ( y ) ≤ y ∈ X \ O x . Proposition 3.6.
For u ∈ H ω and x ∈ X there exists δ > such that [0 , ∋ t → u t := u + δ ( t + t ) ρ x ∈ H ω is a subgeodesic.Proof. Let U ( s, y ) = u Re s ( y ) ∈ C ∞ ( S × X ), where S = { ≤ Re z ≤ } ⊂ C . It is clearthat for small enough δ > u t ∈ H ω , t ∈ [0 , α > ω u t ≥ αω, t ∈ [0 , ω + i∂ S × X ¯ ∂ S × X U has at least n non-negative eigenvalues for all ( s, y ) ∈ S × X . To conclude that ω + i∂ S × X ¯ ∂ S × X U ≥ u t − h ∂ ˙ u t , ¯ ∂ ˙ u t i ω ut ≥ , × X. To show this, we start the following sequence of estimates:¨ u t − h ∂ ˙ u t , ¯ ∂ ˙ u t i ω ut = δρ x − δ (1 + t ) h ∂ρ x , ¯ ∂ρ x i ω ut ≥ δρ x − δ (1 + t ) α h ∂ρ x , ¯ ∂ρ x i ω . After possibly shrinking δ ∈ (0 , O x , where know that ρ x ( y ) = e − k y − x k , y ∈ O x .11n particular, on O x \ { x } we have that h ∂ρ x , ¯ ∂ρ x i ω /ρ x ≃ e − k y − x k k y − x k , which is uniformlybounded. In particular, after possibly further shrinking δ ∈ (0 ,
1) we obtain that¨ u t − h ∂ ˙ u t , ¯ ∂ ˙ u t i ω ut ≥ δρ x − δ (1 + t ) α h ∂ρ x , ¯ ∂ρ x i ω ≥ , what we desired to prove. Theorem 3.7.
Suppose that F : H ω → H ω is a differentiable L isometry with a = 1 . Thenexactly one of the following holds:(i) F is induced by a biholomorphism or anti-biholomorphism f : X → X that preserves orflips the K¨ahler class [ ω ] , respectively.(ii) F ◦ I is induced by a biholomorphism or anti-biholomorphism f : X → X that preservesor flips the K¨ahler class [ ω ] , respectively.Proof. Due to Lemma 2.6, after possibly composing F with I , we only need to worry aboutthe case a = 1 and b = 0. In this case we will show that F is induced by a biholomorphismor anti-biholomorphism g : X → X that preserves or flips the K¨ahler class [ ω ].In the language of Theorem 2.1 applied to F , the first step is to show that G u = G v forall u, v ∈ H ω .We fix x ∈ X and u, v ∈ H ω . We will show that G − u ( x ) = G − v ( x ). Since G u + c = G u for any c ∈ R (Corollary 2.4), we can assume that u ( x ) = v ( x ). First we prove that G − u ( x ) = G − v ( x ) under the extra non-degeneracy condition ∇ u ( x ) = ∇ v ( x ).Let η > w := max( u, v ) + ηρ x ∈ C ω . From our setup it is clear that w ≥ max( u, v ), and the graphs of w , u and v only meet at x . Extending the isometry F to the metric completion, Proposition 2.3 and Proposition 2.5 implies that F ( w ) ≥ max( F ( u ) , F ( v )) , F ( w ) ∈ C ω and F ( u ) , F ( v ) ∈ H ω . Below we will show that F ( w ) and F ( u ) only meet at G − u ( x ), moreover F ( w ) and F ( v ) only meet at G − v ( x ). Finally, we willshow that the graphs of F ( w ), F ( u ) and F ( v ) have to meet at some point of X , implyingthat G − u ( x ) = G − v ( x ), as desired.Let us denote by [0 , ∋ t → u t , v t ∈ E ω the d -geodesics joining u := u with u := w ,respectively v := v with v := w . From Proposition 3.4 it follows that˙ F ( u ) = ˙ u ◦ G u , ˙ F ( v ) = ˙ v ◦ G v . (13)Using (5) there exists a small enough δ > u + δ ( t + t ) ρ x ≤ u t and v + δ ( t + t ) ρ x ≤ v t , t ∈ [0 , t -convexity and (13), weobtain that F ( w ) − F ( u ) ≥ ˙ F ( u ) = ˙ u ◦ G u ≥ δρ x ◦ G u , F ( w ) − F ( v ) ≥ ˙ F ( v ) = ˙ v ◦ G v ≥ δρ x ◦ G v . Due to (12) these two estimates imply the existence of a unique y ∈ X and a unique z ∈ X such that F ( w )( y ) − F ( u )( y ) = 0 and F ( w )( z ) − F ( v )( z ) = 0 . (14)In fact, we need to have that y = G − u ( x ) and z = G − v ( x ). In particular, the graphs of F ( w )and F ( u ) only meet at y , and graphs of F ( w ) and F ( u ) only meet at z .12n case y = z , uniqueness of y and z implies that y ∈ { F ( u ) > F ( v ) } and y ∈ { F ( v ) >F ( u ) } (recall that F ( w ) ≥ max( F ( u ) , F ( v )). This implies that the graphs of F ( w ) andmax( F ( u ) , F ( v )) meet at only two points ( y and z ), away from the compact set { F ( u ) = F ( v ) } . Consequently, using classical Richberg approximation [16, Chapter I, Lemma 5.18],one can take a “regularized maximum” of F ( u ) and F ( v ) to obtain β ∈ H ω satisfying F ( w ) ≥ β ≥ max( F ( u ) , F ( v )) . Since F : H ω → H ω is surjective, there exists a unique α ∈ H ω s.t. F ( α ) = β . Using (12)again, we obtain that max( u, v ) + δρ x = w ≥ α ≥ max( u, v ) . Since ∇ u ( x ) = ∇ v ( x ) and w ( x ) = α ( x ) = max( u, v )( x ), this is a contradiction with thesmoothness of α at x . Consequently, we need to have that G − u ( x ) = y = z = G − v ( x ), asdesired.In case ∇ u ( x ) = ∇ v ( x ), one finds q ∈ H ω (via small perturbation) such that u ( x ) = v ( x ) = q ( x ) and ∇ u ( x ) = ∇ q ( x ) along with ∇ v ( x ) = ∇ q ( x ). Then by the above we havethat G − u ( x ) = G − q ( x ) and G − v ( x ) = G − q ( x ), ultimately giving that G − u ( x ) = G − v ( x ) forany u, v ∈ H ω .Using Theorem 2.1, an integration along the curve t → tu gives that F ( u ) − F (0) = Z ( u ◦ g ) dt = u ◦ g, u ∈ H ω . (15)Returning to the statement of Theorem 2.1, we either have g ∗ ω u = ω F ( u ) , u ∈ H ω , or g ∗ ω u = − ω F ( u ) , u ∈ H ω .Assuming that g ∗ ω u = ω F ( u ) , using (15) we arrive at the identity g ∗ ( i∂ ¯ ∂u ) = i∂ ¯ ∂ ( u ◦ g ).Since after a dilation all elements of C ∞ ( X ) land in H ω , we obtain that actually g ∗ ( i∂ ¯ ∂v ) = i∂ ¯ ∂ ( v ◦ g ) for all v ∈ C ∞ ( X ). According to the next lemma g has to be holomorphic, implyingthat F = L g (see Section 2.3).In case g ∗ ω u = − ω F ( u ) , by a similar calculation we arrive at g ∗ ( i∂ ¯ ∂v ) = − i∂ ¯ ∂ ( v ◦ g ) forall v ∈ C ∞ ( X ). According to the next lemma g has to be anti-holomorphic, giving that F = N g (see Section 2.3), finishing the proof. Lemma 3.8.
Suppose that g : X → X is a smooth map.(i) If i∂ ¯ ∂ ( u ◦ g ) = g ∗ ( i∂ ¯ ∂u ) for all u ∈ C ∞ ( X ) then g is holomorphic.(ii) If i∂ ¯ ∂ ( u ◦ g ) = − g ∗ ( i∂ ¯ ∂u ) for all u ∈ C ∞ ( X ) then g is anti-holomorphic.Proof. We only show (i) as the proof of (ii) is analogous. We start with the followingcomputations expressed in local coordinates: i∂ ¯ ∂ ( u ◦ g ) = i ∂ ( u ◦ g ) ∂z j ∂z k dz j ∧ dz k = i ∂ u∂z a ∂z b (cid:20) ∂g a ∂z j ∂g b ∂z k + ∂g a ∂z k ∂g b ∂z j (cid:21) dz j ∧ dz k + i ∂ u∂z a ∂z b ∂g a ∂z j ∂g b ∂z k dz j ∧ dz k + i ∂ u∂z a ∂z b ∂g a ∂z j ∂g b ∂z k dz j ∧ dz k (16)+ i ∂u∂z b ∂ g b ∂z j ∂z k dz j ∧ dz k + i ∂u∂z b ∂ g b ∂z j ∂z k dz j ∧ dz k . g ∗ ( i∂ ¯ ∂u ) is a (1 ,
1) form we also have that g ∗ ( i∂ ¯ ∂u ) = i ∂ u∂z a ∂z b (cid:20) ∂g a ∂z j ∂g b ∂z k − ∂g a ∂z k ∂g b ∂z j (cid:21) dz j ∧ dz k . (17)Clearly, it is enough to show that g is holomorphic in local coordinate charts. By linearitywe can assume that i∂ ¯ ∂ ( u ◦ g ) = g ∗ ( i∂ ¯ ∂u ) holds for complex valued smooth functions u .Let x ∈ X , and we pick u such that in a coordinate neighborhood of x we have that u ( z ) = z b , b ∈ { , . . . , n } . Then i∂ ¯ ∂ ( u ◦ g ) = g ∗ ( i∂ ¯ ∂u ) gives that ∂ g b /∂z j ∂z k = 0 for all j, k ∈ { , . . . , n } at x . Similarly, after choosing u ( z ) = z b , b ∈ { , . . . , n } in a coordinateneighborhood of x , we obtain that ∂ g b /∂z j ∂z k = 0 for all j, k ∈ { , . . . , n } at x . Since x ∈ X was arbitrary, the terms in the last line of (16) vanish for any choice of u .Repeating this process for u ( z ) = z a z b and u ( z ) = ¯ z a ¯ z b , we conclude that the terms inthe second line of (16) vanish as well, for any choice of u .Revisiting the identity i∂ ¯ ∂ ( u ◦ g ) = g ∗ ( i∂ ¯ ∂u ) one more time, after picking u such that i∂ ¯ ∂u is positive definite in a neighborhood of x ∈ X , we obtain that ∂g a /∂z j = 0 for any a, j ∈ { , . . . , n } at X , implying that g is indeed holomorphic. We start with a lemma about the concatenation of geodesics in E ω : Lemma 4.1.
Suppose that [ − , ∋ t → v t ∈ E ω and [0 , ∋ t → u t ∈ E ω are d -geodesicssuch that u = v ∈ H ω and ˙ u = ˙ v ∈ L ( ω n ) . Then [ − , ∋ t → w t ∈ E ω , the concatena-tion of the curves t → u t and t → v t , is the d -geodesic joining v − , u ∈ E ω .Proof. By possibly changing the background metric, we can assume that u = v = 0. Fromthe L version of [1, Lemma 3.4(ii)], (whose proof is identical to the L version, presented in[1]) we have that d ( v − , = Z X | ˙ u | ω n = Z X | ˙ v | ω n = d (0 , u ) . (18)Next we point out that d ( v − , u ) = d ( v − ,
0) + d (0 , u ) . (19)Indeed, from the triangle inequality we have that d ( v − , u ) ≤ d ( v − ,
0) + d (0 , u ). Thereverse inequality follows from (18) and [14, Theorem 3.1]: d ( v − ,
0) + d (0 , u ) = (cid:18) Z X | u | ω n (cid:19) ≤ d ( v − , u ) . Due to uniqueness of d -geodesic segments, we only need to show that for any a, b ∈ [ − , a < b we have that d ( w a , w b ) = b − a d ( v − , u ) = ( b − a ) d (0 , u ) = ( b − a ) d ( v − , . (20)14ince t → u t and t → v t are d -geodesics, we only need to treat the case a ∈ [ − , b ∈ [0 , d ( v a , u b ) ≥ (cid:18) Z X | ( b − a ) ˙ u | ω n (cid:19) = ( b − a ) d (0 , u ) . The reverse inequality follows from the triangle inequality: d ( v a , u b ) ≤ d ( v a , d (0 , u b ) =( b − a ) d (0 , u ). Proof of Theorem 1.5.
By changing the background metric, we can assume without loss ofgenerality that φ = 0. From (5) it follows that t → φ t + Ct is a d -geodesic for any C ∈ R .As a result, we can also assume that φ ≤ d -geodesic [ − ε, ∋ t → φ t ∈ E ω , as described in the statement of the theorem.First we show that φ − ε ≥
0. This is a simple consequence of the t -convexity. By theresults of [11] (see the discussion near (6)) there exists a set Z ⊂ X of measure zero suchthat for all x ∈ X \ Z we have that t → φ t ( x ) is convex, φ ( x ) = 0, lim t ր φ t ( x ) = φ ( x ) ≤ t ց− ε φ t ( x ) = φ − ε ( x ). Due to t -convexity, we obtain that φ − ε ( x ) ≥ Z .As φ − ε ∈ PSH(
X, ω ), we obtain that φ − ε ≥ φ − ε is usc, it follows that sup X φ − ε < + ∞ , i.e., φ − ε ∈ L ∞ . Using (5) for the d -geodesic joining φ − ε and φ , it follows that φ t ≥ φ − ε − ε − tε sup X φ − ε , t ∈ [ − ε, . Since ( − ε, ∋ t → φ t ( x ) is t -convex for all x ∈ X \ Z , it follows that the above estimateextends to t ∈ [ − ε, φ ∈ E ω \ L ∞ . Proof of Theorem 1.4.
We can assume without loss of generality that φ = 0.To derive a contradiction, we further assume that there exists a metric L symmetry F : V → V , as described in the statement of the theorem.Since V is d -open, it follows that 0 ∈ B (0 , δ ) ⊂ V for some δ >
0, where B (0 , δ )is the d -ball of radius δ centered at 0. As F is a metric L symmetry it follows that F : B (0 , δ ) → B (0 , δ ) is bijective.Let ψ ∈ B (0 , δ ) such that ψ ∈ E ω \ L ∞ . One can find such ψ as a consequence of [12,Theorem 3]. Let [0 , ∋ t → ψ t , F ( ψ t ) ∈ B (0 , δ ) be the d -geodesics connecting 0 and ψ ,respectively 0 and F ( ψ ).Since F is a metric L symmetry, by definition we have that ˙ ψ = − ˙ F ( ψ ). Consequently,according to Lemma 4.1, the concatenation [ − , ∋ t → w t ∈ B (0 , δ ) of the curves t → F ( ψ − t ) and t → ψ t is a d -geodesic. But then t → w t extends t → ψ t at t = 0, giving acontradiction with Theorem 1.5. References [1] R. Berman, T. Darvas, C.H. Lu, Regularity of weak minimizers of the K-energy and applications toproperness and K-stability, preprint, arXiv:1602.03114.
2] B. Berndtsson, D. Cordero–Erauskin, B. Klartag, Y.A. Rubinstein, Complex Legendre duality,arXiv:1608.05541, Amer. J. Math., to appear.[3] B. Berndtsson, D. Cordero–Erauskin, B. Klartag, Y.A. Rubinstein, Complex interpolation of R -norms,duality and foliations, arXiv:1607.06306, J. Euro. Math. Soc., to appear.[4] Z. B locki, On geodesics in the space of K¨ahler metrics, in: Advances in Geometric Analysis (S. Janeczkoet al., Eds.), International Press, 2012, pp. 3–20.[5] Z. B locki, S.Ko lodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer.Math. Soc. 135 (2007), 2089–2093.[6] M. Bridson, A. Haefliger, Metric spaces of non positive curvature. Grundlehren der Math. Wiss. 319(1999).[7] X.X. Chen, The space of K¨ahler metrics, J. Differential Geom. 56 (2000), 189–234.[8] E. Calabi, X.X. Chen, The space of Kahler metrics. II., J. Differential Geom. 61 (2002), no. 2, 173193.[9] J. Chu, V. Tosatti, B. Weinkove, On the C , regularity of geodesics in the space of K¨ahler metrics,Ann. PDE 3 (2017), no.2, 3:15.[10] T. Darvas, Weak geodesic rays in the space of K¨ahler potentials and the class E ( X, ω ), J. Inst. Math.Jussieu 16 (2017), no. 4, 837–858.[11] T. Darvas, The Mabuchi completion of the space of K¨ahler potentials, Amer. J. Math. 139 (2017),1275–1313.[12] T. Darvas, The Mabuchi geometry of finite-energy classes, Adv. Math. 285 (2015), 182–219.[13] T. Darvas, Geometric pluripotential theory on K¨ahler manifolds, arXiv:1902.01982.[14] T. Darvas, C.H. Lu, Uniform convexity in L p University of Maryland [email protected]@math.umd.edu