THE COMPLEX MONGE-AMPÈRE EQUATION IN KÄHLER GEOMETRY

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1 Sietifi Researh of the Istitute of athematis ad Computer Siee THE COPLEX ONGE-APÈRE EQUATION IN KÄHLER GEOETRY Jagielloia Uiversity, Abstrat. We disuss two ases whe the omplex oge-ampère equatio appears i Kähler geometry: the Calabi oeture (with its solutio by Yau) ad the equatio for geodesis i the abuhi spae of Kähler metris, itrodued idepedetly by Semmes ad Doaldso. Itrodutio This is a slightly expaded versio of the talk give o the d Forum of Polish 1. Kähler maifolds Let be a omplex maifold of dimesio. A omplex struture idues a edomorphism of the taget budle J : T T, i loal oordiates give by J ( / x ) / y, J ( / y ) / x It a be exteded i a C-liear way to the omplexified taget budle J : TC TC, so that where : / z, : / z. J ( ) i, J ( ) i Remark. The elebrated Newlader-Nireberg theorem [17] (see also [14]) says that if is a real maifold ad J : T T satisfies J id, the J omes from some omplex struture o if ad oly if [X,Y] J[JX,Y] J[X,JY] [JX,JY] 0 for X,YT.

2 The omplex oge-ampère equatio i Kähler geometry 7 Comig bak to the ase whe is a omplex maifold, let be a hermitia form o. Loally we may write h g k k dz d z h : T T C where g ) is a hermitia matrix of smooth futios (i.e. g g, ( ) 0). ( k Every suh a hermitia form a be assoiated with the (1,1)-form : g k idz d z (oe a easily hek that h ad behave the same way uder holomorphi hage of oordiates). The hermitia metri h gives a Riemaia metri h, whih i tur geerates uiquely defied Levi-Civita oetio o. Oe a the show that k k k g k J 0 0 d 0 g k g k z z loally for some smooth futio g. We the say that the (1,1)-form is Kähler. (I other words, a smooth (1,1)-form is Kähler if, 0 ad d 0). Kähler metris are thus those hermitia metris for whih the Riemaia struture is ompatible with the omplex struture. For ompat omplex maifolds existee of a Kähler metri imposes topologial k ostraits: the Betti umbers b k 0 for 1 k. Namely, is a losed real k-form whih is ot exat (if k d for some the by the Stokes theorem we would have k d( ) 0 whih is a otraditio), ad thus 0 { k } H k (, R). Example (Hopf surfae). : ( C \{0}) /{ : Z}. The is homemorphi 1 3 to S S ad thus b ( ) 0. Therefore, does ot admit ay Kähler metri. Complex (p,q) - forms may be loally writte as f JK J p, K q dz J d z K

3 8 where J,, ), 1 1, ( 1 p for K). We defie the operators p dz J p dz 1 dz (ad similarly : C( p, q) C( p1, q), : C( p, q) C( p, q1) as follows f J K f : dz, ( fdz d z ) : f z dz J d z K f J K f : d z, ( fdz d z ) : f z dz J d z K We the have d, ad sie d 0, we get 0 It is oveiet to itrodue the operator d : i ( ) It is a real operator o (i the sese that it maps real forms ito real forms) depedig however o the omplex struture o. Oe a easily hek that dd i. dd -lemma. O a ompat Kähler maifold a (p,q) -form is d-exat if ad oly if it is dd -exat. This result a be proved usig the Hodge theory ad some (loal) ommutator formulas o Kähler maifolds (see eg [10]).. Calabi oeture Assume that g idz k Rii urvature is give by d z k Ri dd is a Kähler form. Oe a the show that the logdet( g k ) If ~ k g ~ idz d z is aother Kähler form o the k Ri Ri ~ dd

4 The omplex oge-ampère equatio i Kähler geometry 9 where det( g ~ log det( g k k ) ) is a globally defied futio o. We see therefore that the ohomology lass of (1,1)-forms { Ri } is idepedet of. It is i fat the first Cher lass of, we deote it 1( ) (more preisely it is equal to 1 ( ) R / ). From ow o we will assume that (, ) is a ompat Kähler maifold. Calabi [8] oetured that for ay (1,1)-form R whih is ohomologous to Ri (that is R 1 ( )) there exists a Kähler form ~ ohomologous to suh that Ri ~ R. We thus have suh that dd 0 ad R Ri dd for some C ( ), ad we look for C ( ) det( g ) k k dd log 0 det( g ) k k where / z z. k This meas that for some ostat det( g k k ) e det( g k ) or equivaletly ( dd ) e The ostat is uiquely determied, sie by the Stokes theorem ( dd ) Solvig the Calabi oeture is thus equivalet to provig the followig result: Theorem (Yau [0]). Assume that f C ( ), f > 0, is suh that f The there exists a uique (up to a additive ostat) C ( ) dd 0 ad solvig the oge-ampère equatio satisfyig

5 10 ( dd ) f (1) If we osider the Kähler lass of Kähler metris ohomologous to : : { dd : C ( ), dd 0} () the Calabi oeture a be formulated as follows: the mappig ~ Ri ~ ( ) is bietive. 1 the there exists a Kähler metri with vaishig Rii ur- Corollary. If 1 ( ) 0 vature. This result, useful for example i algebrai geometry, is iterestig beause i every sigle ase suh a metri aot be writte expliitly. The Yau theorem is proved i several steps: 1. uiqueess;,. otiuity method reduig the problem to a C a priori estimate; 3. a priori estimate for the L -orm of solutios; 4. a priori estimate for the C -orm of solutios;, 5. a priori estimate for the C -orm of solutios. Uiqueess is a simple osequee of itegratio by parts ad was proved by Calabi already i the 50's (see [3] for a more geeral result). The otiuity method is ofte used i the theory of fully oliear ellipti equatios of seod order, it relies o the impliit futio theorem i ifiitely dimesioal Baah spaes. The uiform a priori estimate was proved i [0] usig oser's iteratio tehique. It should be stressed that i may problems of this kid (eg the Yamabe problem or the problem of existee of a Kähler-Eistei metri) this estimate is ruial. The best result i this diretio was proved usig pluripotetial theory: Theorem C ( ), dd 0, solves (1). The for p > 1 there exists a positive ostat C, depedig oly o (, ), p ad f, suh that L p ( ) os : sup if C See also [4] for a bit differet proof of this result. It is rather uusual that the C -estimate a be derived diretly from the L - estimate. This was doe idepedetly by Aubi [1] ad Yau [0]. This estimate was later improved i [] ad [6] to the followig form:

6 The omplex oge-ampère equatio i Kähler geometry 11 Theorem ([6]). Assume that C 4 ( ), dd 0, solves (1). The there exists a positive ostat C, depedig oly o, o upper bouds for os, sup f ad the salar urvature of, ad o lower bouds for the bisetioal urvature of 1/( 1) ad if f (log f ), suh that C Behaviour o the geometry of i the above result is quite expliit, it is ot so geometri as i the L -estimate. It was show i [18] that the expoet 1/(-1) above is optimal. 3 I the origial proof Yau used Nireberg's estimate for the C -orm. I the early 80's a geeral theory (it is ow alled Evas-Krylov theory) was developed. It allows, to estimate loally the C -orm i terms of the C -orm of solutios of geeral equatios of the form F( D u, Du, u, x) 0 provided that ( F / u k ) 0 ad that F is oave with respet to D u. Coerig the last oditio, the ruial fat for the omplex oge-ampère operator is that 1/ the mappig A (det A) is oave o the set of positive hermitia matries. Usig the Evas-Krylov theory oe a get the followig estimate (see e.g. [5] - this paper otais the whole proof of the Yau theorem): Theorem. For 4 u C (), where is a domai i C, with ( u ) 0 f : det( u ). The for ' there exists (0,1), depedig oly o, o k upper bouds for u, sup u, f, ad o a lower boud for if f, C 1 ( ) C 1 ( ) ad positive ostat C depedig i additio o dist( ', ), suh that u C C, ( ') k set 3. abuhi spae of Kähler metris Let H be the spae of Kähler metris from oe ohomology lass give by (). We a treat it as a ope subset of C ( ) (modulo a additive ostat), so for the taget spae T is equal to C ( ). O we defie the orm T 1 : ( dd ),! T

7 1 1 1 Aordigly, a legth of a urve C ([1,], ) C ([1,] ) is give by l( ) : t t dt 1 where t : ( t, ) ad t : ( / t)( t, ). The above metri, itrodued by abuhi [16], gives a riemaia struture o a ifiitely dimesioal maifold H. It determies a Levi-Civita oetio ad the geodesi equatio turs out to be 1 0 It was show idepedetly by Semmes [19] ad Doaldso [11] that it is equivalet to det 1 ( g k k ) 1 0 Therefore, to fid a geodesi oetig two metris dd 1, equivalet to solvig the homogeeous oge-ampère equatio dd is ( dd ) 1 0 (3) o a ompat Kähler maifold (with boudary) 1 1, where : dd z1 z with the boudary oditio o z1, =1,. Doaldso [11] oetured that ay two metris i H a be oied by a geodesi, ad that the futio d( 1, ) : if{ l() : is a urve i H oiig 1 with } for 1, C - is a distae o H. The latter oeture was proved by X.X. Che [9]. He also 1,1 showed that ay two metris a be oied by a C - geodesi (although it may 1,1 possibly leave H, as the itermediate metris are oly assured to be (almost) C - smooth ad oegative). The existee of C - geodesis remais a ope problem.

8 The omplex oge-ampère equatio i Kähler geometry 13 I geeral, oe should ot expet solutios of a degeerate oge-ampère equatio (suh as (3)) to be C -smooth. O oe had, i the o-degeerate ase we have the followig outerpart of the Yau theorem for domais i C : Theorem ([7]). Let be a smooth, bouded, strogly pseudoovex domai i C (for example a ball). The for f C (), f > 0, ad C () there exists the uique u C (), suh that ( u ) 0, u o ad k det( u k ) f i 1,1 However, whe we oly assume that f 0, the best possible regularity is C, as the followig example of Gameli ad Siboy shows. Example ([13]). Let B:! {(z,w) C : z w 1 } be the uit ball i r! r! C ad set The ( z, w) : z 1/ w 1/, ( z, w)b u( z, w) : (max{ z 1/, w 1/, 0}), ( z, w) B is 1,1 C -smooth (but ot C!), ( u ) 0, det( u ) 0, ad u o B k 1,1 It would suggest that also i (3) the best possible regularity should be C, also amog tori varieties (ote that all the data i the above example depeds oly o z ad w ). However, that problem seems to be more speial, ad for example for tori varieties Doaldso [1] ideed showed that a C -geodesi always exists. Akowledgemet Partially supported by the proets N N ad 189/6 PR EU/007/7 of the Polish iistry of Siee ad Higher Eduatio. k Referees [1] Aubi T., Equatios du type de oge-ampère sur les variétés Kähleriees ompates, C.R. Aad. Si. Paris 83 (1976), [] Regularity of the degeerate oge-ampère equatio o ompat Kähler maifolds, ath. Z. 44 (003),

9 14 [3] Uiqueess ad stability for the oge-ampère equatio o ompat Kähler maifolds, Idiaa Uiv. ath. J. 5 (003), [4] O uiform estimate i Calabi-Yau theorem, Si. Chia Ser. A 48 suppl. (005), [5] The oge-ampère equatio o ompat Kähler maifolds, upublihed leture otes based o the ourse give at Witer Shool i Complex Aalysis, Toulouse, 005, available at [6] A gradiet estimate i the Calabi-Yau theorem, preprit, 008. [7] Caffarelli L., Koh J.J., Nireberg L., Spruk J., The Dirihlet problem for oliear seodorder ellipti equatios. II. Complex oge-ampère, ad uiformly ellipti, equatios, Comm. Pure Appl. ath. 38 (1985), [8] Calabi E., The spae of Kähler metris, Pro. Iterat. Cogress ath. Amsterdam 1954, vol., [9] Che X.X., The spae of Kähler metris, J. Diff. Geom. 56 (000), [10] D ly J.P., Complex Aalyti ad Differetial Geometry, 1997, available at [11] Doaldso S.K., Symmetri spaes, Kähler geometry ad Hamiltoia dyamis, Norther Califoria Sympleti Geometry Semiar, 13-33, Amer. ath. So. Trasl. Ser., 196, Amer. ath. So., Providee, RI, [1] Doaldso S.K., Salar urvature ad stability of tori varieties, J. Diff. Geom. 6 (00), [13] Gameli T.W., Siboy N., Subharmoiity for uiform algebras, J. Fut. Aal. 35 (1980), [14] Hörmader L., A Itrodutio to Complex Aalysis i Several Variables, 3rd ed., North-Hollad, Amsterdam [15] The omplex oge-ampère equatio, Ata ath. 180 (1998), [16] abuhi T., Some sympleti geometry o ompat Kähler maifolds. I, Osaka J. ath. 4 (1987), 7-5. [17] Newlader A., Nireberg L., Complex aalyti oordiates i almost omplex maifolds, A. of ath. 65 (1957), [18] A outerexample to the regularity of the degeerate omplex oge-ampère equatio, A. Polo. ath. 86 (005), [19] Semmes S., Complex oge-ampère ad sympleti maifolds, Amer. J. ath. 114 (199), [0] Yau S.-T., O the Rii urvature of a ompat Kähler maifold ad the omplex oge- Ampère equatio, Comm. Pure Appl. ath. 31 (1978),

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