aa r X i v : . [ m a t h . DG ] O c t ON THE STRICT CONVEXITY OF THE K-ENERGY
ROBERT J. BERMAN
Abstract.
Let ( X, L ) be a polarized projective complex manifold. We show, by a simpletoric one-dimensional example, that Mabuchi’s K–energy functional on the geodesically com-plete space of bounded positive (1 , − forms in c ( L ) , endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some furtherassumptions the strict convexity in question does hold in the toric case. This leads to auniqueness result saying that a finite energy minimizer of the K-energy (which exists on anytoric polarized manifold ( X, L ) which is uniformly K-stable) is uniquely determined moduloautomorphisms under the assumption that there exists some minimizer with strictly positivecurvature current. Introduction
Let ( X, L ) be a polarized compact complex manifold and denote by H the space of allsmooth metrics φ on the line bundle L with strictly positive curvature, i.e the curvature two-form ω φ of φ defines a Kähler metric on X. A leading role in Kähler geometry is played byMabuchi’s K-energy functional M on H , which has the property that a metric φ in H is acritical point for M if and only if the Kähler metric ω φ has constant scalar curvature [25].From the point of view of Geometric Invariant Theory (GIT) the K-energy can, as shown byDonaldson [18], be identified with the Kempf-Ness “norm-functional” for the natural actionof the group of Hamiltonian diffeomorphisms on the space of all complex structures on X, compatible with a given symplectic form.As shown by Mabuchi the functional M is convex along geodesics in the space H endowedwith its canonical Riemannian metric (the Mabuchi-Semmes-Donaldson metric). More pre-cisely, as indicated by the GIT interpretation, M is strictly convex modulo the action of theautomorphism group G of ( X, L ) in the following sense: let φ t be a geodesic in H (parametrizedso that t ∈ [0 , then(1.1) t
7→ M ( φ t ) is affine iff φ (t)= g ( t ) φ , where g ( t ) φ denotes the action on φ of a one-parameter subgroup in G, i.e. φ t is equal tothe pull-back of φ under the flow of a holomorphic vector field on X. However, the boundaryvalue problem for the geodesic equation in H does not, in general, admit strong solutions [24].In order to bypass this complication Chen introduced a natural extension of M to the largerspace H , consisting of all (singular) metrics φ on L such that the curvature ω φ is definedas an L ∞ − form. The advantage of the latter space is that it is geodesically convex (in thesense of metric spaces; see [15] and references therein). It was conjectured by Chen [13] andconfirmed in [3] that M is convex on H , . However, the question whether M is strictly convexmodulo the action of G on H , was left open in [3]. In this short note we give a negativeanswer to the question already in the simplest case when X is the Riemann sphere and themetrics are S − invariant. Theorem 1.
Let L → X be the hyperplane line bundle over the Riemann sphere. There existsan S − invariant geodesic φ t in H , such that M ( φ t ) is affine, but φ t is not of the form g ( t ) φ .
1N THE STRICT CONVEXITY OF THE K-ENERGY 2
In terms of the standard holomorphic coordinate z on the affine piece C of X the geodesic φ t in the previous theorem can be taken so that φ t is equal to the Fubini-Study metric φ onthe lower-hemisphere, flat on a t − dependent collar attached to the equator and then glued toa one-parameter curve of the form g ( t ) ∗ φ in the remaining region of the upper hemi-sphere.The failure of the strict convexity of M on H , modulo G appears to be quite surprising inview of the fact that the other canonical functional in Kähler geometry - the Ding functional D - is strictly convex on H , modulo G. In fact, the Ding functional (which is only definedin the “Fano case”, i.e. when L is the anti-canonical line bundle over a Fano manifold) is evenstrictly convex modulo the action of G on the space of all L ∞ − metrics on L with positivecurvature current [9] .One important motivation for studying strict convexity properties of the Mabuchi func-tional M on suitable completions of H comes from the Yau-Tian-Donaldson conjecture.
Inits uniform version the conjecture says that the first Chern class c ( L ) of L contains a Kählermetric of constant scalar curvature iff ( X, L ) is uniformly K-stable (in the L − sense) relativeto a maximal torus of G [10, 17]. The “only if” direction was established in [8] when G is trivial(and a similar proof applies in the general case [5]). The proof in [8] uses the convexity of M onthe finite energy completion E of H . The remaining implication in the Yay-Tian-Donaldsonconjecture is still widely open, in general, but a first step would be to establish the existenceof a minimizer of M in the finite energy space E , by generalizing the variational approach tothe “Fano case” introduced in [4]. Leaving aside the challenging question of the regularity of aminimizer one can still ask if the minimizer is canonical, i.e. uniquely determined modulo G ? (as conjectured in [16]). The uniqueness in question would follow from the strict convexity of M on E modulo the action of G. However, by the previous theorem such a strict convexitydoes not hold, in general. On the other hand, it would be enough to establish the followingweaker strict convexity property: t
7→ M ( φ t ) is constant = ⇒ φ ( t ) = g ( t ) φ , (the converse implication holds if ( X, L ) is K-stable). This may still be too optimistic, but herewe observe that this approach towards the uniqueness problem can be made to work in thetoric case if one assumes some a priori positivity of the curvature current of some minimizer. Theorem 2.
Let ( X, L ) be an n − dimensional polarized toric manifold. Assume that ( X, L ) is uniformly K-stable relative the torus action. Then there exists a finite energy minimizer φ of M and the minimizer is unique modulo the action of C ∗ n under the assumption that thereexists some finite energy minimizer φ whose curvature current is strictly positive on compactsof the dense open orbit of C ∗ n in X. Relations to previous results.
In view of its simplicity it is somewhat surprising thatthe counterexample in Theorem 1 does not seem to have been noticed before. The key pointof the proof is a generalization of Donaldson formula for the Mabuchi functional M in thesmooth toric setting to a singular setting (see Lemma 5 and Lemma 7), showing that(1.2) M ( φ ) = F ( u ) , where the non-linear part of the functional F only depends on the non-singular part (in thesense of Alexandrov) of the Hessian of the convex function on the moment polytope of ( X, L ) , corresponding to the metric φ. This leads, in fact, to a whole class of counter-examples to the The space E was originally introduced in [22] from a pluripotential point of view and, as shown in [15], E may be identified with the metric completion of H with respect to the L − Finsler version of the Mabuchi-Semmes-Donaldson metric on H N THE STRICT CONVEXITY OF THE K-ENERGY 3 strict convexity in question, by taking φ t to be any torus invariant geodesic φ t , emanatingfrom a given φ ∈ H , whose Legendre transform u t is of the form u t = u + tv, for a convex and piece-wise affine function v. Such a curve defines a geodesic ray associatedto a toric test configuration for ( X, L ) and the fact that φ t ∈ H , then follows from generalresults (see [14, 27, 28, Section 7]). A by-product of formula 1.2 is a slope formula for theK-energy along toric geodesics of finite energy (see Section 7).The functional F has previously appeared in a series of papers by Zhou-Zhu (see Remark 8).In particular, it was shown in [31, Section 6] (when n = 2) that a minimizer u of F is uniquelydetermined, modulo the complexified torus action, under the stronger assumption that thereexists some minimizer u of F which is C ∞ − smooth and strictly convex in the interior (whileour assumption is equivalent to merely assuming that u is C , − smooth in the interior).We also recall that in the toric surface case (i.e when n = 2) it was shown by Donaldson[19] that uniform K-stability is equivalent to K-stability. Moreover, the Yau-Tian-Donaldsonin the latter case was settled by Donaldson in a series of papers culminating in [21] (as aconsequence, any minimizer of M is then smooth and positively curved). Theorem 2 shouldalso be compared with the general uniqueness result for finite energy minimizers of M on anyKähler manifold which holds under the assumption that there exists some minimizer whichis smooth and strictly positively curved. The proof of the latter result, which generalizes theuniqueness result in [3], exploits a weaker form of strict convexity which holds on the linearizedlevel around a bona fide metric with constant scalar curvature. Acknowledgment.
This work was supported by grants from the ERC and the KAW foundation.I am grateful to Long Li for comments on the first version of the paper.2.
Proofs
We start with some preparations. To keep things as simple as possible we mainly stick tothe one-dimensional situation (see [2] for the general convex analytical setup and its relationsto polarized toric varieties).2.1.
Convex preparations.
Let φ ( x ) be a lower semi-continuous (lsc) convex function on R (taking values in ] − ∞ , ∞ ]) . Its point-wise derivative φ ′ ( x ) exists a.e. on R and defines anelement in L ∞ loc ( R ) . We will denote by ∂φ the subgradient of φ, which is a set-valued map on R with the property that ( ∂φ )( x ) is a singleton iff φ ′ ( x ) exists at x. Similarly, we will denote by ∂ φ the measure on R defined by the second order distributional derivative of φ. By Lebesgue’stheorem we can decompose ∂ φ = ∂ s φ + φ ′′ , where ∂ s φ denotes the singular part of the measure ∂ φ and φ ′′ ∈ L loc ( R ) denotes the regularpart (wrt Lebesgue measure dx ), which coincides with the second order derivative of φ almosteverywhere on x. We set(2.1) φ ( x ) := log(1 + e x ) and P + ( R ) := { φ : φ convex on R , φ − φ ∈ L ∞ ( R ) } ( ∂ φ is a probability measure for any such φ ). Given a function φ in P + ( R ) we will denote by u its Legendre transform which defines a finite lsc convex function u on [0 , (which is equal N THE STRICT CONVEXITY OF THE K-ENERGY 4 to ∞ on ]0 , c ) u ( y ) := ( φ ∗ )( y ) := sup x ∈ R xy − φ ( x ) Since φ = u ∗ the map φ u gives a bijection P + ( R ) ←→ { u : u convex on [0 , } ∩ L ∞ [0 , which is an isometry wrt the L ∞ − norms. Moreover, φ is smooth and strictly convex on R iff u is smooth and strictly convex on ]0 , , as follows from the formula(2.2) φ ( x ) = xy − u ′ ( y ) , if x = u ′ ( y ) and u is differentiable at y (and vice versa if φ is replaced by u ) . Moreover, if u istwo times differentiable at y = 0 and u ′′ ( y ) > then φ ′′ is differentiable at x and(2.3) φ ′′ ( x ) = 1 /u ′′ ( y ) Complex preparations.
Let ( X, L ) := ( P , O (1)) be the complex projective line P endowed with the hyperplane line bundle O (1) . Realizing P as the Riemann sphere (i.e. theone-point compactification of the complex line C ) a locally bounded metric Φ on O (1) may,in the standard way, be identified with a convex function Φ( z ) on C such that Φ − Φ ∈ L ∞ ( C ) , Φ ( z ) := log(1 + | z | ) , where Φ corresponds to the Fubini-Study metric on O (1) , which defines a smooth metricwith strictly positively curvature ω on P (coinciding with the standard SU (2) − invarianttwo-form on P ) . Moreover, the metric Φ on O (1) has semi-positive curvature ω Φ on P iff Φ( z ) is subharmonic on C . More precisely,(2.4) ω Φ | C = i π ∂ ¯ ∂ Φ := i π ∂ Φ ∂z∂ ¯ z dz ∧ ¯ dz We will denote by H S b the space of all bounded (i.e L ∞ ) metrics Φ on O (1) → P withsemi-positive curvature which are S − invariant (wrt the standard action of S ) . Setting x := log | z | gives a correspondence H S b ←→ P + ( R ) , Φ( z ) φ ( x ) := Φ( e x ) As is well-known, under the Legendre transform the Mabuchi-Donaldson-Semmes metric on H T corresponds the standard flat metric induced from L [0 , . It follows that a geodesic Φ t in H S b corresponds to a curve φ t in P + ( R ) with the property that the corresponding curve u t := φ ∗ t of bounded convex functions on [0 , is affine wrt t. In particular, taking φ to be defined by 2.1, the following curve defines a geodesic in H S b : φ t ( x ) := ( u + tv ) ∗ , u ( y ) := φ ∗ ( y ) = y log y + (1 − y ) log(1 − y ) , where v is the following convex piece-wise affine function on [0 ,
1] : v ( y ) := 0 , y ∈ [0 ,
12 ] , v ( y ) = −
12 + y, y ∈ [ 12 , N THE STRICT CONVEXITY OF THE K-ENERGY 5
Lemma 3.
Assume that t ∈ ]0 , . Then φ t defines an S − invariant metric on O (1) → P which is C , − smooth on P with semi-positive curvature and C ∞ − smooth and strictly posi-tively curved on the complement of a t − dependent neighborhood of the equator. More precisely,in the logarithmic coordinate x ∈ R we have φ t ( x ) = φ ( x ) when x ≤ and φ t ( x ) = φ (0)+ x/ when x ∈ [0 , t ] and φ t ( x ) = φ ( x − t ) + t/ when x ≥ t. Proof. Step 1: φ t is in C , ( R ) . This step only uses that v is convex and bounded on [0 , . Let v ( j ) be a sequence ofsmooth strictly convex function on [0 , such that (cid:13)(cid:13) v ( j ) − v (cid:13)(cid:13) L ∞ [0 , → . Set u ( j ) ( t ) := u + tv ( j ) and φ ( j ) := u ( j ) ∗ . Since the Legendre transform preserves the L ∞ − norm we have (cid:13)(cid:13) φ ( j ) − φ (cid:13)(cid:13) L ∞ ( R ) → . By construction, u ( j ) t is smooth and satisfies ( u ( j ) t ) ′′ ≥ ( u ) ′′ ≥ /C > . Hence, by 2.3, ( φ ( j ) t ) ′′ ≤ C and letting j → ∞ thus implies that ∂ φ = φ ′′ ≤ C, showing that φ t is in C , ( R ) . Step 2:
Explicit description of φ t We fix t > and observe that the map y u ′ t ( y ) induces two diffeomorphisms (withinverses x φ ′ t ( x )) (2.5) u ′ t : Y − :=]0 ,
12 [ → X − :=] − ∞ , , Y + :=] 12 , → X − :=] t, ∞ [ This follows directly from the fact that u ′ t is strictly positive on Y ± and converges to and t when y → / from the left and right, respectively. We claim that this implies, by generalprinciples, that the restriction of φ t to X ± only depends on the restriction of u t to Y ± . Indeed,if x = u ′ t ( y ) for y ∈ Y ± then formula 2.2 shows that φ t | X ± only depends on the restriction of u to X − . Hence, the restriction of φ t to X − is given by u ∗ = φ and the restriction of φ t to X + is given by the Legendre transform of u ( y ) + t ( y − / , which is equal to φ ( x − t ) + t/ . Finally, it follows from the diffeomorphism 2.5 (using, for example, that φ ′ t is increasing) that φ ′ t = 1 / on [0 , t ] . (cid:3) Remark . A more symmetric form of the geodesic φ t may be obtained by setting ˜ φ t ( x ) :=2 φ t ( x ) − x, which has the property that ˜ φ t ( x ) := ˜ φ ( x ) := log( e − x + e x ) when x ≤ and ˜ φ t ( x ) = ˜ φ (0) when x ∈ [0 , t ] and ˜ φ t ( x ) = ˜ φ ( x − t ) when x ≥ t. Geometrically, ˜ φ t defines ageodesic ray of metrics on O (2) → P , expressed in terms of the trivialization of O (2) over C ∗ ⊂ P induced from the embedding C ∗ → C → P defined by F ( z ) := ( z − , z ) ∈ C , where X is identified with the closure F ( C ) of F ( C ) in P and O (2) with the restrictionof O P (1) to F ( C ) . A direct calculation reveals that ˜ φ t ( x ) (and hence also φ t ( x )) is in fact C , − smooth when viewed as a function on R × R . This implies that the Laplacian of thecorresponding local potentials over P × D ∗ (where D ∗ denotes the punctured unit disc withholomorphic coordinate τ such that t := − log | τ | ) is locally bounded, i.e. the geodesic hasChen’s regularity [12] in the space-time variables. It should also be pointed out that ˜ φ t canbe realized as the geodesic ray, emanating from the Fubini-Study metric, associated to thetoric test configuration of ( X, L ) := ( P , O (2)) determined (in the sense of [19, 14, 28]) by thepiece-wise affine function ˜ v ( y ) = max { , y } on the moment polytope [ − , of ( P , O (2)) (asin [14]). Using this realization the C , − regularity also follows from the general results in[27, 14] which show that the Laplacian (or equivalently complex Hessian) of the correspondingpotential is locally bounded over X × D ∗ . Indeed, in the toric setting boundedness of thecomplex Hessian is equivalent to boundedness of the real Hessian, i.e. to C , − regularity.2.2.1. The K-energy.
Let ( X, L ) be a polarized compact complex manifold. We recall thatthe K-energy functional was originally defined by Mabuchi [25] on the space H of all smooth N THE STRICT CONVEXITY OF THE K-ENERGY 6 metrics Φ on L with strictly positive curvature by specifying its differential (more precisely,this determines M up to an additive constant). Chen extended M to the space H , consistingof all (singular) metrics φ on L such that the curvature ω Φ of Φ is defined as an L ∞ − form[13]. The extension is based on the Chen-Tian formula for M on H which may be expressedas follows in terms of a fixed Kähler form ω on X : (2.6) M ( u ) = (cid:18) ¯ Rn + 1 E (Φ) − E Ric ω (Φ) (cid:19) + H ω n ( ω n Φ ) , ¯ R := nc ( X ) · [ ω ] n − [ ω ] n , where(2.7) H µ ( µ ) := Z X log (cid:18) µµ (cid:19) µ and E and E Ric ω are defined, up to an additive constant, by their differentials on H : (2.8) d E | Φ = ( n + 1) ω n Φ , d E Ric ω | Φ = nω n − ∧ Ric ω with Ric ω denoting the two-form defined by the Ricci curvature of ω (see [3] for a simpledirect proof of the Chen-Tian formula). The extension of M to H , is obtained by observingthat both terms appearing in the rhs of formula 2.6 are well-defined (and finite) when Φ ∈ H , . We note that the functional appearing in the first bracket of the formula is continuous wrt the L ∞ − norm on H , . Indeed, it follows readily from the definitions that both E and E Ric ω areeven Lip continuous wrt the L ∞ − norm.In the present setting where X = P we can, for concreteness, take ω = ω Φ , whoserestriction to C is equal to a constant times e − dz ∧ d ¯ z. Conclusion of proof of Theorem 1.
The proof will follow from the following extensionto H S , of a formula due to Donaldson when Φ ∈ H S [19, Prop 3.2.8]. Lemma 5.
Assume that Φ is in H S , . Then M (Φ) = L ( u ) − Z [0 , log( u ′′ ( y )) dy, L ( u ) = 12 ( u (1) + u (0)) − Z u ( y ) dy where u ′′ ∈ L loc denotes the non-singular part of ∂ u. Proof. Step 1:
Assume that Φ ∈ H S , . Then M (Φ) = L ( u ) + Z R φ ′′ ( x ) log φ ′′ ( x ) dx In the case when Φ ∈ H S (or more generally when u is continuous on [0 , and smoothand strictly convex in the interior) this follows from Donaldson’s formula [19]. To extend theformula to the case when Φ ∈ H S , first observe that(2.9) Z φ ( x ) φ ′′ ( x ) dx < ∞ , as follows directly from estimating φ ′′ ≤ Cφ ′′ ≤ Ae −| x | /B and φ ( x ) ≤ | x | + C. Hence, we canrewrite the Chen-Tian formula 2.6 as(2.10) M (Φ) = E (Φ) + Z R φ ′′ ( x ) log φ ′′ ( x ) dx, N THE STRICT CONVEXITY OF THE K-ENERGY 7 where E (Φ) = (cid:18) ¯ Rn + 1 E (Φ) − E Ric ω (Φ) (cid:19) + 2 Z R φ ( x ) φ ′′ ( x ) dx, Now take a sequence Φ j ∈ H S such that k Φ j − Φ k L ∞ → (which equivalently means that k u j − u k L ∞ [0 , → and ω Φ j ≤ Cω Φ , i.e.(2.11) φ ′′ j ( x ) ≤ Cφ ′′ ( x ) We claim that(2.12) E (Φ j ) → E (Φ) . Indeed, as pointed out above the first term appearing in the definition of E is continuouswrt the L ∞ − norm. To handle the second term first observe that, since k Φ j − Φ k L ∞ ( X ) → , the probability measures φ ′′ j ( x ) dx converge weakly towards φ ′′ ( x ) dx and hence, for any fixed R > , lim j →∞ Z | x |≤ R φ ( x ) φ ′′ j ( x ) dx = Z | x |≤ R φ ( x ) φ ′′ ( x ) dx Moreover, the uniform bound 2.11 gives lim sup R →∞ lim sup j →∞ Z | x |≥ R φ φ ′′ j ( x ) dx ≤ C lim sup R →∞ Z | x |≥ R φ ( x ) φ ′′ ( x ) dx = 0 Hence, letting first j → ∞ and then R → ∞ proves 2.12.Now take a sequence Φ j ∈ H S such that k Φ j − Φ k L ∞ → which equivalently means that k u j − u k L ∞ [0 , → . By Donaldson’s formula E ( φ j ) = L ( u j ) and since both sides are continuous wrt the convergence of Φ j towards Φ this concludes theproof of Step 1. Step 2:
Let φ be a convex function on R such that ∂ φ is a probability measure which isabsolutely continuous wrt dx. Then(2.13) Z R φ ′′ ( x ) log φ ′′ ( x ) dx = − Z [0 , log( u ′′ ( y )) dy, if the left hand side is finite (and then u ′′ ( y ) > a.e.).This formula is a special case of McCann’s Monotone change of variables theorem [26,Theorem 4.4]. But it may be illuminating to point out that a simple direct proof can begiven in the present setting when φ is of the form φ t appearing in Lemma 3. Indeed, then ρ := φ ′′ = 0 on a closed intervall I of R and φ ′ diffeomorphism of the complement I onto ]0 , −{ / } . Since ρ log ρ = 0 if ρ = 0 the formula 2.13 then follows directly from making thechange of variables y = φ ′ ( x ) on R − S. (cid:3) Now, let Φ t be the geodesic in H S , defined by the curve φ t appearing in Lemma 3. Since v is piece-wise affine we have u ′′ t = u ′′ a.e on R and hence the previous lemma gives M (Φ t ) = − Z [0 , log( u ′′ ( y )) dy + t L ( v ) which is affine in t. Moreover, φ t is not induced from the flow of a holomorphic vector field(since this would imply that v is affine on all of [0 , . This concludes the proof of Theorem 1.
N THE STRICT CONVEXITY OF THE K-ENERGY 8
Remark . The functional E in formula 2.10 coincides with the (attractive) Newtonian energyof the measure µ = ∂ φ : E ( µ ) = 14 Z R | x − y | µ ( x ) ⊗ µ ( y ) and the continuiuty property of E used in the in Step 1 can be alternatively deduce fromthe fact that E is continuous on the space P ( R ) of all probability measures with finite firstmoment (endowed with the L − Wasserstein topology). This point of view is further developedin the higher dimensional toric setting in [1].2.4.
Proof of Theorem 2.
In this higher dimensional setting we will be rather brief andrefer to [1] for more details. Let ( X, L ) be an n − dimensional toric manifold and denote by P the corresponding moment lattice polytope in R n which contains in its interior. We willdenote by dσ the measure on ∂P induced from the standard integer lattice in R n (which iscomparable with the Lebesgue measure on ∂P ) [19]. The n − dimensional real torus acting on ( X, L ) will be denoted by T. As above we can then identify a T − invariant metric Φ on L withpositive curvature current with a convex function φ ( x ) on R n (whose sub-gradient maps into P ) and, via the Legendre transform, with a convex function u on P. We will denote by ∂ φ the distributional Hessian of φ and by ( ∇ φ )( x ) the Alexandrov Hessian of φ which is definedfor almost all x (on the subset where φ is finite).Assume that ( X, L ) is uniformly K-stable relative to the torus T (in the L − sense). Con-cretely, this means (see [23]) that there exists δ > such that for any rational piece-wise affineconvex function u on P, (2.14) L ( u ) := Z P udy − c Z ∂P u ≥ δ inf l ∈ ( R n ) ∗ (cid:18)Z P ( u − l ) dy − inf P ( u − l ) (cid:19) , c := Z P dy/ Z ∂P dσ, where the inf ranges over all linear functions l on R n (which, geometrically, may be identifiedwith an element of the real part of the Lie algebra of the complex torus). We note that,by a standard approximation argument, the inequality 2.14 holds for the space C ( P ) of allconvex functions u on P such that u ∈ L ( P ) ∩ L ( ∂P ) (where u | ∂P ( y ) is defined as the radialboundary limit of u ) . The uniform K-stability implies, by [29, 23], that M is coercive relativeto T, i.e. there exist C > such that the following coercivity inequality holds on H T : M (Φ) ≥ inf g ∈ C ∗ n J ( g Φ) /C − C, where J denotes Aubin’s J − functional. The functional M admits a canonical extension tothe space E of all (singular) metrics on L with positive curvature current and finite energy(namely, the greatest lsc extension of M from H to E , endowed with the strong topology[6, 7]). The coercivity of M combined with the results in [6] (which show that M is lsc wrtthe weak topology on E ) implies that there exists a T − invariant minimizer Φ of M on thespace E ( X, L ) of all (singular) metrics on L with positive curvature current and finite energy.A generalization of the argument used in the proof of Lemma 5 gives the following lemmawhich extends Donaldson’s formula in [19] to the finite energy setting (the proof is given in[1]): Lemma 7.
Assume that Φ ∈ E ( X, L ) T and M (Φ) < ∞ . Then u ∈ C ( P ) and (2.15) M (Φ) = F ( u ) := L ( u ) − Z P log( ∇ u ( y )) dy, In fact, if u is convex on P and in L ( ∂P ) , then automatically u ∈ L ( P ) . N THE STRICT CONVEXITY OF THE K-ENERGY 9 where ∇ u denotes the Alexandrov Hessian of u and both terms are finite (in particular, ∇ u ( y ) > a.e. on P ) . Remark . The functional F on C ( P ) has previously been studied in a series of papers byZhou-Zhu (see [30, 29]). In particular it was shown in [30] that F admits a minimizer u. Butthe point of the previous formula is that it identifies F with the Mabuchi functional on thespace E ( X, L ) T . As a byproduct this gives a new proof of the existence of a minimizer u of F . Let now Φ and Φ be two given minimizers of M in E ( X, L ) T and denote by Φ t thecorresponding geodesic in E ( X, L ) T (which corresponds to u t := u + t ( u − u ) under theLegendre transform). By the previous lemma the function t
7→ M (Φ t ) decomposes in twoterms, where the first term is affine in t and the second one is convex. Since M (Φ t ) isconstant (and in particular affine) it follows that the second term, t
7→ − Z P log(det ∇ u t ( y )) dy is also affine. But this forces, using the arithmetic-geometric means inequality, that(2.16) ∇ u = ∇ u a.e. on P .
As a consequence the previous function in t is, in fact, constant. Since M (Φ t ) is also constantin t formula 2.15 forces L ( u t ) = L ( u ) for all t. Setting v := u − u this means that L ( v ) = 0 . Now, if v is convex, then it follows form the assumption of uniform relative K-stability that v is affine and hence Φ and Φ coincide modulo the action of C ∗ n . All that remains is thus toshow that v is convex. To this end we invoke the assumption that the distributional Hessianof φ satisfies ∇ φ ≥ C K I on any given compact subset K of R n . We claim that this implies that u ∈ C , loc ( P ) . Indeed,since Φ has finite energy it has full Monge-Ampère mass and hence the closure of the sub-gradient image ( ∂ Φ )( R n ) is equal to P. It follows (just as in the proof of Lemma 3) that ∂ u = ∇ u ≤ C − K I on the closure of ( ∂ Φ )( K ) in P. Since K was an arbitrary compact subset of R n it follows that u ∈ C , loc ( P ) . The proof of the theorem is now concluded by invoking the following lemma(see [26, Lemma 3.2]):
Lemma 9.
Let u and u be two finite convex functions on an open convex set P ⊂ R n suchthat u ∈ C , loc ( P ) and the Alexandrov Hessians satisfy 2.16. Then u − u is convex. A generalized slope formula for the K-energy.
We conclude the paper by observingthat a by-product of Lemma 7 is the following generalization of the slope formula for the K-energy in [11] (which concerns the case when Φ t is defined by a bona fide metric on a testconfiguration) to the present singular setting: Proposition 10. (Slope formula) Let Φ t be a geodesic ray in E ( X, L ) T such that Φ ∈H ( X, L ) T and M (Φ t ) < ∞ for any t ∈ [0 , ∞ [ . Then lim t →∞ t − M (Φ t ) = L ( v ) < ∞ where u t = u + tv is the curve of convex functions in L ( ∂P ) corresponding to Φ t underLegendre transformation. N THE STRICT CONVEXITY OF THE K-ENERGY 10
Proof.
Since M (Φ t ) < ∞ Lemma 7 shows that u t = u + tv ∈ L ( ∂P ) for all t ≥ , where v := u − u . Moreover, since u t is convex for any t ≥ it also follows that v is convex and v ∈ L ( ∂P ) . Now, since ∂ u is invertible we can, denoting the inverse by A ( y ) , write Z log(det ∇ ( u + tv )( y )) dy = C + Z P log(det(1 + tA ( y ) ∇ v ( y )) dy, which is finite for any t (by Lemma 7). Moreover, since ∇ v ( y ) ≥ we have, when t ≥ , that ≤ Z P log(det( I + tA ( y ) ∇ v ( y )) dy ≤ Vol ( P ) n log t + Z P log(det( I + A ( y ) ∇ v ( y )) dy, where all terms are finite. Hence, dividing by t and letting t → ∞ concludes the proof of theproposition. (cid:3) In the terminology of [10, 11, 4] this formula shows that the slope of the Mabuchi functionalalong a finite energy geodesic is equal to the Non-Archimedean Mabuchi functional of thecorresponding (singular) Non-Archimedean metric. It would be very interesting to extend thisslope formula to the non-toric setting. Indeed, this is the key missing ingredient when tryingto extend the variational approach to the (uniform) Yau-Tian-Donaldson conjecture in the“Fano case” in [4] to a general polarized manifold ( X, L ) , in order to produce a finite energyminimizer of M . References [1] Berman, R: A probabilistic tropical analog of the Yau-Tian-Donaldson conjecture. (In preparation).[2] Berman, R.J, Berndtsson, B: Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fanovarieties. Ann. Math. de la fac. des sciences de Toulouse. (2013) vol. 22 n° 4[3] Berman, R.J.; Berndtsson, B: Convexity of the K-energy on the space of Kähler metrics and uniquenessof extremal metrics. J. Amer. Math. Soc. 30 (2017), no. 4, 1165–1196[4] Berman, R.J; Boucksom, S; Jonsson, M: A variational approach to the Yau-Tian-Donaldson conjecture.arXiv:1509.04561[5] Berman, R.J; Boucksom, S; T. Hisamoto; Jonsson, M. Work in progress.[6] R.J.Berman ; P: S. Boucksom; Eyssidieu, V. Guedj, A. Zeriahi: Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties. Crelle’s Journal (to appear).[7] Berman, R.J; Darvas, T; Lu, C.H: Convexity of the extended K-energy and the large time behaviour ofthe weak Calabi flow. Geometry & Topology 21 (2017) 2945–2988[8] R.J. Berman, T. Darvas, C.H.Lu: Regularity of weak minimizers of the K-energy and applications toproperness and K-stability. arXiv:1602.03114[9] Berndtsson, B: A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems inKähler geometry. Invent. Math. 200 (2015), no. 1, 149–200.[10] S. Boucksom, T. Hisamoto and M. Jonsson: Uniform K-stability, Duistermaat-Heckman measures andsingularities of pairs. Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841.[11] S. Boucksom T. Hisamoto and M. Jonsson: Uniform K-stability and asymptotics of energy functionals inK¨ahler geometry (JEMS, to appear). arXiv:1603.01026[12] Chen, X.X. The space of K¨ahler metrics. J. Differential Geom. 56 (2000), no. 2, 189–234.[13] Chen, X.X: On the lower bound of the Mabuchi energy and its application , Int. Math. Res. Not. 2000,no. 12, 607-623[14] Chen, X.X., Tang, Y. Test configuration and geodesic rays. In Geometrie differentielle, physique mathe-matique, mathematiques et societe. I. , Asterisque No. 321 (2008), 139– 167.[15] T Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015)[16] T Darvas, Y A Rubinstein: Tian’s properness conjectures and Finsler geometry of the space of Kählermetrics, J. Amer. Math. Soc. 30 (2017) 347[17] R. Dervan. Uniform stability of twisted constant scalar curvature K¨ahler metrics. Int. Math. Res. Not.IMRN 2016, no. 15, 4728–4783.
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