AbstractLet .X;!/ be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi,the set H0 of Kähler forms cohomologous to ! has the natural structure of an infinite-dimensional Riemannian manifold. We address the question of whether any two pointsin H0 can be connected by a smooth geodesic and show that the answer, in general,is “no.”
1. IntroductionLet X be a connected, compact, complex manifold, and let !0 be a smooth Kählerform on it. It was discovered by Mabuchi, and rediscovered by Semmes and Don-aldson, that the set H0 of smooth Kähler forms cohomologous to !0 and the setH of smooth, strongly !0-plurisubharmonic functions on X carry natural infinite-dimensional Riemannian manifold structures (see [M], [S], [D1]). — A functionu W X ! R is (strongly) !0-plurisubharmonic if !0 C i@@u � 0 (resp., > 0). —Mabuchi shows that, in fact, H is isometric to the Riemannian product H0 � R,and both he and Donaldson point out that understanding geodesics in these spacesis important for the study of special Kähler metrics. Donaldson then raises the obvi-ous question of whether any pair of points in H (or H0) can be connected by a smoothgeodesic. In this paper we give a negative answer.
THEOREM 1.1Suppose that .X;!0/ is a positive-dimensional, connected, compact Kähler manifoldand that h W X !X is a holomorphic isometry with an isolated fixed point such thath2 D idX . Then there is a Kähler form !1 2H0 which cannot be connected to !0 bya smooth geodesic.
Concretely, one can take X to be a torus Cm=� , !0 to be a translation-invariantKähler form, and h induced by reflection z 7! �z in C
According to Semmes [S], geodesics in H (and therefore in H0) are related toa Monge–Ampère equation as follows. Let S D ¹s 2 C W 0 < Im s < 1º, and let ! bethe pullback of !0 by the projection S � X ! X . With any smooth curve Œ0; 1� 3t 7! vt 2 H , associate the smooth function u.s; x/ D vIm s.x/, .s; x/ 2 S � X . SetmD dimX . Then t 7! vt is a geodesic if and only if u satisfies
.! C i@@u/mC1 D 0: (1.1)
Since !Ci@@u, restricted to fibers ¹sº�X , is positive, we have that (1.1) is equivalentto rk.!C i@@u/�m; and so a smooth geodesic connecting 0; v 2H gives rise to an!-plurisubharmonic u 2 C1.S �X/ solving
rk.! C i@@u/�m;
u.sC �;x/D u.s; x/ for � 2R; .s; x/ 2 S �X; (1.2)
u.s; x/D
´0 if Im s D 0;
v.x/ if Im s D 1.
Recall that if a .1; 1/ form ˛ on a complex manifold is given in local coordinatesbyP˛jkdzj ^ d Nzk , its rank rk˛ at z is the rank of the matrix .˛jk.z//. Therefore
Theorem 1.1 follows from the following more precise result.
THEOREM 1.2If .X;!0/ and h are as in Theorem 1.1, then there is a v 2H for which (1.2) admitsno real valued solution u 2 C 3.S �X/. One can choose v so that h�vD v.
When mD 1, the v’s in Theorem 1.2 even form an open subset of the space ofh-invariant functions in H , but we do not know if this holds when m> 1.
The idea that symmetries help in the analysis of solutions of Monge–Ampèreequations is not new. The first examples of irregularity of certain boundary valueproblems in C
m were constructed by Bedford and Fornaess using symmetries (see[BF]). Our approach, based on the study of the so called Monge–Ampère foliation,is different from theirs. The symmetry will be used to identify a leaf of the foliationassociated to a C 3-solution u of (1.2). By analyzing the first-order behavior of thefoliation about this particular leaf we obtain a condition on the Hessian of u at .1; x0/,where x0 is an isolated fixed point of h. The proof is concluded by finding a boundaryvalue v which is incompatible with this condition.
Studying solutions of the homogeneous Monge–Ampère equation through theassociated foliation is not new, either. This approach first appeared in [Be1], [Be2],[L1], and [L2], and still seems to be the only way to prove smoothness of the solution.
SPACE OF KÄHLER METRICS 1371
More recently, in [D2] Donaldson used the foliation method in a variant of the bound-ary value problem (1.2) to prove (resp., disprove) regularity, depending on the bound-ary data. Donaldson studied the Monge–Ampère equation onD�X , withD �C theunit disk, but his analysis does not apply if we replaceD by something noncompact—as S in our case—or multiply connected.
Generalized solutions to (1.2) and to rather more general boundary value prob-lems for the homogeneous Monge–Ampère equation (1.1) are known to exist (see [C],with complements in [Bl]).
Most of this paper is devoted to the proof of Theorem 1.2. In Section 4 we willdiscuss the implications of the theorem on the precise regularity of geodesics in H .
2. The Monge–Ampère foliationLet Y be an .mC 1/-dimensional complex manifold, and let ! be a real .1; 1/-formon it, of class C 1; d! D 0. If rk! � m, then the kernels of ! form an integrablesubbundle of T Y , and so Y is foliated by Riemann surfaces, whose tangent spacesare the kernels of !. The foliation is of class C 1. If w is a locally defined potentialof !, that is, i@@wD !, then the section @w of T �1;0Y is holomorphic along any leafof the foliation (see [BK, Theorem 2.4] for a more general statement). Applying thiswith ! replaced by ! C i@@u, we obtain the following.
PROPOSITION 2.1Suppose that u 2 C 3.S �X/ is a real solution of (1.2). Then there is a foliation Fu onS �X , of class C 1, whose leaves are Riemann surfaces, tangent to Ker.!C i@@u/. Ifw is a locally defined potential of !, then @.wC u/ is holomorphic along the leavesof Fu.
Fu is called the Monge–Ampère foliation associated with u. We will also needthe following.
PROPOSITION 2.2Suppose that h W X!X is a holomorphic map with a fixed point x0. If h2 D idX , thenthere are holomorphic local coordinates zj centered at x0 in which h is expressed as.zj / 7! .˙zj /. All signs will be minus if x0 is an isolated fixed point.
ProofThis is again not new. For any compact group of holomorphic transformations, localcoordinates centered at any fixed point can be found in which the transformations arelinear (see, e.g., [BM, p. 19]), and each linear transformation in this compact groupis diagonalizable. In our case, the eigenvalues of the linearization of h must be ˙1,
1372 LEMPERT and VIVAS
which means that in an eigenbasis it is indeed given by .zj / 7! .˙zj /. Clearly, thefixed point is isolated only if all the signs are minus.
From now on we assume that X;!0, and h are as in Theorem 1.2, and we denoteby x0 an isolated fixed point of h. Let Qh D idS �h W S � X ! S � X . We also fixv 2H such that h�vD v.
PROPOSITION 2.3If a real u 2 C 3.S �X/ solves (1.2), then Qh�uD u and S � ¹x0º is one leaf of theMonge–Ampère foliation Fu.
ProofFirst observe that the solution to (1.2) is unique. Indeed, if RD ¹s 2C W 1 < jsj< eº,then any solution u of (1.2) defines a solution U.s; x/ D u.i log s; x/ of the homo-geneous Monge–Ampère equation on R � X . The latter being a compact manifoldwith boundary, uniqueness for the boundary value problem on it is standard. One canargue as follows (see [D1, Corollary 7] for a slightly weaker statement). Denote by� the pullback of !0 by the projection R �X!X . The rank condition implies thatthe signature of �C i@N@U is constant in R � X ; since �C i@N@U is semipositiveat boundary points, it must be semipositive everywhere. Now suppose that U 0 alsosolves rk.�C i@N@U 0/ � m, and suppose that U D U 0 on @R � X . Upon adding aconstant we can assume that U � 0. Then with � > 1 and U 00 D �U 0 �U , we have
.��C i@N@U C i@N@U 00/mC1 D �mC1.�C i@N@U 0/mC1 D 0:
As ��C i@N@U is positive on the fibers ¹sº �X , by [D1, Lemma 6], U 00 attains itsminimum on @R � X . (Donaldson speaks of maximum because he works with theoperator N@@ rather than @N@.) But U 00 � 0 on @R � X , and hence everywhere, and soU � �U 0; letting �! 1 gives U � U 0. Reversing the roles of U and U 0 we see thatU D U 0.
Since u and Qh�u solve the same boundary value problem, we conclude that theyare indeed equal. It follows that Ker.! C i@@u/ is invariant under Qh�, and so isKer.!C i@@u/j.s;x0/ � T.s;x0/.S �X/, for any s 2 S . Proposition 2.2 shows that theonly Qh�-invariant complex lines in T.s;x0/.S �X/� TsS ˚ Tx0X are TsS ˚ .0/ andlines contained in .0/˚ Tx0X . Since u j .¹0º �X/ is strongly !0-plurisubharmonic,Ker.!C i@@u/j.s;x0/ must agree with TsS ˚ .0/ when s D 0. By continuity, the samemust hold for s in a connected neighborhood S 0 � S of 0, and so S 0 � ¹x0º is con-tained in a leaf of Fu. By analytic continuation, S � ¹x0º is itself a leaf.
In the rest of this section, for a real solution u D Qh�u 2 C 3.S � X/ of (1.2)we will study the first order behavior of Fu about the leaf S � ¹x0º. We fix local
SPACE OF KÄHLER METRICS 1373
coordinates zj ; j D 1; : : : ;m, on a neighborhood V �X of x0, as in Proposition 2.2;we can choose them so that, in addition, !0jx0 D i
Pdzj ^ d Nzj jx0 , and the local
coordinates map V on a convex set in Cm. Using the coordinates we identify V with
its image in Cm and x0 with 0 2Cm. Then S�V is identified with a subset of S�Cm,
and occasionally we shall write z0 for the s coordinate of a point .s; z1; : : : ; zm/ 2S � V .
Exhaust S by compact subsets, say, by rectangles:
Sr D®s 2 S W jRe sj � r
¯; r 2 .0;1/:
Given r , if a 2 V is sufficiently close to x0 D 0, then the leaf of Fu j Sr �X passingthrough .0; a/ is the graph of a C 1-function fa W Sr ! V , holomorphic on intSr . Forexample, f0 � 0. Let the components of fa be faj W Sr ! C, j D 1; : : : ;m, and letfa0.s/D s. Write
! j S � V D i
mXj;kD0
!jkdzj ^ d Nzk; with !jk D 0 if j or k D 0:
That the image of .faj /mjD0 is tangent to Ker.! C i@@u/ means that
mXjD0
®!jk
�s; fa.s/
�C uzj Nzk
�s; fa.s/
�¯f 0aj .s/D 0; k D 0; 1; : : : ;m: (2.1)
Further, if !0 j V D i@@w0 andw.s; x/Dw0.x/, then @.wCu/ is holomorphic alongthe (interior of the) leaves, that is,
wzj�s; fa.s/
�C uzj
�s; fa.s/
�; j D 0; 1; : : : ;m; (2.2)
depend holomorphically on s 2 intSr .Replace a 2Cm by ta 2Cm, t 2 Œ0; 1�, and replace fa by fta W Sr.t/! V , where
limt!0 r.t/D1; then differentiate (2.1) and (2.2) with respect to t at t D 0. Notethat 'j D df.ta/j =dt jtD0 are continuous functions on S , holomorphic on S , and that'0 � 0. The symmetry of !;u implies that odd-order derivatives of !jk and u, withrespect to zl ; Nzl ; l � 1, all vanish at .s; 0/. Hence differentiating (2.1) gives
mXlD1
®uzl Nzks.s; 0/'l.s/C u Nzl Nzks.s; 0/'l.s/
¯C
mXjD1
®ıjk C uzj Nzk .s; 0/
¯'0j .s/D 0;
(2.3)
k D 1; : : : ;m; and (2.2) gives that, for j D 1; : : : ;m,
1374 LEMPERT and VIVAS
mXkD1
®w0zj zk .0/C uzj zk .s; 0/
¯'k.s/C
mXkD1
®w0zj Nzk .0/C uzj Nzk .s; 0/
¯'k.s/
are holomorphic. Note that on the leftPw0zj zk .0/'k.s/ is automatically holomor-
phic. Therefore if uzz ; uz Nz , and ' denote the .m�m/-matrices .uzj zk /; .uzj Nzk /, andthe column vector .'j /, j; k D 1; : : : ;m, then the latter says that
uzz.s; 0/'.s/C�I C uz Nz.s; 0/
�'.s/D .s/ (2.4)
is holomorphic.Now u.s; 0/ depends only on Im s. If we let
P D I C uz Nz.i; 0/D I C vz Nz.0/ and QD uzz.i; 0/D vzz.0/; (2.5)
then (2.4) implies, in light of (1.2), that
.s/D
´'.s/ if Im s D 0;
P'.s/CQ'.s/ if Im s D 1:(2.6)
On the other hand, restricting (2.3) to real s gives
'0.s/DA'.s/CB'.s/; s 2R; (2.7)
where
AD�u Nzzs.0; 0/D�A�; B D�u Nz Nzs.0; 0/DB
T : (2.8)
That A is skew adjoint follows from the translational invariance of the self-adjointmatrix u Nzz.s; 0/, while the symmetry of B is obvious. Finally, we note that
'.0/D a: (2.9)
The equations (2.6), (2.7), and (2.9) describe the first-order behavior of Fu about theleaf S � ¹x0º. To summarize, we obtained the following.
PROPOSITION 2.4For any a 2Cm, the solution of the initial value problem (2.7), (2.9) can be extendedto a continuous function ' W S ! C
m, holomorphic on S , and there is another con-tinuous function W S!C
m, also holomorphic on S , that satisfies (2.6).
3. The proof of Theorem 1.2We consider a real solution u 2 C 3.S �X/ of the boundary value problem (1.2), withv 2H satisfying h�vD v, and we choose local coordinates zj centered at an isolatedfixed point x0 of h as in Section 2. The main point of the proof is the following.
SPACE OF KÄHLER METRICS 1375
LEMMA 3.1Let P;Q be defined by (2.5). Assume that P and Q have a common eigenbasis con-sisting of real vectors �1; : : : ; �m 2Rm. If the eigenvalues of
RD .I CP 2 �QQ/P�1
are simple, then at least one of them is ��2.
One can even show that all the eigenvalues are ��2, but this is irrelevant to ourpurposes. We need the following auxiliary result.
PROPOSITION 3.2Let AD�A� and B DBT be m�m complex matrices. If the block matrix
M D
�A B
B A
�(3.1)
has an eigenvector .xy/ with eigenvalue � 2C, then either �2 2R or jxj D jyj.
ProofOne computes
M 2 D
�C D
�D� E
�; with C � D C;E� DE:
Denoting by h ; i the Euclidean inner product on Cm, for �; � 0; �; �0 2Cm we have
to obtain �2.jxj2 � jyj2/D �2.jxj2 � jyj2/, and the claim follows.
Proof of Lemma 3.1The assumptions imply that the �k’s are eigenvectors of P and Q, with eigenvaluesconjugate to the eigenvalues of P , respectively, Q. As the eigenvalues of P are real,it follows that P D P , Q, and Q commute among themselves. Define A;B by (2.8),and let .x
y/ be an eigenvector of the matrix (3.1), with eigenvalue � 2 C. We apply
Proposition 2.4. With a D x C y it is easy to write down the solution of the initialvalue problem (2.7), (2.9):
1376 LEMPERT and VIVAS
'.s/D xe�s C ye�s;
first for s 2 R, and then by analytic continuation, for all s 2 S . Upon replacing theeigenvector by a multiple, we can arrange that aD xCy ¤ 0. From the first equationin (2.6), .s/D xe�s C ye�s . Substituting in the second, and noting that s D s � 2iwhen Im s D 1, we obtain
P.xe�2i�e�s C ye�2i�e�s/CQ.xe�s C ye�s/D xe�s C ye�s : (3.2)
If � 2R, we divide through by e�s to get QaD .I �Pe�2i�/a. Combining thisequation with its complex conjugate yields
RaD .e�2i�C e2i�/a: (3.3)
If � …R, then comparing the coefficients of e�s and e�s in (3.2) yields
Qx D .I � e�2i�P /y; Qy D .I � e�2i�P /x:
Again, we combine these equations and their complex conjugates to get
QQx D .I � e�2i�P /.I � e2i�P /x; QQy D .I � e�2i�P /.I � e2i�P /y;
or
Rx D .e�2i�C e2i�/x; Ry D .e�2i�C e2i�/y: (3.4)
By (3.3) or (3.4), one eigenvalue of R is e�2i� C e2i�. If � happens to be real orimaginary, then e�2i�C e2i� ��2, as claimed.
It remains to take care of the case when � is neither real nor imaginary. But in factthis case cannot occur. Indeed, Proposition 3.2 would imply jxj D jyj. Now �1; : : : ; �mare eigenvectors of R, with different eigenvalues; in particular, e�2i� C e2i� is asimple eigenvalue. Therefore by (3.4) x and y are proportional to each other, and to areal vector �j . At the price of passing to a multiple, we can assume x 2Rm and theny D ˛x, j˛j D 1. We have �
A B
B A
��x
˛x
�D �
�x
˛x
�;
whence Ax C ˛Bx D �x and Bx C ˛Ax D ˛�x. Multiplying the latter by ˛ andsubtracting from the former we obtain 0D .�� �/x, or � 2R, a contradiction. Withthis the proof of the lemma is complete.
It is easy to construct self-adjoint (resp., symmetric) matrices P > 0 and Q thathave a common real eigenbasis, but contrary to the claim of Lemma 3.1, the eigen-
SPACE OF KÄHLER METRICS 1377
values of .I C P 2 � QQ/P�1 are simple and < �2. For example, one can takeP D I and Q diagonal, with real eigenvalues q1 > > qm > 2. Theorem 1.2 wouldthen follow from Lemma 3.1, if we could produce a strongly !0-plurisubharmonicv 2 C1.X/ such that h�v D v, and in the local coordinates constructed, vz Nz.x0/DP � I , vzz.x0/DQ. This can indeed be done, as we now prove.
LEMMA 3.3Let .X;!0/ be a Kähler manifold with almost complex structure tensor J W TX !TX , let h W X ! X be a holomorphic isometry that fixes x0 2 X , and assume h2 DidX . Let q be a real quadratic form on Tx0X , invariant under h�. If its Hermitianpart
q1;1.�/D�q.�/C q.J �/
�=2; � 2 Tx0X;
satisfies q1;1.�/C !0.�; J �/ > 0 for nonzero � 2 Tx0X , then there is a strongly !0-plurisubharmonic v 2 C1.X/ such that h�v D v, dv.x0/D 0, whose Hessian at x0is q.
The lemma is essentially the same as [D2, Lemma 8], except for the extra ingre-dient h. We also had to impose the condition on q1;1. The proof to follow is a slightvariation on Donaldson’s proof.
ProofWe choose local coordinates zj centered at x0 in which h is expressed as .zj / 7!.˙zj /. It is clear that in some neighborhood U of x0 there is a w 2 C1.U / whichhas all the properties of v, except it is not defined on all of X . Indeed, if
q DXj;k
ajkdzjd Nzk CReXj;k
bjkdzjdzk j Tx0X;
then w D .Pajkzj Nzk C Re
Pbjkzj zk/=2 will do. By shrinking U and by scaling
the coordinates we can arrange that i@@w > .ı � 1/!0 on U , with some ı 2 .0; 1/,and that the coordinates map U on a neighborhood of ¹z 2Cn W
Pjzj j
2 � 1º.Next choose a smooth function ' W Œ�1;1/! Œ0; 1� such that
'.t/D
´0 if t > 0;
1 if t <�1:
Then .t/ D '." log t /, " > 0, defines a smooth function on Œ0;1/, supported onŒ0; 1�, such that
.t/D 1 if t < e�1="; and tˇ 0.t/
ˇ; t2ˇ 00.t/
ˇ<C";
1378 LEMPERT and VIVAS
with some C independent of ". Thus the smooth function
vD
´ .jzj2/w on U ,
0 on XnU;
satisfies dv.x0/D 0, h�vD v, and its Hessian at x0 is q. Further, on U
!0C i@@v > !0C i �jzj2
�@@w �C 0"!0 > .ı �C
0"/!0;
with C 0 independent of ". If " is small enough, v therefore has all the required prop-erties.
4. The smoothness of geodesicsEven without delving into technicalities of infinite-dimensional analysis, it is ratherclear that if X;!0, and v 2 H are as in Theorem 1.2, then by this theorem thereis no C 3-geodesic connecting 0 and v in H . It emerged from a conversation withBerndtsson that there is not even a C 2-geodesic, and in fact not even a geodesic ofSobolev class W 1;2. (This space is the largest in which the geodesic equation can beconsidered.) But to prove this stronger result, we will have to delve into technicalities.Modulo technicalities though, the proof is just standard bootstrapping for ordinarydifferential equations.
One technicality involves replacing H by Banach spaces. From now on C k.X/will denote the real Banach space of real functions of class C k on X . If k D 2; 3; : : :and l D 0; 1; : : : , let
Hk D®v 2 C k.X/ W !0C i@@v > 0
¯; F l D
®� 2 C l1;1.X/ W�> 0
¯;
be open subsets of C k.X/, respectively, of the Banach space C l1;1.X/ of real .1; 1/-forms of class C l . We also write C l1;0.X/ for the Banach space of .1; 0/ forms ofclass C l . Another technicality is the notion of vector-valued Sobolev spaces. Let Bbe a Banach space, and let J D Œ0; 1�. The notions of measurability, integrability, andLp-integrability of functions J ! B are rather obvious, and were codified in [Bo].The weak derivative of ˇ 2 L1loc.intJ IB/ is a function 2 L1loc.intJ IB/, denotedby P, such that
RJ' D �
RJP'ˇ for all ' 2 C1.J IR/, supp' � .0; 1/. The only
Sobolev space of B-valued functions of interest to us is
W 1;2.J IB/D®ˇ 2 C.J IB/ W ˇ has a weak derivative in L2.J IB/
¯:
Mabuchi’s Riemannian metric on H extends to a Riemannian metric on H2. Theequation for geodesics ˇ W J !H involves a function
F W F 0 �C 01;0.X/! C.X/; F.�;f /D hf;f i�; (4.1)
SPACE OF KÄHLER METRICS 1379
the pointwise squared length of f , measured in the Hermitian metric determinedby �. By [M, (3.1)] or [S, (1.5)] the geodesic equation is
R.t/D F�!0C i@@ˇ.t/; @ P.t/
�(4.2)
and makes sense for ˇ W J !H2.
PROPOSITION 4.1For l D 0; 1; : : : , F defines a C1-map F l �C l1;0.X/! C l.X/. Furthermore, givena compact K � F l , there is a constant c such that��F.�;f /��
C l .X/� ckf k2
C l1;0.X/; for � 2K;f 2 C l1;0.X/:
ProofIt suffices to prove when the second argument f in F.�;f / is supported in a coor-dinate neighborhood N �X . If in local coordinates
�D iX
�jkdzj ^ d Nzk and f DX
fjdzj ; then
F.�;f /DX
�jkfj Nfk in N;
and 0 elsewhere, with .�jk/ denoting the inverse of the matrix .�jk/. From thisformula both claims of the proposition follow.
By Proposition 4.1, F.!0C i@@ˇ; @ P/ 2L1.J IC k�2.X// if ˇ 2W 1;2.J IHk/.Hence the geodesic equation (4.2) can be considered for ˇ 2 W 1;2.J IHk/ in theweak sense: we require that the weak derivative of P 2 L2.J IC k.X// � L2.J IC k�2.X// should be F.!0C i@@ˇ; @ P/.
THEOREM 4.2If X;!0, and v are as in Theorem 1.2, and k � 4, then there is no ˇ 2W 1;2.J IHk/
that would satisfy ˇ.0/D 0, ˇ.1/D v, and the geodesic equation (4.2) in the weaksense.
ProofOf course, it suffices to prove when k D 4. We will show that for any weak solutionˇ 2W 1;2.J IH4/ of (4.2) the function
w.t; x/D ˇ.t/.x/; .t; x/ 2 J �X;
is in C 3.J �X/. In general, w 2 C l.J �X/ if (and only if, but we will not need thereverse implication) ˇ 2 C l�j .J IC j .X// for j D 0; 1; : : : l . Indeed, when l D 0, this
1380 LEMPERT and VIVAS
follows directly from the definition of continuity. The case l > 0 can be reduced tothe case l D 0 by observing that w 2 C l.J �X/ if �1 �lw 2 C.J �X/ for smoothvector fields �i , each tangential either to the fibers ¹tº �X or to the fibers J � ¹xº.
By the definition of W 1;2.J IH4/,
ˇ 2 C�J IC 3.X/
�: (4.3)
Further, by Proposition 4.1 the map t 7! F.!0 C i@@ˇ.t/; @ P.t// 2 C2.X/ (defined
for almost every t 2 J ) is integrable. Hence by (4.2), t 7! P.t/ 2 C 2.X/ is contin-uous, or rather has a representative, continuous on J (namely, the one obtained byintegrating R). In other words,
ˇ 2 C 1�J IC 2.X/
�: (4.4)
Feeding this and (4.3) back into (4.2), by Proposition 4.1 we obtain R D F.!0 Ci@@ˇ; @ P/ 2 C.J IC 1.X//, or
ˇ 2 C 2�J IC 1.X/
�: (4.5)
Finally, feeding this and (4.4) back into into (4.2) once more gives R 2 C 1.J IC.X//,whence ˇ 2 C 3.J IC.X//. The latter, together with (4.3), (4.4), and (4.5), impliesw 2 C 3.J �X/, and therefore u.s; x/Dw.Im s; x/, .s; x/ 2 S �X , is in C 3.S �X/.However, (4.3)–(4.5) also imply that the two sides of (4.2), viewed as elements ofC.X/, agree for every t 2 J , not just weakly. By the computations in [S] or [D1], thismeans that rk.!0 C i@@u/�m, and, of course, the other equations in (1.2) are alsosatisfied. But if v 2H is suitably chosen, by Theorem 1.2 the solution of (1.2) cannotbe in C 3.S �X/, and consequently ˇ cannot be in W 1;2.J;H4/.
Acknowledgments. We are grateful to B. Berndtsson and Z. Błocki for sparking ourinterest in the subject, and to V. Tosatti for spotting an algebra error in the first versionof this paper. The first author is also grateful to the Université Pierre et Marie Curie,Paris, where some of this research was done.
References
[Be1] E. BEDFORD, Extremal plurisubharmonic functions for certain domains in C2, Indiana
Univ. Math. J. 28 (1979), 613–626. MR 0542948.DOI 10.1512/iumj.1979.28.28043. (1370)
[Be2] , Stability of envelopes of holomorphy and the degenerate Monge–Ampèreequation, Math. Ann. 259 (1982), 1–28. MR 0656649.DOI 10.1007/BF01456826. (1370)
[BF] E. BEDFORD and J. FORNAESS, Counterexamples to regularity for the complexMonge–Ampère equation, Invent. Math. 50 (1978/79), 129–134. MR 0517774.DOI 10.1007/BF01390286. (1370)
[BK] E. BEDFORD and M. KALKA, Foliations and complex Monge–Ampère equations,Comm. Pure Appl. Math. 30 (1977), 543–571. MR 0481107. (1371)
[Bl] Z. BŁOCKI, “On geodesics in the space of Kähler metrics” in Advances in GeometricAnalysis, Adv. Lect. Math. (ALM) 21, Int. Press, Somerville, Mass., 2012, 3–20.(1371)
[Bo] S. BOCHNER, Integration von Funktionen deren Werte die Elemente einesVektorräumes sind, Fund. Math. 20 (1933), 262–276. (1378)
[BM] S. BOCHNER and W. T. MARTIN, Several Complex Variables, Princeton Math. Ser.,Princeton Univ. Press, Princeton, N.J., 1948. MR 0027863. (1371)
[C] X. X. CHEN, The space of Kähler metrics, J. Differential. Geom. 56 (2000), 189–234.MR 1863016. (1371)
[D1] S. K. DONALDSON, “Symmetric spaces, Kähler geometry and Hamiltonian dynamics”in Northern California Symplectic Geometry Seminar, Amer. Math.Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, R.I., 1999, 13–33.MR 1736211. (1369, 1372, 1380)
[D2] , Holomorphic discs and the complex Monge–Ampère equation,J. Symplectic Geom. 1 (2002), 171–196. MR 1959581. (1371, 1377)
[L1] L. LEMPERT, La métrique de Kobayashi et la représentation des domaines sur laboule, Bull. Soc. Math. France 109 (1981), 427–474. MR 0660145. (1370)
[L2] , Solving the degenerate complex Monge–Ampère equation with oneconcentrated singularity, Math. Ann. 263 (1983), 515–532. MR 0707246.DOI 10.1007/BF01457058. (1370)
[M] T. MABUCHI, Some symplectic geometry on compact Kähler manifolds, I, OsakaJ. Math. 24 (1987), 227–252. MR 0909015. (1369, 1379)
[S] S. SEMMES, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114(1992), 495–550. MR 1165352. DOI 10.2307/2374768. (1369, 1370, 1379,1380)
Lempert
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA;
