on the ricci curvature of a compact kähler manifold and ... the ricci curvature of a... · on the...

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere

Equation, I*

SHING-TUNG YAU Stanford University

0. Introduction

Let M be a compact Kahler manifold with Kahler metric C,,, g,rdz'@ddzI. Then it is well known that the Ricci curvature of M can be computed in the following way. If R,; dz' C3 dZJ is the Ricci tensor, then

In particular, the (1,l) form ( & i / 2 ~ ) I ~ , ~ R , i d z ' ~ d f ' is a closed (1, 1) form

which is equal to ( - f i /27r )&log det (gS7)]. A well-known theorem of S. S. Chern [6] shows that the cohomology class of this (1,l) form depends only on the complex structure of M and is equal to the first Chern class of M.

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M.

More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

Rlr dz' A dZJ represents the first Chern class of M, In fact, if (d?/27r) then we can find a smooth function F defined on M such that

If XI , , Rl jdz '@d2' is the Ricci tensor of some Kahler metric Ci,j gijdz'@ddz1, then

"The research f o r this paper was done while the author was a Visiting Member at the Courant Institute of Mathematical Sciences. Reproduction in whole or in part is permitted for any purpose of the United States Government.

Comniunicitticins on Pure and Applied Mathematics, Vol. XXXI, 339-41 1 (1978) @ 197X John Wiley & Sons. Inc. 00 10-36401781003 1-0339/$01 .OO

340 SHING-TUNG YAW

and the above equation shows that

Since M is compact and det (g,r)/det (g,;) is a globally defined function, the maximal principle shows that log det (g,r)/det (g ,T) - F is a constant. Hence, for some constant C>O,

(0.3) det (is:) = C exp {F) det (g,r) .

If ;& giidzi A d.2' is cohomologous to 4 J - 1 giTdzi ~ d 9 , then for some smooth function Q we have gii = gij +aZcplaziaZi. In this case, the above equation is equivalent to the following equation:

= C exp {F) det ( g s r ) .

Conversely, if for some constant C>O, we can solve (0.4) so that the solution cp is smooth and C,,J (gii +a*cp/az'dZ') dz'@&f defines a Kahler metric, then we can solve the conjecture of Calabi. It turns out, by integrating (0.4), that there is only one choice of C:

(0.5) C exp { F } = Vol ( M ) . I In this paper, we shall solve equation (0.4) with the (necessary) compati-

bility condition (0.5). In view of its application in Kahler geometry, we generalize (0.4) to the following more general case.

Let L, and L, be holomorphic Hermitian line bundles over M. Let sl, . * , s, be holomorphic sections of L , and t , , - - , t, be holomorphic sections of L,. Then for hl 2 0, * - , h, 2 0, * * * , k, 2 0, we study the equation

where F is a smooth function defined on M x R such that aF/acp 2 0.

ON THE RICCI CURVATURE OF A COMPACT KAHLER MANIFOLD 341

We assume that we can find a smooth function JI defined on M such that

(0.7) (1~~)~'1+$-. - ~+)spJZh~)()fllZkl+* * -+)tq)2kq)-1exp{F(x, J/)}=Vol ( M ) .

It is also necessary to assume that the right-hand side of (0.6) cannot tend to infinity too fast. Hence we assume that, for m =dim M,

c

and for some q >0, the function ( l t l ) z k l + - * + + Itq(2kq)-q-1/m

(0.9) )A log (JtlJ2k1 + * * * + 1 tq)2kq))(m-1)'m is integrable over A4 and over every analytic disc of M.

We shall show that, under the conditions (0.7)-(0.9), equation (0.6) has a bounded solution cp which is smooth and defines a Kahler metric

outside the variety defined by the si and the variety defined by the ti. Near the variety defined by the si, the Kahler metric is uniformly bounded from above and near the variety defined by the ti the Kahler metric grows upward at most like a "pole". Furthermore, the function 50 is unique up to a constant.

Equation (0.6) is considerably more complicated than equation (0.4) and will be proved only at the end of the paper. However, when the right-hand side of (0.6) specializes to the form exp{q+F(x)}, then equation (0.6) is much easier than (0.4) and can be used to deduce the existence and uniqueness of a canonical Kahler-Einstein metric on a compact Kahler manifold with ample canonical line bundle. This was also conjectured by E. Calabi.

It should be mentioned that Calabi proved the uniqueness of the solution of (0.4) a long time ago. He also proved that when the right-hand side of (0.4) is close enough to a constant, then (0.4) has a smooth solution. (This part of Calabi's work was also made precise by T. Ochiai.) His later paper [ 5 ] on affine hyperspheres is very important for the study of the Monge-Ampere equation. The computation of the Laplacian of the third derivatives originated from there and was shown to us by L. Nirenberg. Furthermore, in [l], T. Aubin published a proof of Calabi's conjecture assuming the original Kahler manifold has non-negative holomorphic bisectional curvature. This class of manifolds is rather restrictive. Moreover, he used the variational method and his procedure is rather difficult to comprehend.

342 SHING-TUNG YAU

My proof of Calabi's conjecture was completed in the middle of 1976. A large amount of work was done while I was working on the real Monge- Ampere equation (cf. [8], [9]). For example, Pogorelov's second order estimate in the real Monge-Ampere equation had direct bearing on my work here. Indeed, besides the uniform estimate, the higher order estimates were discovered at that time. A special case of equation (0.6) when the right side is the function exp { i+ F ( x ) } had been studied independently by Aubin [2].

The plan of this paper consists in starting from special cases of the theorem to make it easier for the reader to comprehend what is going on before treating the more involved equation (0.6). We use the continuity method and the required estimates are worked out for equation (0.4) in Sections 2 and 3. We shall give applications in the second part of the paper.

I am very grateful to all my teachers and my friends for their constant encouragement during this work. Among them I should mention E. Calabi, S. S. Chern, S. Y. Cheng, S. Kobayashi, J. J. Kohn, B. Lawson, L. Nirenberg and Y. T. Siu. Professor Chern kindly lectured on this paper and made significant improvements on its presentation. His help and his encouragement can hardly be exaggerated. My joint papers with Cheng have definite influence on the present paper. I am also very grateful to L. Nirenberg for many helpful conversations on the complex Monge-Amp6re equation, especially on the complex analogue of Calabi's computation.

I dedicate this paper to my mother Leung Yeuk Lam and to the memory of my father Chiou Chen Ying. My father worked in both philosophy, economics and Chinese literature. The depth of his insight had a tremendous influence on me. He died before he was well-known. My mother took over the most difficult job of teaching and feeding seven children in a poor family. Without my parents' persistent teaching, I would never become a mathematician.

Most of the results of this paper and their applications 1151.

have been able to

were announced in

1. Notations and Local Formulae

Let M be a rn-dimensional Kahler manifold with Kahler metric ds2= xi,, g,r dz'C3ddz'. The Hermitian condition of the Kahler metric shows that, for all i, j ,

be the Kahler form, then the Kahler condition is that

(1.3) d H = O ,


or, for all i, j , k ,

(1.4)

The connection form w! of the KBhler metric is characterized by the fact that it is of type ( 1 , O ) and it satisfies the following equation (of preserving scalar product):

where

- w : = w : (1.7) 11 1 1 .

The matrix ( g i r ) satisfies the equation

Hence

The curvature form is given by

Differentiating (1.8), we have

(1.11)

Substituting (1.11) into (l.lO), we obtain

344 SHING-TUNG YAU

Hence is of type (1.1) and we can set

(1.13)

where

(1.14)

or

(1.15)

By Bianchi's identity

(1.16)

we have

(1.17)

and

(1.18)

On the other hand,

(1.19)

so that

(1.20)

Therefore the tensor R i j k T is symmetric in the first and third indices, as well as in the second and fourth indices.

The Ricci tensor is


By direct computation, we get

(1.22) a log det (gij) = giT agil

Hence,

-

i.i

Therefore, the Ricci tensor is given by

(1.23) a* log det (gij) R.7= -

IJ a 2 azj

The curvature tensor will be an obstruction for the commuting of covariant differentiations. Thus let us set up the terminology for covariant differentiation.

Let f be any real-valued function defined on M. Then we define

(1.24) df=Cf io i+Cf+ i i , I 1

(1.26)

By exterior differentiating (1.24), one verifies that

(1.27)

For convenience, we shall write Af as c,fi;. Then, as is well-known, 26 is the Laplace-Beltrami operator of the Kahler manifold M. In local coordi- nates, it is given by C,,, gLTazf/az1 d?.

Exterior differentiating (1 .29 , we have

c ( dfij - c f i k o : - & fkjw:) A 0; k

(1.28)

346 SHING-TUNG YAU

Hence we define

(1.29)

(1.30)

By (1.28), we have

(1.31) f i j k = f i k j 1

(1.32) f i y E = f i n ; 9

(1.33) f - i k l -f q k - = - z f , R ' - qk . I

Continuing in this way, we can find the commutation formulae of the higher covariant derivatives of f. For example,

(1.21) n n

The main fact that we shall use is that, when we apply covariant differentiation successively with respect to two indices without bar, we can interchange these two indices. The same principle applies when the two indices have a bar at the same time. If one of the indices has a bar and the other does not, then after commuting them, we still have terms which are contraction of a curvature tensor with the covariant derivatives of f whose order is the order of the original one minus two.

2. Estimates up to Second Order

From now on, we assume that M is a compact Kahler manifold. We shall study the equation

det ( giT+- ) (det (gi;))F1 = exp { F } , azi azT (2.1)

where F is assumed to be in C3(M) . We are going to look for solutions (p of (2.1) such that (g i r+a2(p /dz i az;) is

a positive definite Hermitian matrix. The tensor Ci,j ( g i r +a2(p/azi aZ i ) dz'@dZ'


will define another Kahler metric on M. We shall denote it by xi,, g:rdzl@&' so that

g! :=g:+- . a2 cp 'I 82, azj

Before proving the existence of cp, we need a priori estimates of cp. Therefore we assume that c p ~ C5(M). Our first aim will be to obtain the estimates of cp up to second derivatives. This will be achieved under the normalization

q = O . I, (2.3)

Differentiating (2.1), we have

(2.4)

where (gfii) is the inverse matrix of (gii). Now we differentiate (2.4) again and obtain

Let A' be the normalized Laplacian associated with the metric Cij gT; dz' 8 dz'. Then

348 SHING-TUNG YAU

Since the left-hand side of (2.6) is independent of the choice of coordinate system, we can compute the right-hand side by picking a coordinate system at one point so that gir = S i i , agi,-lazk = 6’gir/6’2’ = 0. Then inserting (2.5) and (1.9) into (2.6), we have

Note that each term on the right-hand side is a scalar function and is independent of the choice of the coordinate system if we contract the terms suitably with the tensor {g t i } .

If we choose another coordinate system so that g,, = S,, and qI; = Sl,q1;, then we have

(2.8) g’l’ = Sll(l + qI;)-*

and

Combining (2.7) and (2.9), we see that

On the other hand,

(2.1 1)


Let C be a positive constant. Then we have

(2.12) -C exp {-Cq} c gfiicpi(Aq)7

- C exp {-Cq} c gfii(Aq)icpy i,i

i.i

-Cexp{-Cq}A'q(m +Aq)+ exp{-Cq}A'(Aq) . Applying Schwarz' inequality to the first three terms of (2.13), we have

A'(exp{-Cq}(m+Aq))z -(m+Aq)-l exp{-Cq} gfir(Aq)i(Aq)T k,i

(2.13) - c exP - Cp>(A'co>(m + Aq) + expi - Cq}A'(Aq).

In computing the right-hand side of (2.13), we assume that gii=Sii and 'pi;= qir6,, at a point. Then, using (2.10), we have

-(m + A q ) - ' C g'ii(Aq)i(Atp)T+A'(Aq)

(2.14)

350 SHING-TUNG YAU

(Here we use the fact that q k r i = (P&i = (PEtk = q , ~ ~ and qrkk; = q k K ; = q k ; ~ , see Section 1). Hence (2.14) and (2.15) imply

(2.16)

Inserting (2.16) into (2.13), we obtain

A’(exp{-Cq}(m +Aq))Zexp{-Cq} (2.17)

’ Inserting (2.11) into (2.17), we get

+ exp {-cq} inf ( R ~ F ~ ~ ) i # I

(2.18) 1

-Cexp{-Cq}m(m+Aq)

- C exp {- Cq}m(m + Aq) + C exp { -Cq} (m + A q ) c - i l+q i :

e x p { - C q ) ( m + ~ q )

Now let us notice the following inequality:

(2.19)

which can be verified by taking the (rn - 1)-th power of both sides. Therefore, by (2.1),

(2.20)

35 1 ON THE RICCI CURVATURE OF A COMPACT KAHLER MANIFOLD

Choose C so that

(2.21) C + inf > 1 . i f f

Then it follows from (2.18) and (2.20) that

A'(exp {-Ccp}(m +Acp))

(2.22) - C exp {-Ccp}m(m +Acp)

Using (2.22), we can now estimate exp{-Ccp}(m+Acp). In fact, it must achieve its maximum at some point p so that the right-hand side of (2.22) is non- positive. At this point,

0 2 A F - m2 inf (Rinr) - Cm( m + Acp) i + f

(2.23)

(Rini)) exp [x} ( m + Acp)l+l/(m-l) m - 1

Hence ( m + Acp)(p) has an upper bound C1 depending only on sup, (-AF),

ay + b implies either y1+'/(m-1)S2ay or y1+"("-')52b. In either case, we have an upper estimate of y in terms of a and b.)

Since exp {-Ccp}(m + Acp) achieves its maximum at p, we have the following inequality

sup, linfizl (Ri:fi)I, C m and sup, F. (It is clear that the inequality Y'"'(~-')< - -

(2.24)

We are going to make use of (2.24) to estimate sup, IcpI. First of all, since Acp + m = x i (1 + pi;) =

Let G(p, q ) be the Green's function (cf. [lo]) of the operator A on M. Let K be a constant (depending only on M) such that

gi7g;7 > 0, we can estimate sup, cp as follows.

(2.25) G ( p , q ) + K Z O .

352 SHING-TUNG YAU

Then (2.3) shows that

(2.26)

Therefore,

(2.27) supcpSmsup ( G ( p , q ) + K ) d q . M p E M I M

The inequalities (2.3) and (2.27) also imply

(2.28) s sup $0 Vol ( M ) - cp + sup cp Vol (M) ( M I ( M

= 2m Vol (MI sup (G(p, 4 ) + K ) dq paM I M

To estimate infM cp, we offer two different proofs. One works for rn = 2 and the other works in general.

For m = 2 , we renormalize cp so that ~ ~ p ~ q ( P - 1 . (This is possible because we have estimated SUP,^ already so that we still have an estimate

Let p be any positive number greater than or equal to 1. Then

ON THE RICCI CURVATURE OF A COMPACT U H L E R MANIFOLD 353

When m = 2. we find

(2.30)

Multiplying (2.30) by exp {F) and integrating with respect to the volume form of the original metric Ci,i gi;dzi@ddZj, we have

(2.31)

Since cp is negative, (2.31) implies that, for p%2,

where C, depends only on sup F. Now applying the Sobolev inequality (this is true on a compact manifold

because we can cut the manifold into a finite number of pieces which are diffeomorphic to a domain in Euclidean space and we can apply the standard Sobolev inequality in Euclidean space), we can find a constant C, depending

354 SHING-TUNG YAU

only on M such that

sc31, IcpIP+PClC,I M IcpIp-l.

Since cpS-1, we derive from (2.33) another constant C4 depending only on M such that

(2.34) I, lcpI’PSC4P2(I, lcpl’)1

In order to use (2.34), we need to find an estimate of IqI2. But this

IcpI (see (2.28)), the Poincarh inequality and

We now claim that we can find a constant C5 depending only on M such

I follows from the estimate of (2.32). l M

that

(2.35)

for all integers p 2 1. In fact, let po be the first integer such that, for p 2 po,

(2.36)

Then we choose C5 so that (2.34) is valid for 1 S p 5 2p,. (This is possible by (2.34), the Lz-estimate of cp and the Holder inequality.) We are going to prove (2.34) for all p 2 1 by induction.

There are two cases. If p + 1 is divisible by two, then from (2.34) we have

(2.37)

where the last inequality follows by the induction hypothesis. If we apply (2.36) to (2.37), then (2.35) is valid with p replaced by p+ 1.


If p + 1 is not divisible by two, then, by (2.34), the Schwarz inequality and the induction hypothesis,

(2.38)

Applying (2.36) to (2.38), we verify (2.35) with p replaced by p + 1. In conclusion, (2.35) is valid for all p. Therefore,

(2.39)

Since the Stirling’s formula tells us that

(2.40)

for all positive integers p, we find

(2.41) m

exp { k q 2 ) 5 (2a)-1’2(kCge)Pp-1’2. p=o

When

(2.42) k < Ci2e-’ ,

the right-hand side of (2.41) is finite and we have obtained an estimate of

exp {kq’} with any k satisfying (2.42).

We are now in a position to use (2.24) to estimate SUP, Iql. Rewrite I, (2.24) as

356 SHING-TUNG YAU

where

- (2.44) m S f S C , exp{Csup cp}exp{-inf cp} .

M

Then by the Schauder estimate (see [13] p. 156 inequality (5.5.23)), there is a constant c6 depending only on M and CI exp { C sup cp} such that

(2.45)

Since (2.27) and (2.28) already provide estimates for sup cp and 1, (cpl, we

have a constant C, depending only on M and C such that

(2.46) sup [ V q l ~ C,(exp {-C inf q}+ 1). M

Let q be a point in M where cp(q)=inf cp. Then in the geodesic ball, with center q and radius -4(inf cp) C;’(exp {-C inf cp}+ l)-*, cp is not greater than iinfcp. Since we may assume -inf cp to be large, we may assume that the radius of this geodesic ball is smaller than the injectivity radius of M. Therefore, the integral of exp{kq2} in this ball is not less than C8 exp {$k(inf cp)’}(-$ inf(p)2mC;2m(exp { -C inf c p } + l)-2m, where C, is a positive constant depending only on M. Since (2.40) gives an estimate of the integral of exp{kcp2} over M, the last quantity is estimated. This of course implies an estimate of linf,cpl.

Let us now give an estimate of linfMcpl without assuming rn = 2. Let N be any positive number. Then (2.18) shows that

(2.47) - N exp {-Ncp}rn(m + Acp) +

Choose N so that

(2.48)

357 ON THE RICCI CURVATURE OF A COMPACT K.&HLER MANIFOLD

Then, by (2.20),

There is a constant C, depending only on s u p F and m such that

-F (2.50) $N exp ( , 1 } ( m +Aq)m/(”’-l)Z 2Nm(m + A @ ) - NC, .

Inserting (2.48), (2.49) and (2.50) into (2.47), we find

A ’ ( e x p { - N q } ( m + A q ) ) Z e x p { - N q } (2.5 1)

Therefore. +Nexp{-Nq}rn(m+Aq).

2 exp {-Nq} exp {F}

+ N exp { - N q } exp {inf F}m(m + A q )

11

I1

+ m2N exp {inf F) (2.52)

- mN exp {inf F) exp { - N q } A q

+ m Z N exp {inf F )

+ m exp {inf F)(-A exp { -Nq}+ N 2 exp {-&} IVq12)

2-Cloexp{-Nq}+m exp{inf F)(-Aexp{-Nq}+N2exp{-Nq}IVq12),

where Clo depends only on N, F and M. Integrating (2.52), we obtain

JV exp {-+Nq}12 = +NZ L exp {-Nql I V d 2 (2.53)

d $Clom-’ exp {-inf F}

358 SHING-TUNG YAU

We claim that for each N satisfying (2.48), the inequalities (2.53) and

exp{-Nq} (depending on N, F and M ) . We

Suppose there exists a sequence {q i } satisfying (2.28) and (2.53) such that

(2.28) furnish an estimate of

are going to prove this statement by contradiction.

1imi+,- JM exp {-Nq,} = CQ. Then we define

i,

It follows from (2.53) that the sequence IV exp {-iN+}lz is uniformly

bounded from above by a constant depending only on N, F and M. Since

exp {-N+i}= 1 for all i, this last fact implies (see [13]) that a subsequence

of {exp(-$NGi}} converges in L'(A.4) to some function f in L2(M). We assume this subsequence is {exp {-$N&}} itself.

I, I

On the other hand, we know that, for any A > 0,

(2.55) Vol {X 1 A S Iq 1 (x)} 5 -

Furthermore,

(2.56) Vol{x 1 ASexp{-~N~i}}=Vol N exp{-Nqi}.-pi].

Since 1imi-= exp {-fNqi}= m, we conclude that, for i large enough,

By (2.28) [I, lqi1} is uniformly bounded and the inequality (2.57) then

implies that

(2.58) limVol{x t-m 1 ~ ~ e x p { - $ ~ ~ ~ ) = o


for all A > 0. Clearly,

Vol{x I A Sf}SVol{x I $A SIf-exp{-$N@i}l}+Vol{x [ $ A Sexp{-$N@i}}

(2.59)

Since exp{-$N@i} converges to f in L'(M), (2.58) and (2.59) shows that, for all A > 0,

(2.60) Vol {x I A Sf)= 0 I

Since f is the L2-limit of exp{-$N@i}, (2.59) implies that f is zero almost

everywhere. This is a contradiction because f'= 1 which is a consequence

of (2.54) and the L2 convergence of exp(-$N&} to f. Therefore we have arrived at the conclusion that whenever N verifies (2.53), (2.28) and (2.48)

exp{-Ncp} has an estimate from above depending only on N, F and M.

We can now repeat the previous argument to find an estimate of linf,cpJ. Inequality (2.46) is valid for arbitrary dimension. As before, we find a geodesic ball with radius -; inf cp C;'(exp {-C inf cp}+ l)-' (which is not greater than the injectivity radius) such that cp is not greater than $inf cp in this ball. Then we choose N so large that (2.48) is satisfied and that N is greater than 4mC. Since the integral of exp {-lVcp} in the above geodesic ball is not less than

IM I,

C12 exp {-$N inf cp}(-$ inf qIZm C;'(exp {-c inf cp)+ ~ ) - ~ m ,

where Clz is a positive constant depending only on M, we have an estimate of -inf, cp.

Since we have already found an estimate of sup cp, this gives an estimate of sup, Iq(. The inequalities (2.46) and (2.24) then give estimates of sup, IVcpl and sup, (fn+Acp). On the other hand, since (S,,+cp,,) is a positive definite Hermitian matrix, we can find upper estimates of 1 t cpir for each i. The equation nLl (1 + cp,;) = exp {F} then gives a positive lower estimate of (1 + cpi ; ) for each i. In particular, the metric tensor z,,, (g , i + a2cp/az' a5') dz' Q? df' is uniformly equivalent to zI,l g,T dz'8ddz".

In conclusion, we have proved the following

360 SHING-TUNG YAU

PROPOSITION 2.1. Let M be a compact Kahler manifold with metric tensor

2 Ci,j gi:dzi@ddiJ. Let cp be a real-valued function in c"(M) such that cp = 0

and xi,, (gi;+a2rp/azi a?') dzi@ddZi defines another metric tensor on M. Suppose det (gi;+a2cp/azi 85') det (gi;)-l = exp {F). Then there are positive constants C , , C,, C, and C,, depending on inf, F, sup,F, inf AF and M such that sup, Jrp I 5 C1, sup, IVql S C, , 0 < C, 5 1 + c p i ~ 5 C, for all i .

I,

3. Third-Order Estimates

In this section, we estimate the third derivatives of ViTk assuming cp solves

Following E. Calabi [ 5 ] , we consider the function the equation (2.1) and F is C3(M).

(3.1)

We are going to compute the Laplacian of S. For convenience, we shall introduce the following convention. We say that A - B if ( A - B ( d C,&+ C,, where C, and C2 are constants that can be estimated. We also say that A r B if \A - B l S C,S + C,&+ C, , where C, , C, and C, are constants that can be estimated.

As in the last section, we shall diagonalize our metric tensor and the Hessian ( c p , ~ ) at a point that is under consideration. Then by the computation which will be carried out in the Appendix A, we have

(3.2)

On the other hand, by (2.7)

where c6 is a constant that can be estimated. Therefore by letting C, be a large positive constant, we find

where C,, C,, C, are positive constants that can be estimated.

ON THE RICCI CURVATURE OF A COMPACT KAHLER MANIFOLD 36 1

At the point where S+C,Acp achieves its maximum, (3.4) shows that

(3.5) C8( S + C7 A cp) S C, + C, C7 Acp . Since we have estimated Acp already, (3.5) gives an estimate of the quantity sup, (S + C, Ap) and hence of sup, S. This in turn gives estimates of ( P i F for all i , j , k.

Let M be a compact Kuhler manifold with metric tensor PROPOSITION 3.1.

gi;dzi@ddZi. Let cp be a real-valued function in C5(M) such that and

defines another metric tensor on M. Suppose

det (g1;)-' = exp {F} .

4. Solution of the Equation

With the estimates of Section 1 and Section 3, we can now solve the

If R is the Kahler form of M, then the above equation is equivalent to equation det [gii+(d2(p/azi aij)] det (gij)-' = exp {F), where F E C3(M).

(4.1) (R + = (exp {F))Rm .

Hence integrating (4.1), we see immediately that

Conversely, we shall now prove that if F E Ck(M) with k Z 3 and F satisfies (4.2), then we can find a solution cp of (4.1) where p E Ck+'+(M) for any 0 5 a < 1. (Ck-',a (M) are the functions whose ( k - 1)-derivatives are Holder continuous with exponent a.)

We are going to use the continuity method. We consider the set

the equation det

(4.3) -1

= Vol (M) [ exp { t f l ] exp { t f l has a solution in Ckt'."(M)].

362 SHING-TUNG YAU

Obviously O E S. Hence we need only to show that S is both closed and open in [0, 13. This will imply that 1 E S and that our original equation has a solution in Ck+lzu(M).

To see that the set S is open, we use the standard inverse function theorem (cf. [14]). Let 8 = {Q E Ck+l*"(A4) 1 1 + qir > 0 for all i and I M Q = 0)

and B = f~ Ck-',"(M) f =Vol A4 . Then 8 is an open set in the Banach

space Ck+'@(A4) and B is a hyperplane in the Banach space Ck-',"(A4). We have a map G mapping 8 into B:

{ IL I

(4.4) G(rp)=det

The differential of G at a point Qo is given by det (g i r+ aZQo/azi azj) det(gir)-'Aao, where A*,, is the (normalized) Laplacian of the metric 1 (g,r+d2Lpn/8zi azi) d z ' @ d f ' . The tangent space of B is the space of

-functions such that f = 0. Ck-1 ," I, It is well-kncwn that the condition for Aqp0(p = g to have a weak solution

g dV, = 0. Hence the condition for

det (g,;)-lA,,cp = f

to have a weak solution is that f = 0. The Schauder theory (cf. [13]) makes

sure that Q E C k + l * u ( M ) when f~ Ck-l,s(M). The solution is clearly unique if

we require that Q = 0. Hence the differential d G of G at cp0 is invertible

and G maps an open neighborhood of 'Po to an open neighborhood of G(Q,,) in B. this proves that the set S is open.

It remains to prove that the set S is closed, If {t,} is a sequence in S, then we have a sequence Q ~ E C ~ + ' * ~ ( M ) such that

I, L

) det (gi7)-' = Vol ( M ) (I, exp {t,F})-' exp { t ,F} .

Normalizing, we can assume qq = 0.

ON THE RICCI CURVATURE OF A COMPACT K;iHLER MANIFOLD 363

Differentiating the above equation, we have

(4.5) [exp {tqF} det k f ) l ,

where (g6') is the inverse matrix of (g l ;+d2cp , /az ' d F J ) for each q. Proposition 2.1 shows that the operator on the left-hand side of (4.5) is

uniformly elliptic. Proposition 3.1 shows that the coefficients are Holder continuous with exponent a for any 0 5 a 5 1. The Schauder estimate then gives an estimate on the C2,"-norm of acp,/azp. Similarly we can find the CZ70r-norm of dcpq,'aiP. From this information we kqow that the coefficients of the elliptic operator on the left-hand side of (4.5) have better differentiability. The Schauder estimate again provides better differentiability of dcp,/dzp and acp,/d?'. Iterating, one finds Ck+'+' -estimates of cp, . Therefore the sequence {cp,} converges in the Ck+13a-norm to a solution of the equation

det ( gi;)-' = Vol ( M ) (I exp { t0F})-' exp { t,F) ,

where to = 1imq+- t , . Hence S is closed.

THEOREM 1. Assume that M is a compact Kahler manifold with metric r

C g i ; d z i @ d i ' . Ler F be Ck(M) with k 2 3 and exp{F)=Vol(M). Then Jh4

there is a function cp in Ck+'."(M) for any OS'aCl such that 2 C (giy+d2cp/dzi 132') d z i @ d f i defines a Kahler metric and

As a consequence of Theorem 1, we can prove the conjecture of Calabi [3]. Let 1 gg dx" €3 d f a be a Kahler metric on a complex manifold M. Let 1 R a ~ d r o r @ d f P be the corresponding Ricci tensor. Then Chern [6] proved that the ( 1 , l ) form (i/27r) 1 Rap dz" A d I P represents the first Chern class of M. Calabi asked the following question: Given a (1 , l ) form (i/27r)C Rapdzar\dZP which represents the first Chern class of M, can one find a Kahler metric on M so that 1 kp dz" a d z P is the Ricci tensor of this metric. Calabi himself observed the uniqueness of this problem. Namely, there is at most one Kahler metric in each cohomology class which does this job. He also observed the local version of this problem. He proved that, if the tensor C Rap dz"@dZP can be realized, then all tensors which are close to this one and which represent the first Chern class can also be realized.

364 SHING-TUNG YAU

To see how Calabi's conjecture follows from Theorem 1, we notice the following well-known fact (see (1.10)):

(4.6)

Since we assume that ( i /27~) 1 R,B dz, A dZp represents the first Chern class, we conclude immediately from Chern's theorem that

(4.7)

for some smooth real-valued function f .

1 (gap +a2p/az" aZB) d z " @ d P defines a Kahler metric and that According to Theorem 1, we can find a smooth function p so that

(4.8) a2p ) det (gap)-' = C exp { f } ,

where C is a constant chosen to satisfy the equation

(4.9)

From (4.6), (4.7) and (4.8), it is easy to see that C tensor of C (gap +a2cplaza a i p ) dz"@dZP.

R , p dz" @ d Z P is the Ricci

THEOREM 2. Let M be a compact Kahler manifold with Kahler metric C gap dz" @ dZP. Let C R a ~ dz"@ d Z P be a tensor whose associated (1 , 1 ) form (i/27r) R e p dz" A d i p represents the first Chern class of M. Then we can find a Kahter metric gapdzU @dZ@ whose Ricci tensor is given by C R a p dz" @ d P . Furthermore, we can require that this Kahler metric has the same Kahler class as the original one. In this case, the required Kahler metric i s unique.

Remark. The uniqueness was proved by Calabi [4] and will also be indicated and proved in Theorem 3.

5. Complex Monge-Ampere Equation with Degenerate Right-Hand Side

Let L be a line bundle over a compact Kahler manifold M. Let s be a (non-trivial) holomorphic section of L. Suppose L is equipped with a Hermitian metric so that we have a globally defined function (sI2 on M. Then,


for k20, we shall study an equation of the form

(5.1)

where giT d z i @ dZi is a Kahler metric and F is a smooth function such that

,. (5.2) exp {F) = V O ~ ( M ) .

JM

In order to solve (5.1), we approximate the equation by the following non-degenerate elliptic equation:

(5.3) det ( gi7 + - a:rrzj)det (gi7)-l = C,(lslz+ E ) ~ exp { F } ,

where E > 0 is a small constant and

(5.4)

It was proved in Section 4 that (5.3) has a smooth solution (P, such that (gii+a2cp,/dzi 82') is positive definite and

cp,=o. I, (5.5)

We are going to prove that when E approaches zero, cp, tends to a solution of (5.1). This depends on estimates of cps which are independent of E .

Since the right-hand side of (2.28) depends only on the Green's function of the fixed Kahler metric, we can find a constant C, independent of E such that

In order to estimate inf, cpE and AcpE we notice that, when s # 0,

(5.7) A I O ~ ( J S ~ ~ + E ) Z - " I 2 (A log 1 ~ 1 ' ) . ( s ( Z + E

366 SHING-TUNG YAW

By direct computation, one knows that, when sZ 0, A log Is( ' depends only on the fixed Hermitian metric of L. In fact, it is the trace of the first Chern form of L with respect to the tensor X I , gl;dzi@ddzJ. Hence Alog ( I s ~ * + E ) is uniformly bounded from below by a constant independent of E .

Let A be the normalized Laplacian of the metric I,,, (gi;+ (82cp,/8z1 d 2 ' ) d z ' @ d Z J . Then according to (2.22), we have

By (5.7), A log Section 2 that the following inequality holds:

E ) is bounded from below. Hence one can conclude as in

where C, is a constant independent of E.

inequality. We find If m = 2, we multiply (5.8) by (lsI2+ E ) ~ exp {F) and integrate the resulting

Here C, is the trace of the first Chern form of L with respect to the tensor Ci,i g i r dz'@ dz'.

For any 6 > 0 , we can find a constant C, such that

(5.11) m+Aq,Si(rn+Acp,)2+Cg.


Hence by choosing C +infi+[ (Rirlr) L 1, we can obtain an estimate of the form

where C3 is a positive constant independent of E.

independent of E and hence also an estimate of supM Acp, independent of E.

Then by Schwarz’ inequality, we have

One can now proceed as in Section 2 to find an estimate of infMcp,

For m > 2 , we proceed as follows. Let p be any non-negative number.

-C(lsl*+ E ) ~ exp {-Ccp,}A’,cp,.

For s f 0, A: log IsI2 is dominated from below by the trace of the first Chern form of L with respect to (gi~+82cpe/8zi 82’) dzi@d2j. Hence there is a positive constant C, independent of E such that

(5.14)

As in (5.7), we have

(5.15)

Hence

368 SHING-TUNG YAU

Inserting (5.16) into (5.13), we obtain

- (5.17)

= -rnC(ls12+ E ) P exp {-Cq,}

Using the arithmetic-geometric mean inequality and (5.3), we deduce that

(5.18)

x exp ( -:} exp 1- ~ c p , ) .

Multiplying (5.18) by (lsI2+ E ) ~ exp {F) and integrating, we find a positive constant C, independent of E such that

In (5.19), C and p are arbitrary non-negative constants. By iterating (5.19), we see that, for all fixed p 2 0 , we can find (large) constants C and C, such that

On the other hand, by (5.8), we can find positive constants C, and C,


independent of E such that

Multiplying (5.21) by (Is/'+ E ) ~ exp {F) and integrating, we obtain

where C9 is independent of E. Since rn + A q , > O , it follows from (5.22) that we can find a positive constant Clo independent of E such that

(5.23) JM (Is/'+ E)k+l(m +Acp) exp{-Ccp,}S CloJM E ) ~ exp {-Ccp,}.

Integrating by part in (5.23), we derive

(5.24)

Hence

370

(5.26)

where Cll is independent of E. (We use the fact that IV lsI2l2 is dominated by Isl")

By making use of (5.20), ( l s12+~)k exp{-Cq,} is dominated by I, I, (lsI2+ E)k+l exp {-CCp,).

Therefore we can find a constant C,, independent of E such that C,, independent of E such that

(5.27) JM l ~ [ ( l s 1 2 + &)(k+1)/2 exp { - ~ C ~ & } ] I Z ~ c'./ ( \ s 1 2 + &)'+I exp {-cp,). M

Since (5.5) and (5.6) provide an estimate of lqEI independent of E, we

see applying the arguments of Section 2 that (5.27) gives an estimate of

(Is['+ E ) ~ + ' exp {-Ccp,} independent of E. (Suppose there is no estimate.

(Is\'+ E ~ ) ~ + ' exp {-Ccp,}

I,

JM I, Then one can find a sequence si + 0 such that

tends to infinity. Then one defines

exp {-c+,}= exp{-cq,,~( I, (1sl2+ Ej)k+1 exp {-eve,})-' . By (5.27), (Is\'+ E ~ ) ( ~ + ~ ) ' ~ exp {-&jjj} converges in L2(M) to a non-negative

function f. Using the estimate of 1p.l and restricting to sets of the form


Is/ 2 l /n, one shows that f is zero almost everywhere and hence obtains a contradiction.)

As in Section 2, inequality (5.9) and an upper estimate of cp, give an estimate of (Vp,( (exp {-C inf q,}+ 1)-' independent of E. Hence, for some geodesic ball B(x, R ) of radius R = C1,(-inf cp,)(exp {-C inf q,}+ l)-', (P, is not greater then 4 inf c p E . (Here C13 is a positive constant independent of E ,

and R is less than the injectivity radius of M.) Over this geodesic ball, the integral of ((s('+ E)'+' exp {-Ncp,} is not less than

C,, exp {-4N inf cp,} ra(k+l) dr I," (5.28)

exp {-4N inf cp,}[CI3(-inf cp,)(exp {-C inf q,}+ l)-l]ak+n+l , >- c14 - - a k + a

where C,4 and a are positive constants independent of E.

By choosing N large enough, inequality (5.28) and the estimate of

( (s I2+ E ) ~ + ' exp { -Nq , } provide an estimate of -inf cpE independent of E

and inequality (5.9) gives an upper estimate of rn + Acp, independent of E.

Let us now show that I(cpE)i;kk(2 has an upper bound independent of E on compact subsets of the complement of the divisor of s. In fact, we can follow the arguments of Section 2 to define S, to be c g',"g',"g'k'(cpE)ii((P~)~~~. Let p be a non-negative smooth function in M whose support is disjoint from the divisor of s. Then over the support of p, the metric tensor cL,, (gij+ a2cp,/dzi dZJ) d z i @ d Z j is uniformly bounded from above and below by some positive multiple of gii d z ' @ dZj, the constants being independent of E .

(This is because we have an upper estimate of rn + Acp, and, over the compact set under consideration, the volume form has a fixed positive volume form as a lower bound.)

From inequality (2.10), we can find positive constants C15 and C16 independent of E such that

I,

372 SHING-TUNG YAU

Because the new metric is uniformly equivalent to the old one on the support of p, the right-hand side of (5.30) can be estimated by a constant independent of E . Since lsI>O on the support of p, we can therefore find an estimate of

pS, independent of E. Since the function p is chosen quite arbitrary, we

see that we have found an L,-estimate of S, over any compact subset of M which is disjoint from the divisor of s. - , 2,) 1 xi ( z i / '5 R } be a coordinate chart in the complement of the divisor of s. Then we claim that we can estimate S,(O) in terms of the L,-norm of S, over the coordinate ball. First of all, we note that, using the computations of Section 3, we know that there are positive constants C1, and C, , independent of E and cp,, such that the inequality

I, Let B ( R ) = {(z, ,

(5.31)

holds on B ( R ) . By choosing C, , larger if necessary, we may assume that the function

5, = S, + C,, AcpE + C18(Ci Izi ('+ 1) is ppitive. We can now solve the Dirichlet pro)lem to find a smooth function S : = s , on the boundary of B ( R ) and A:Sz = 0. By the maximum principle, S, I sE > 0 in B ( R ) .

By using the fact that A: is the normalized Laplacian of a Kahler metric, we see that A: can be written as

where gk,r= g,,+d'cp,/dz' a,? and (g'Li) is the inverse matrix of (gl.,;). Since the metric z1,' g',,idz_'@d.T' is uniformly equivalent to I,,, 8,; dz '@dZJ on B ( R ) , we know that s, is a solution of a uniform elliptic equation of divergence form whose elliplicity is estimated.

Using Moser's Harnack inequality [ 1, we conclude that there is a positive constant CI9 independent of cp, and E such that

(5.31')

We are going to prove that the right-hand side of (5.31') can in turn be

estimated by I 8. To achieve this, let u be a non-decreasing C'-function

defined on the real line such that u ( t > = O for tSO. Then, for 7 < R, we can 3 B ( R )

define JI on the real line to be +(s) = ta(7- t ) dt. It is easy to check that


$ ( r ) = $((Ci I Z ~ ~ ~ ) * ’ ~ ) is in C’,’(B(R)) and that $ ( r ) vanishes outside a compact subset of the interior of B ( R ) .

By direct computation, we have

Since $ ( r ) -has compact support in the interior of B ( R ) , we can multiply (5.32) by gE det (g&) and integrate with respect to the Euclidean volume form dE to obtain

Since uZ0, and u‘1-0, it follows immediately from (5.33) that

Therefore, by the uniform ellipticity of A’, we can find a positive constant C,, independent of u, T, qE such that

(5.35)

Letting T tend to R in (5.351, we see that (5.35) is also valid when T is replaced by R. Now choose u SO that u(t) = 1 when t Z E and u’(t)S 1 / ~ for

& u ( ~ - r ) dE 5 c,, I,,., &’(T - r ) dE .

374 SHING-TUNG YAU

all t. Then (5.35) implies

Letting F + 0 in (5.36), we see that d E can be estimated from above . ,

s,. Since $ I dB(R) = 9, 1 aB(R) and g, > O , we conclude from (5.31’) by I aB(R) that there is a positive constant CZ1 independent of cpe and E such that

(5.37)

Since C,, can be chosen to be independent of R when B ( R ) lies in a fixed coordinate chart in the complement of the divisor of s, we can integrate (5.37) to find an estimate of sE(0 ) in terms of the L’-norm of $ over a ball in the coordinate chart. Combining this with the previous L’-estimate of sE, we conclude that we have an estimate of S, on every compact subset of the complement of the divisor of s.

Therefore we have a uniform Lipschitz estimate of (~p,)~; over any compact subset of the complement of the divisor of s. It is now easy to estimate the higher derivatives of c p E . In fact, we can fix a local coordinate system in M so that (5.3) holds with cp replaced by rp,. Then we differentiate (5.3) with respect to a/azk. We see that arpE/azk satisfies an equation of the form

(5.38) A: -r; = 7 [log (C,((sl2+ E ) ~ exp { F } det (gi;))]. (3”2’.) a:

In our coordinate system, the operator A, is an elliptic operator whose coefficients are the coefficients of the inverse matrix of (gir + acpF/dzi afj). Since we have Lipschitz estimates of these coefficients over compact subsets of M which are disjoint from the divisor of s, the Schauder estimate (see [13]) shows that all higher derivatives of rp, can be estimated over these sets.

By letting E + O , we can now conclude that {cp,} has a subsequence converging to a solution rp of (5.11) such that cp is smooth outside of the divisor of s and { l c p i ~ l } is bounded for all i , j .

THEOREM 3. Let L be a holomorphic line bundle over a compact Kahler manifold M. Let s be a holomorphic section of L. Let Cii gii dzi@ dZi be the Kahler metric of M. Then, for any k 2 0 and any smooth function F with


exp {F}= Vol ( M ) , we can find a solution cp of the equation I, = ( s I z k exp {F) det (gir)

with the following properties: Outside the divisor of s, cp is smooth and, ozler M, {Icplrl} is bounded for all i , j . Furthermore, any function Jr satisfying the above properties must be equal to cp plus a constant.

We have only to prove the last statement. Let A' be the Laplacian of the metric XI,, (g,;+a2cp/dz' dT')dz'@ddT'. We claim that, if f is a function such that {Ifi;l} is bounded over M for all i, j , then

Proof:

(5.39)

Indeed, using our previous notation, we have

(5.40) (A:f)(lslz+ E ) ~ exp {F} = 0 . I If we let c(g,),r be the (i, T)-th cofactor of the matrix (giT+a2cp,/azi azi),

then (5.40) is equivalent to

Since c(g:)i i and a2f/azi 321 are bounded independent of E, we can use the Lebesque dominated convergence theorem to obtain (5.39) from (5 .41) .

Now let $ be another solution of (5 .1) satisfying the properties mentioned in the theorem. Then we have

(5.42) det ( g i i + (pi ; ) = det ( giT + $ i ~ )

Rewriting (5.42), we have

(5.43) det(gir+cpiT+($-cp)iT)det(g,T+cpiT)-l= 1 .

Using the arithmetic-geometric mean inequality, we have

1 - [ m + A ' ( $ - c p ) ] Z l . m (5.44)

376 SHING-TUNG YAU

Therefore, $ - c p is subharmonic with respect to the operator A'. Since \$irl and Jcpiil are both bounded, I(4-cp)gl is also bounded over M and $ - c p is a C'-function over M. By adding a constant to 9-9, we may assume that $ - cp 2 0. Then applying (5.39) to ($- cp)', we obtain

Since ($ - c p ) 2 0 and A'($ - cp) 2 0, we conclude from (5.45) that Of($ - cp) = 0 and $-cp is a constant.

6. Complex Monge-Amphe Equation with More General Right-Hand Side

In this section, we study equations of the form

= exp {F(x , cp)} det (g i i ) ,

where gii dz'@ddzJ is a Kahler metric defined on M, and F is a smooth function defined on M x R with aFlat 2 0.

We shall look for real-valued smooth functions cp satisfying (6.1) such that 1 (g,;+d2cp/dz' a,?') d z ' @ d z J defines another Kahler metric. If such a function cp exists, then integrating (6.1), the integral of the right-hand side is equal to the volume of M. Therefore, a certain compatibility condition must be satisfied for the solvability of (6.1). The condition is that there exists a smooth function $ defined on M such that

Relying on (6.2), we are going to use an iteration method to solve (6.1).

Let cp and yS be two smooth solutions of (6.1) such that both First of all, let us prove the uniqueness of the solution of (6.1). LEMMA.

and

define Kahler metrics on M . Then cp-4 is a constant.

O N THE RICCl CURVATURE OF A COMPACT KAHLER MANIFOLD 377

Proof of Lemma 1: Using the fact that both cp and Cp satisfy (6.1), we can immediately derive the following equation:

Let A, be the normalized Laplacian of the metric

Then it follows from the we have the inequality

arithmetic-geometric mean inequality and (6.3) that

(6.4) rn+A,(cp-Cp)Zrnexp .

By the mean valued theorem we have

where t(x) is a number between inf ( cp (x ) , Cp(x)) and sup (cp(x), Cp(x)). Since aF/at 2 0, we can combine (6.4) with (6.5) to conclude that whenever

cp(x)- Cp(x) is strictIy positive, A,(cp - S;)(x) is non-negative. If supxEM (cp - Cp)(x) > 0, then, since M is compact and cp - Cp is smooth,

the standard proof of the maximal principle shows that cp-Cp is a constant function. Therefore we conclude that either cp-Cp is a constant function or cp(x)S +(x) for all x E M. Clearly we can interchange the roles of cp and Cp in the above assertion. Combining these two inequalities, we can deduce Lemma 1.

After this preparation, we proceed to solve (6.1). By Theorem 1, we can find a smooth function cpo such that C,,i(gi~+a2cpo/az' a i l ) d z ' @ d f J defines a Kahler metric and

If we define

378 SHING-TUNG YAIJ

and

then clearly both Cpo and Go satisfy equation (6.6) with cpo replaced by Cpo and yo, respectively. Furthermore,

(6.9) G o b ) 2 $(XI 2 G o ( 4

for all x on M.

Hence we can define The set A = {(x, t) I x E M, Cpo(x) 2 t I &(x)} is a compact subset of M x R.

aF (6.10) k = SUP - ( ~ , t ) + 1 > 0

(x, t ) E A at

Using this definition, we are going to prescribe our iteration scheme as follows. For each i 2 1, we define C p i and Gi as the smooth solutions of the following equations:

(6.11) det =exp{k@,+F(x, @,-J- k@i-l}det(&p),

(6.12) det ( ~ ~ + + ) = e x p ( k y , + F ( x , az az $i-l)-k@i-l}det(g+),

so that both

define Kahler metrics. To make sure that such functions (pi and Gi exist, we need the following

LEMMA 2 . Let M be a compact Kahler manifold with Kahler metric g i i dz'@ d2'. Let F ( x ) be any smooth function defined on M. Then, for any

constant k > 0, there exists a unique smooth function cp such that

det gg + - = exp { kcp + F} det ( g i r ) ( 8 2 Z J )

and ,&j g i i dzi @ dz" defines a Kahler metric.


Proof of Lemma 2: As in Theorem 1, we can use the continuation method where the one parameter family of equations is

The fact that k > O makes the proof of Lemma 2 much easier. In fact, the estimate of supM Icpl can be derived easily by the maximum principle as follows. When cp achieves its maximum at a point xo, we must have

Hence exp {kcp(xo)+ tF(x,,)}S 1. This implies immediately sup cp = cp(xo)S -(t/k)F(xo). Similarly one can draw an estimate of inf,cp. The uniqueness part of Lemma 2 follows easily from Lemma 1 noting that k # 0.

Lemma 2 justifies our construction of the iteration scheme. We claim that, for all i 2 0,

For i = O , we use the fact that & satisfies equation (6.6) and inequality (6.9) to conclude that

At the point where Cpl-Cpo achieves its maximum,

)C,det ( g @ + _ p ) . a2 (Po az az

Hence it follows from (6.14) that sup, (Cpl - Cpo)5 0. Similarly one can also prove that sup, (fo - GI> 5 0.

To show that CplSCpl, we divide (6.11) by (6.12) to obtain

) det (gp += (6.15)

=exp{k&- k&+F(x, &,)-F(x, Go)- k(cpo-$o))

380 SHING-TUNG YAU

Since Cpo2=Go and k is defined by (6.10), it follows that

Therefore,

At the point where Cpl-'yl achieves its minimum, the right-hand side of (6.17) is non-negative and hence Cpl Z G 1 .

Suppose now that we have proved - -

(6.18) c p n ~ C p j - l ~ ; j ~ @ . " , - cp,-1= - -=z cpo -

for all j < i. We claim that (6.18) remains valid when j is replaced by i. In fact, applying (6.1 1) twice, we have

(6.19)

= exp { kCpi - kCpi-, + F(x, C p i - l ) - F(x, - k(Cpi-l - .

Since Cpo2 (piP22 CpiPl Z Go and k is defined by (6.10), we see that

)det ( g a p 2 e x p { k @ , - kc&-,}

Hence the maximal principle shows that @ i - l Z @ i . Similarly one can show that @i- l 5 C p i .

- -

To prove that Cpi 2 &, one just divides (6.11) by (6.12) and obtains

= exp {k(pi - kGi + F(x, C p i P l ) - F(x , Cpi- l ) - kCpi_l + .

Using CpiPIZ@i-l, one can repeat the above argument to show that 'pi2&. Therefore the inequality (6.13) is valid in general and both C p i and Cpi are uniformly bounded.

Our next step is to find a uniform estimate of a2Cpi/az" 82". As in Section 2, it suffices to estimate A&.


Let A, be the normalized Laplacian operator associated with the metric c,,p(g,p +a2Cpi/dza a Z p ) dz"@ddzP. Let C be any positive constant such that C+infi+l (Rinr)> 1. Then by the same computation as in (2.18), we have

Since SUP,

[13]) that, for some constant C1 independent of i, has been estimated, it follows from Schauder's estimate (cf.

(6.23) M

As in (2.20),

1 2 (rn + (exp {- -k (qi - qi - l ) +- 1 F(x, q i - l ) ] (6.24) l + ( @ , ) , ~ m - 1 m - 1

where C, is a positive constant independent of i. Noting again that SUP, lqil has been estimated, it follows from (6.22),

(6.23) and (6.24) that there are positive constants C,, C,, C, and c6 , independent of i, such that

382 SHING-TUNG YAU

At the point where exp {-C@,}(m +A@,) achieves its maximum, the right-hand side must be non-positive and so

C6 m +sup A@, ( M

1 " M

m i exp [- sup (pi x C, m +sup (6.26)

m - 1 M

+ C, m +sup ACpipl + C, m +sup ( M ) ( M

Choose a constant C, such that, for all x Z 0 ,

sup cpi x - c, . I [ m y 1 M (6.27) x l+l'(m-l) 2 (C, + 2 C, + 2 C,) C; ' exp -

Then (6.26) implies that we can find a positive constant C,, independent of i, such that

(6.28) ( M m+supA@, 5- m+supA@ipl)+C, . M

By iteration, (6.28) gives

m+supA@,S(;)' m+supA@, +2C8. (6.29) M ( M I Therefore we have found uniform estimates for a"(pi/aza aZp. To find uniform estimate of aZ@,/aza di@ azy, let

(6.30)

where (g',"') is the inverse matrix of (gar+az@,/az= 35'). By a computation similar to that of (3.12), we have

(6.30') Ai(Si + C9A@i)2 CloSi - C 1 1 J s a - C12,

where C,, Clo, Cll and C,, are positive constants independent of i.


Since [A@,[ has been estimated, it follows from the maximum principle that

(6.31) sup S. 5% d s d F l +&+ C9 SUP \ A & \ . M ' - G o ClO M

It should be noted that in (6.30), we can choose Clo to be arbitrarily large if we are allowed to increase C, and C1,. In particular, we may assume that CI1/CloS~. Then (6.31) shows that

sup S. <-sup 1 Si-l+4c"+4C9 sup IA+il. M ' -3 M Cl 0 M

(6.32)

By iteration, we can find a uniform estimate of Si and hence a uniform estimate of a2Cpi/azm a z p azy.

Letting i tend to infinity in (6.11), we can then obtain a solution of (6.1). The Schauder estimate guarantees the solution to be smooth.

THEOREM 4. Let M be a compact Kahler manifold with Kahler metric gi; d z i @ dZi. Let F be a smooth function defined on M X R with aF/at I 0.

r Suppose that, for some smooth function II, defined on M, J exp {F(x , II,(x))} =

M Vol(M). Then there exists a smooth function cp on M such that

= exp {F(x, cp(x))} det (gii)

and (gii + a'cplaz' a i i ) dz'@ dZJ defines a Kahler metric. Furthermore, any other smooth function satisfying the same property differs from cp by only a constant.

As a consequence of Theorem 4, one deduces that, on a Kahler manifold with ample canonical line bundle, one can find a (unique) Kahler-Einstein metric whose Ricci tensor is the negative of the metric tensor.

In fact, by hypothesis, the negative of the first Chern class of M is represented by some positive (1, 1)-form a Ci,i gi; dz' A di ' . Take this form as our Kahler form. Then by Chern's theorem [6], the closed ( 1 , 1)-form -a? log det ( g i r ) also represents the first Chern class of M. Hence we can find a smooth function f such that

(6.33) as log det <gii) = 42 1 gijdzi A dZj +a; i f . i,i

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Now by Theorem 4, we can solve the equation

(6.34) =exp{cp-f}det (s i r )

so that (6.34) we have

(gii+d2cp/dzi 82') d z ' @ & defines a Kahler metric. By (6.33) and

(6.35)

This last equation simply says that the Ricci tensor of the metric C,,,(g,r+ a2&z' a?) dz'@,dZ' is the negative of the metric tensor itself. Hence we have found the metric that we are looking for.

To see that the metric is unique (that is to say, that the metric depends only on the complex structure of M ) we notice that if I,,, &-dz'@ddzl is another such metric, then its Kahler form must represent the negative of the first Chern class of M. Hence we can find a smooth function (Ir defined on M such that glr = g,, +d2(Ir/az' 132 . On the other hand, the fact that the Ricci tensor of s l y dz' @ d2' is the negative of the metric tensor can be written as

= -aalogdet (gil)+aaf --a&,

where the last equation follows from (6.33). Equation (6.36) shows that the function

is a pluriharmonic function. Since we are on a compact manifold, this function must reduce to a constant exp {c } . Therefore,

(6.37) = exp {$ + c - f ) det (giT) .

The function $ + c then satisfies an equation of the form of (6.34). Lemma 1


shows that c p - $ is a constant. Hence,

THEOREM 5. Let M be a Kahler manifold with ample canonical line bundle. Then there is a Kahler-Einstein metric whose Ricci tensor is the negative of the metric tensor. Furthermore, a metric of this form is unique and depends onty on the complex structure of M.

7. Degenerate Complex Monge-Ampere Equation with General Right-Hand Side

In this section, we combine the main results of the last two sections. Let L be a line bundle over a compact Kahler manifold M. Let s be a (non-trivial) holomorphic section of L. Suppose L is equipped with a Hermitian metric so that the function 1sI2 is globally defined on M. Then, for k20, we study equations of the form

(7.1) = exp {F(x , c p ) } det ( s i r ) ,

where F is a smooth function defined on M x R with aFlat Z 0. As in Section 6, we look for solutions cp of (7.1) such that (cpii( is

uniformly bounded on M and cp is smooth outside the divisor of s. (In fact, cp is going to be the uniform limit of a sequence of smooth functions (cp,} so that det (giT+a2’pa/azi 32’) det (gii)-’ converges uniformly to

exp {F(x , cp)}. Clearly a necessary condition is obtained by integrating (7.1). The result is that there is a function t,b whose partial derivatives are uniformly bounded on M so that

(7.2) exp {F(x, 9)) = Vol (M) . I, With this sole assumption on the existence of $, we proceed to solve (7.1).

We approximate (7.1) by the following non-degenerate elliptic equation:

(7.3) = C,(lsI2+ E)’ exp {F(x, c p ) } det (gi;) ,

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where E > O is a smooth constant and

(7.4)

Here we choose {&} to be a sequence of smooth functions such that (CI, converges to + uniformly on M and that sup, Jd*ICl,/dz' 82'1 is uniformly bounded on every coordinate chart.

According to Theorem 4, we can find smooth-solutions qE of (7.3) such that 1 i . j (gii+d2q,/dz' 35') dz'@ddi' defines a metric. By examining the proof of Theorem 4, we can also assume that we have an estimate of SUP, (cp,( in the following way.

Let qE and & be two smooth solutions of the equation

(7.5) = C,(lsI2+ E ) ~ exp {F(x , &)} det (gii)

such that

Then the arguments of Theorem 4 show that

On the other hand, for the unique solution of (7.5) with cp = 0, we

can find an estimate of SUP, 191 which is independent of E. (This is seen in the proof of Theorem 3. Note that boundedness of AJI, is needed.) In particular, we can choose Cp, to be bounded from above by a constant independent of E , and & to be bounded from below by a constant independent of E .

Hence according to (7.7), we can find a smooth solution cpE of (7.3) whose supremum norm is bounded from above by a constant independent of E and whose matrix (giF+d2cpe/dzi 6'2') is positive definite.

Let us now proceed to estimate AqE from above. Let As be the normalized Laplacian of the Kahler metric (gi~-d2cpEldzi 6'9) dz' @ d2'.

I,


Then, as in (5.8), we have

(7.8)

Choose C so that C+infi.l (R,ni)2$Chl. Then noting (5.7) and the fact that sup, [ c p B ( is uniformly bounded, we can find positive constants C1, C,, C3 and C, independent of E such that

A’(exp {- Ccp,}( m + AqP,) ) 2 - C , - C,( rn + Acp, )

- C3 IVq, I + C,(m + AqE)m/(m-l) . (7.9)

On the other hand, by Schauder’s estimate and the estimate of supM ]cp,I, there are positive constants C, and C, such that

Combining (7.9) and (7.10) and the estimate of sup, lcp&l, it becomes clear that we can apply the maximum principle to find an upper estimate of m + Acp,. Therefore, since Ci,i (giT + a’ cpE/azi afj) dz‘ 8 d f i defines a Kahler metric, we have uniform estimates of (a2cp,/azi a f J ( on every coordinate chart of M.

Using the uniform estimate of a2cp,/azi a,?, one can now follow the arguments of Section 5 to provide higher derivative estimates of qE on compact subsets of the complement of the divisor of s. Letting E -+ 0, we have then proved the following theorem.

THEOREM 6. Let L be a holomorphic line bundle over a compact Kahler manifold M whose Kahler metric is given by Ci,i girdzi8ddi i . Let s be a holomorphic section of L. Let F be a smooth function defined on M x R such

388 SHING-TUNG YAU

that aFlatZ0. Suppose, for some function I/J whose partial derivatives la2$/azi aziJ are bounded on every coordinate chart of M, we have

lslZk exp {F(x , I/J(x))=Vol (M). Then we can find a solution cp of the

equation det (gir+a2cp/azi a,?) = exp {F(x , cp(x)} with the following properties. Outside the divisor of s, cp is smooth and (gir+a2cplazi aZi) dzi@ddii defines a Kiihler metric. Over M, {Iqiil} is bounded for all i, j . Furthermore, any solution satisfying the above properties must be equal to cp plus a constant.

Proof: We have only to prove the last statement. Let A' be the normalized Laplacian of the metric I,,, ( gii + 32cplaz 82') dz 63 dz'. Then we claim that if f is a C'-function on M such that, for all i, j , la2f/dzi aFi( is bounded on every coordinate chart of M, then

I,

(7.11) A'(f2) lS12k exp M x , c p b ) } = 0 . {x I f(x)>o)

Indeed, let A: be the normalized Laplacian of the Kahler metric (g,r+d2qE/dz''82') dz'Bd2' . Then, for all 6 > O such that the boundary

of {x I f ( x ) L S} is a C'-manifold, we know that by Stoke's theorem,

&(f2)(ls12+ exp {F(x, cp,(x)} (x lfcx,zs

can be expressed in terms of the boundary integral of 2faflan. Here alan is the normal of the sets {x l f ( x ) = S} taken with respect to our metric xl,f (g, +8*q,/az' 8.2') dz'@ddzf. Since { z I f ( x ) >0} is an increasing union of the sets {x I f(x)Z6}, we know that the measure of {x 16 > f(x)>O} taken with respect to ( I s ~ ~ + E ) ~ exp{F(x, cp,(x)} d V tends to zero as S tends to zero. Applying the co-area formula (see [ l l ] ) to the set {x 16 > f(x)>$S}, it is easy to see that we can find a sequence (6,) such that the product of 8, with the codimension one volume of {x I f(x) = S , } tends to zero as 8, tends to zero. Considering this together with the previous fact, we conclude that

A:(fz)(ls12+ E ) k exp ~ x , cp,(x)~ { z If(x)LGJ

tends to zero as Si tends to zero. Hence we have

Letting E + 0 as in Theorem 3, we see that (7.11) follows from (7.12).

ON THE RICCI CURVATURE OF A COMPACT K h L E R MANIFOLD 389

Suppose now that @ is another solution of (7.1) with all the properties described in the theorem. Then

(7.13) det (gir + cpi7 + (4 - (P)~T) det (giT + = exp {F(x , @) - F(x, cp)}

Consider the set f l= {x I Cp(x)-q(x)>O); if it is nonempty, then the arithmetic-geometric mean inequality shows that the inequality

(7.14) A’(@ - cp) 2 rn exp [F(x , @(x)) - F(x, cp(x))]

holds on R. (Note that aF/at is used here.) Applying (7.11) to the function (4 - cp)’, we conclude immediately that

(7.15)

A’(@ - cp)’ lslZk exp {F(x, cp(x)} = 0 = b Combining (7.14) and (7.15), we see that V’(@ - cp) = 0 on fl and @ - cp is a

constant on each component of R={x I @(x)-cp(x)>O}. Since + -cp is continuous, this is possible only if fl is empty or Q = M. In the first case, @(x) S cp(x) for all x E M. In the second case, @ - cp is a constant. By reversing the role of @ and cp, we conclude easily that, in any case, @ - cp is a constant.

8. Complex Monge-Ampere Equations with Meromorphic Right-Hand Side

Let L, and L, be two holomorphic line bundles over a compact Kahler manifold M. Let s1 and s2 be two (non-trivial) holomorphic sections of L, and Lz that are equipped with Hermitian metrics so that we have globally defined functions lsllz and lszlz on M. Then, for klBO and k z B O , we shall study equations of the form

where gij d z ’ @ dT’ is a Kahler metric and F is a smooth function such that

390 SHING-TUNG YAU

As before we approximate (8.1) by the equation

where

In order to prove that the solutions qE of (8.3) converge on the complement of the divisors of s1 and sZr we consider the expression ( I S ~ J ~ + S ) ~ exp{-Cq,}(rn+Acp,) with p 2 0 .

Letting A; be the normalized Laplacian of the metric (gii+ a2cp,/azi 82') dz'addzj, we compute the Laplacian of the above expression as follows:


By applying the same reasoning as in (2.15), we have

As in (5.15), we have a positive constant C1 which is independent of E such that

(8.7)

Combining (8.6) and (8.7) and computing as before, we can find positive constants C, and C3 which are independent of E such that

Clearly, for any fixed p, we can choose C large enough so that

With this choice of C, we consider the point where (ls212+ E ) ~ exp {-Cq,}X (rn +AqE) achieves its maximum. At such a point, the left-hand side of (8.8) is

392 SHING-TUNG YAU

non-positive and so is the right-hand side of (8.8). Therefore, at this point,

It follows easily from (8.9) that, for some positive constant C, independent of E. we have

It is easy to see from (8.2) that k 2 < 1 . Hence the third term in the right-hand side of (8.10) will be the dominating term. If we choose p to be (rn-l)/rn+(k,/rn)+Cq with q 2 0 in (8.10), we see that, for some positive constant C, independent of E ,

We are going to estimate the supremum norm of 1cp.I. First of all, we recall that when we solve (8.3), we can normalize our solution cp, so that ..

cpE = 0. Then, using the inequality Acp, + rn > 0, we have an upper estimate

of c p E . Hence it remains to find a lower estimate of c p E .

x exp {F) dV, we have Integrating (8.8) with respect to the volume from (ls2I2+ ~ ) - ~ z ( l s ~ 1 ~ + E ) ~ ,

- pC, + inf Ririi exp {inf F } , # I 1

(Is2/' + E)~-(m-2)k,/(m-l)

x ( m + A L ( P , ) ~ / ( ~ - ' ) exp {- Ccp, 1

ON THE RICCI CURVATURE OF A COMPACT K ~ ~ H L E R MANIFOLD 393

5 C, exp {sup F} (ls2I2+ ~ ) ~ - ~ z ( l s , l ~ + E ) ~ , exp {-Ccp,} I, + mC exp {sup F} ((s2I2+ E ) ~ - ~ Z ( ( S ~ ( ~ + E ) ~ , exp {-Ccp,}(m + Acp,)

Using (5.7), we can find a positive constant c6 which is independent of E

such that

Hence, for any positive constant L,

+ LC, exp (sup F) (ls2I2+ E ) ~ - ~ ~ ( ) s ~ ) ~ + E ) ~ , exp {-Ccp,} JM

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(Js2I2+ E ) P - ~ ~ Z ( ( S J ~ + ~ ) ~ ~ 1 exp {-Ccp,} s-"

For any p Z 0 , we choose a constant C large enough so that C21, C3(C-pC1+infi.,Rinr)~~C3C. Then we choose S so that

(rn - 1)6m"m-"CC;' exp {sup F - inf F } = .

Substituting (8.15) into (8.12) and keeping (8.14) in mind, we see that we can find positive constants C, and c g which are independent of E and C for which

In order to make use of (8.16) to derive an integral estimate exp{-Cq,},

we shall assume that the integral I ~ ~ 1 - ~ ~ ~ a is finite. Then by choosing p so that

C,(p - k2 3- C7)-l = 4, we shall prove in the Appendix B that, €or some positive canstant C,, independent of 8 and C,

I


Since we have an estimate of Icpel, we can use the method of Section 5

to find an estimate of

which is independent of E.

Note that in (8.11) and (8.17) we may choose two different values of C as long as they are larger than some fixed constant. From (8.11) and the supremum estimate of cp&, we derive that, for any q 2 0,

5 C5(Cm-l + 1) exp {C sup cp,}(sup (Is2I2 + ~ ) q exp { - ~ p , } ) ~ ,

where C is any positive constant so that

From (8.17), we conclude that when ( p - k , ) ( p - k 2 + C 7 ) = i and N is a large constant, we can find a positive constant C,, independent of E such that

(ls2I2+ E)P-k2(Jsl12+ E ) ~ , exp {-Ncp,} (8.19)

= (l~,1~+ ~ ) ~ 7 ( l s , l ~ + E ) ~ , exp {-iVq,}S C,, . Jh4

Recall that C, is independent of N also. Using (8.18) and (8.19) we shall show that, for any q>O,

( lszlz+ E ) ~ exp{-cpE} has an upper bound which is independent of E when E

tends to zero. In fact, if this assertion were not true, we could find a sequence of

numbers { E ~ } and a sequence of points {xi} in M such that E + O and (Isz12 (xi)+ E ~ ) ~ exp {-cp,,)(x,) = sup, (lszl2+ E ~ ) ~ exp {-cp,,} tends to infinity. Without loss of generality, we may assume that {xi} converges to a point xo in M.

Suppose the sequence { E ; ~ Isi\’ (xi)} is bounded independent of i. Then, clearly, ~fexp{-cp,,}(x,) tends to infinity when xi tends to xo. On the other hand, using inequality (8.18) and the L’-estimate of cp,, we can apply the

396 SHING-TUNG YAU

Schauder estimate to find a constant C , , independent of i such that

It follows from (8.20) that there is a constant CI4 independent of i such that

SUP (Wag (Is2IZ+ Ei)' - vo,,II M

(8.21)

5 c,4 - ( m -1)Im-(k,/m)-Cq sup (ls212+ ei)q exp {-cp,,}

Clearly we may assume that SUP, [q log ()s2I2 + E ~ ) - qC,] 2 0. Then proceed- ing as in Section 2, we can now conclude that for some constant eI4, independent of i,

(8.22) I"' ( ls2I2+ E J ~ exp {-cp,,}

In (8.19), the number N can be made as large as we want (where C12 depends on N ) . Therefore, (8.22) shows that SUP, ((s2I2+ E ~ ) ~ exp {-cp,,} is bounded from above independently of i.

The above contradiction shows that, by passing to a subsequence, we may assume that limi.+m E ; ~ lsz12 (xi) = w.

For each x i , let B ( x i , S i ) be a geodesic ball around xi such that, €or each x E B ( x i , ail, (8.23) 1 lS2l2 (Xi) 2 IS2l2 (x) 24 lS2 l2 (Xi) .

Let C,* be the supremum of /V Is2I2 I on M. Then one can easily prove that we may assume

(8.24) 1 s. 2- lS2lZ ( X i ) . ' - 2c15

We only need inequalities (8.23) and (8.24) with suitably large CIS. Hence we may assume that ai is smaller than the injectivity radius of M.


It is easy to derive from (8.18) that, over the ball B ( x i , & ) ,

By applying the Schauder estimate on the ball B(xi , $), we can find a constant c 1 6 which is independent of i such that

(8.26)

Since we have an L’-estimate of I c p E , ) , it follows from (8.26) that there 1. is a constant C17 , independent of i, such that

(8.27)

Since s1 is holomorphic, one can find positive constants 1 and CI8, independent of i, such that, for any (small) r > O and X E M ,

(8.28)

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As before, we may assume that sup, [q log (Is,\” + ei) - qs, J 2 0. Then proceed- ing as in Section 2, we can now conclude from (8.23), (8.27) and (8.28) that there is a constant CI9, independent of i, such that

I”’ (I~Zl’(xi))~7(5 Is21’(xi)+ Ei)-”” (ls2I2+ ~ i ) ‘ exp {-~p,,}

Since limi, EL’ Is21’(xi)=03 and the right-hand side of (8.29) can be estimated even when N is large, we see that the quantity sup, (Is21z+ EJ‘

X exp {-cp,,} can be estimated by a constant independent of i. In conclusion we have proved that, for any q > 0, -cpE + q log ((s2l2+ E ) is

bounded from above by a constant independent of E . In particular, -cp, is uniformly bounded over compact subsets of the complement of the divisor of sz . From (8.18) and the supremum estimate of c p E , we see that both 19.1 and IAcp,l are uniformly bounded over compact subsets of the complement of the divisor of s,. The arguments of Section 5 now show that one can find uniform estimates of { ( c p , ) i T k } over compact subsets of the complement of the divisors of s1 and s2. Hence we have proved the following theorem.

THEOREM 7. Let L, and L, be two holomorphic line bundles over a compact Kahler manifold M whose Kahler metric is given by giTdzi@dZi. Let s1 and s2 be two holomorphic sections of L1 and L,, respectively. Let F be a

smooth function defined on M such that Is112kl Is21-2k2 exp { F } =Vol ( M ) ,

where k, and k, are two non-negative integers. Suppose that, for m =dim M, I,

<*. Then we can solve the equation


so that

(i) cp is smooth outside the divisors of s1 and s2 with sup, cp <w,

(ii) (qIi) is a bounded matrix outside the divisor of s2 and, for any q>O,

(iii) for any q > 0 , the function cp - q log ls2I2 is bounded from below, (iv) the matrix (gIr+a2cp/az' a 2 ) is positive definite outside the complement

Acp is bounded on M, 2(m-l)m-'+Zk2m-'+q IS21

of the divisors of s1 and s2.

Furthermore, if we assume that ( I ~ ~ 1 ~ ) ( ~ - ~ ) ~ - ' - ~ 2 ~ - ' - ~ < w for some q > 0,

then any two solutions of the equation which has the above properties (i), (ii) and (iv) must differ from each other by a constant. If we also know that (IS212)(1-m)m-k2m-'-q is integrable over every analytic disc of M, then the unique solution cp is bounded from below on M.

We have only to prove the last sentence. Suppose t,h is another solution of the equation which possess properties (i), (ii) and (iv) stated in the theorem. Then clearly we have

Proof:

On the other hand, we can rewrite det (giT+d2+/'lazi 13.2~) as

Therefore the arithmetic-geometric mean inequality shows that

(8.31) m+A:(+-cpE)Z m [ C i 1 Is112k1 IS^(^+ E ) - ~ , (Is212+ ~ ) ~ 2 ] " ~ ,

where A: is the normalized Laplacian of the metric

Let k be any constant. We claim that, over the domain a ,& = {x E I 4 4 x ) - PE (x) > k),

(8.32)

400 SHING-TUNG YAU

In fact, let S > O be a positive constant. Then the domain f l E , k , S = {x 1 J/(x)-cp,(x)> k - S log Is2]’} is disjoint from the divisor of s2 by property (i). Hence both (Jli7) and ( ( (P~)~T) are bounded on f l . ,k ,S and we can apply the divergence theorem on to find

Outside the divisor of s2, A; log lsz12 is simply the trace of the first Chern form of L, with respect to the metric

Hence for some constant C2n, independent of 6, we have

Using property (ii) and the assumption that

can find a constant C21, independent of S, such that

IS^^^)(^-^)^-^-^^^-'- q<w, we

(8.35)

It follows easily from (8.33), (8.34) and (8.35) that

lim J ( J / - q , - k ) A : ( $ - c p E ) 5 0 . n e , k . S

s-0 (8.36)

Using the definition of flZB,k,S, we see that, over aE,k,&, $- cp. - k is bounded by a constant independent of S when S is small. The function ($-cp,-k) X &($ - cp,) is therefore uniformly integrable and we can apply Lebesque’s dominated convergence theorem to prove (8.32).


LO.

When E tends to zero, the integral on the left-hand side tends to zero. Let K be any compact set of the complement of the divisor of s2, then by (8.31) and (8.37), we have

X ~ S ~ ) - ~ ~ ~ ( ~ S ~ ( ~ + E ) ~ ~ ] ~ ’ ~ + b ~ ( $ - ~ ~ ) } = o .

(Note that (8.31) is used strongly here.) Let f l k ={x I t(r(x)-cp(x)> k } . Then (8.38) gives

(8.39)

where A’ is the normalized Laplacian of the metric

As in (8.31), we know that A’(JI-cp)ZO and hence we conclude from (8.39) that A‘(+ - rp) = 0 on nk. When the arithmetic-geometric mean inequality becomes equality, the eigenvalues of the corresponding matrix are equal. Therefore, $1~7 = cpi; everywhere on a,. Since k is arbitrary, JIii = ( p i ~

everywhere on the complement of the divisor of sz. One can now prove that Q - JI = constant by simply letting first E tend to zero and then 6 tend to zero in (8.33). This shows that

IVY$ - q)I2 < -lime-,o (4 - pE - k ) A:($- Q,) 6.. which is equal to zero by (8.37).

It remains to prove that the unique solution Q is bounded from below.

402 SHING-TUNG YAU

From (8.11) and the estimate of (ls2I2+ E ) ~ exp {-cp,} for any q > 0, we know that

(8.40) SUP [(m + Aqe)((s21' + E ) ~ + ( ~ - ~ ) ~ - ' + ~ Z ~ - ' 1 5 c z z M

for any 4'0, where C,, is a constant independent of E .

Let x be any point on the divisor of s2. Let D, be an analytic disc passing through x such that s2 is not zero on aD,. Then Icpel is uniformly bounded on aD, when E tends to zero.

Since [gir+(cpe)ir] is positive definite, it follows from (8.40) that when we restrict qE to D,, the absolute value of its Laplacian is estimated by (Isz12 + E)-q- (m- l )m- l -k ,m- ' . On the other hand, the assumption guarantees the integrability of (Is2I2 + E ) - ~ -(m-l)m-l-kzm-l over D, for some ij > 0.

We can now apply Green's formula to 0, to prove that supDx I cpEl can be estimated from above by a quantity involving supaDx IpEJ and

Therefore, by choosing q < Lj, we obtain an estimate of IqE 1 on 0,. Since x is arbitrary, we can conclude the boundedness of cp.

9. The General Case

In this section, we write down a general theorem. Let M be a compact Kahler manifold with Kahler metric gii dz'8.5'.

Let t l , r2, * 9 , I ~ ~ + ~ be non-zero non-negative functions defined on M with the following properties. For each ti, we can find holomorphic sections sl, s2, * * , sI of some Hermitian holomorphic line bundles such that ti = c=, I ~ ~ 1 ~ ~ j , where ki 2 0 for each j .

Then we consider the equation

where F is a smooth function defined on M x R with aF/at LO.

i,lI defined on M such that As in the previous section, we assume that we can find a smooth function

(9.2) t l tz . . t,, . - . t;;+,,,F(x, 4) = Vol (M) . I,


If m =dim M, we also assume that

(9.3)

and, for some q > O ,

is integrable over M and over every analytic disk of M. The arguments of Section 8 can easily be modified to prove the solvability

of (9.1) when either F is independent of cp or F(x, cp) = exp {kcp} with k > 0. The arguments of Section 6 can then be modified to solve (9.1) in general. (The required comparison theorem can be treated as in the end of Section 8.)

THEOREM 8. Let M be a compact Kahler manifold. Suppose that, in equation (9.1), the ti are functions satisfying the above mentioned properties. Then we can find a solution cp of (9.1) such that

(i) cp is smooth outside the zeros of the ti and supM Icp(<m, + i ) ( l - m ) / m is (ii) SUP, (AV)(tnl+l * * tnl+nJq+l/m ( ( A log tn,+l * * * tn,+nzl

bounded,

Kahler metric. (iii) outside the zeros of the ti, Ci,j (gir+dzcp/dzi dZi) dzi@dZi defines a

Furthermore, any solution of (9.1) satisfying the above three properties differs from cp by a constant.

Appendix A

In this appendix, we carry out the computation needed in Section 3. In

From the commutation formula of Section 1, we have other words, we prove formula (3.2) here.

404 SHING-TUNG YAU

Since Proposition 2.1 shows that the metric tensor Ca,p g&,- dz" @ d Z P is uniformly equivalent to gap dz* @ dfP, we see from (3.2) that

Similarly, one can also prove that

Let us now write (2.5) as

Then differentiating, we have

Diagonalizing and simplifying, we obtain


Appendix B

In this appendix, we carry out the proof of (8.17) promised in Section 8. Integrating by parts in (8.161, we have

-&I- k,) I (~s2~2++)P-kz-1((sl~2++E)kiexp(-C(p,)V l ~ , 1 ~ . V ( p ,

- kl

M

e)P-kz(lsl12+ &Ik1-' exp {-C(p,)V (s1I2 * V(p,

(ls2I2+ E)P-k2(ls112+ & I k , exp {-CCp,) IVcp,12

+ ( p - k,)C,C-'

+ klC,C-' ((s2I2+ e)P-k2-1(lsl12+

(Is212+ e)P-kz-Z(lsl12+ & I k i exp {-Ccp,) IV ls2I2 1' I, (B.1)

exp {-Ccp,) V Is1l2 9 V lszlz

408 SHING-TUNG YAU

ON THE RICCI CURVATURE OF A COMITZ qm KAHLER MANIFOLD 409

Since we have assumed that C 7 ( p y k2+C,)-l=$, we know that (p - k 2 ) ( p - k2+ C,)-’ = 4 and (B.2) shows tHf at

To estimate the right-hand side of (B constant C, independent of E such ti

.3), we use (5.7) again to find a positive .hat

Integrating by parts in (B.4), we have

410 SH ING-TUNG YAU

Multiplying (€3.5) by akl and sub we obtain

stituting the resulting inequality into (B.3),

Since we assume IsJZmk2<m, p ve can find a constant Cl0 independent I, of E such that

Then by Holder’s inequality, we have

Added in proof (February 21, 1978): Some de after the completion of this paper may be recoi offered two proofs of Calabi’s conjecture, one €or c one for the general case. After I finished my proc

VeloPments on the Problem d e d here. In Section 2, I OmPlex dimension two and lf, I was kindly invited by

ON THE RICCI CURVATURE O F A COMPACr KAHLER MANIFOLD 41 1

Professor E. Calabi to Philadelphia to present a talk in 1976 in which Professor J. Kazdan was present.. After a few months, J. Kazdan was able to simplify my two-dimensional proof. This simplification will also appear in this issue. Then Professor J. Kazdan was invited by Professor T. Aubin to present in Paris my two-dimensional proof and his simplification. A t the end of 1977, Professor Aubin and Professor J. P. Bourguignon, who was kindly giving a talk on my paper in the Bourbaki seminar, independently generalized my two-dimensional proof to higher dimensions. I wish to thank these colleagues for their interest in my wotik and for giving another complete proof of the general case.

Bibliography

[l] Aubin, T., Mitriques Riemianniennes et courbure, J. Diff. Geom. 4, 1970, pp. 383-424. [2] Aubin, T., Equations du type Monge-Ampere sur les uariitis Kuhleriennes compactes, C.R.

[3] Calabi, E., The space of Kiihler metrics, Proc. Internat. Congress Math. Amsterdam, 1954,

[4] Calabi, E., On Kiihler manifolds with oanishing canonical class, Algebraic Geometry and Topology, A symposium in honor of S. Lefschatz, Princeton Univ. Press, Princeton, 1955,

[ 5 ] Calabi, E., Improper affine hyperspheres and generalization of a theorem of K . Jiirgens, Mich.

[6] Chern, S. S., Chavacteristic classes of Hermitian manifolds, Ann. of Math. 47, 1946, pp.

Acad. Sc. Paris, 283, 1976, pp. 119-121.

Vol. 2, pp. 206-207.

pp. 78-89.

Math. J. 5, 1958, pp. 105-126.

85-121. [7] Chern, S. S., Complex Manifold Without Potential Theory, Princeton, N.J., van Nostrand,

1967. [8] Cheng, S. Y. and Yau, S . T., On the regularity of the Monge-Ampire equation

[9] Cheng, S . Y. and Yau, S. T., On the regularity of the solution of the n-dimensional

[ 101 de Rham,G., Vari&% Diflerentiables; Formes, Courants, Formes Harmoniques, Paris, Hermann,

[ll] Federer, H., Geometric Measure Theory, Springer-Verlag, 1969. [12] Moser, J., A new proof of De Giorgi's theorem concerning the regularity problem for elliptic

differential equations, Comm. Pure Appl. Math., Vol. 13, 1963, pp. 457-468. [ 131 Morrey, C. B., Multiple Integrals in the Calculus of Variation, Springer-Verlag, 1966. [14] Schwartz, J. , Nonlinear Functional Analysis, N.Y., Gordon and Breach, 1968. [15] Yau, S. T., On Calabi's conjecture and some new results in algebraic geometry, Nat. Acad.

det (a*u/ax, ax,) = F(x, u), Comm. Pure Appl. Math., Vol. 30, 1977, pp. 41-68.

Minkowski problem, Comm. Pure Appl. Math., Vol. 29, 197'6, pp. 495-516.

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Sci. U.S.A., 74, 1977, pp. 1798-1799.

Received June, 1977

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