The Poisson Equation and Hermitian-Einstein Metrics on Holomorphic Vector Bundles over Complete Noncompact Kähler Manifolds

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1 The Poisson Equaion and Hermiian-Einsein Merics on Holomorphic Vecor Bundles over Complee Noncompac Kähler Manifolds LEI NI ABSTRACT. Auhor: Please supply an absrac. 1. INTRODUCTION The exisence of Hermiian-Einsein merics on compac Kähler manifolds has been well-undersood afer he work of [9] and [27]. In [20] we sudied i on complee noncompac Kähler manifolds which have posiive lower bound on he specrum of he Laplacian operaor. In [21], i was shown ha here is a relaion beween solving he Poisson equaion and he Poincaré-Lelong equaion. In his paper we shall illusrae he relaion beween solving he Poisson equaion (hea equaion and he Hermiian-Einsein equaion (Hermiian-Einsein hea equaion. The general idea is ha by firs solving he Hermiian-Einsein equaion on he deerminan line bundle, which is a Poisson equaion, ogeher wih some desirable esimaes, one can use he soluion as a barrier o consruc a soluion o he corresponding nonlinear equaion on he vecor bundle. (In [22], we will show ha here is a similar relaion beween he Poisson equaion and he Kähler- Ricci flow equaion on he complee Kähler manifolds wih bounded nonnegaive bisecional curvaure. There, he auhors use he soluion o he Poisson equaion o reduce he curvaure esimae, herefore he long ime exisence, o a simple use of he maximum principle of Karp-Li on complee Riemannian manifolds, which considerably simplifies he previous work of Shi [24]-[25]. By exploring his idea we give a simple and unified proof on he exisence of Hermiian-Einsein merics on complee Kähler manifolds under various assumpions. The main resuls can be found in Theorems 2.9 and There is an analogy beween he Kähler- Einsein merics on he angen bundle and he Hermiian-Einsein merics on general holomorphic vecor bundles. The previous proved resul in [20] corresponds o heexisenceresul of Cheng-Yau [7] on he Kähler-Einsein merics on bounded sricly pseudoconvex domains in C m. The resuls in Theorems 2.9 and 679

2 680 LEI NI 2.10 of his paper are more in he spiri of he resuls proved by Bando-Kobayashi [2], Joyce [13] and Tian-Yau [26] on he exisence of Ricci fla Kähler merics. The various Sobolev inequaliies play an imporan role in boh our proof and he previous consrucions in [26]. In he second par of his paper we sudy he Laplacian hea equaion (we will jus call i hea equaion and is relaion o he Hermiian-Einsein hea equaion. For he resuls on he hea equaion we focus on he case when he manifold M is a complee Riemannian manifold wih nonnegaive Ricci curvaure. (There are similar resuls for he complee Riemannian manifolds wih L 2 -Sobolev inequaliy. See, for example, Theorem 3.6. In Secion 3 we firs derive some momen ype esimaes (afer Nash s [18] for he hea equaion (cf. Theorem 3.1. The esimaes follow direcly from some general resuls of Li-Yau [17] on he hea kernels. The derivaion is more direc and simpler han Nash s original argumen for he Euclidean case, which depends on an enropy esimae. The relaion beween he asympoic spaial behavior of he iniial daa and he long ime behavior of he hea equaion can hen be esablished afer he momen ype esimaes. Among ohers we show he following resul. Le f be a nonnegaive coninuous funcion on M. Consider he hea equaion ( u(x, = 0, u(x, 0 = f(x. Le u(x, be is nonnegaive soluion. Then (i 0 u(x, s ds < if and only if 0 r k(x, r dr < ; (ii lim u(x, = 0 if and only if lim r k(x, r = 0; (iii u(x, = O(log( + 1 if and only if k(x, r = O(log(r + 1; (iv u(x, = O( d if and only if k(x, r = O(r 2d, for any d wih n/2 1 < d<. Here as in [21] k(x, r = 1 f(ydv y. V x (r Bx(r The above saemen can be hough as a parabolic analogue of he resuls proved in [21]. In paricular, he second par of he above saemen and Corollary 3.3 give parial generalizaions of a resul of Li [14] on he large ime behavior of he hea equaion, since we assume neiher ha he manifold is of maximum volume growh nor he boundedness of he iniial daa. In [22], surprisingly, we find a similar momen ype esimae for he Kähler-Ricci flow. I urns ou ha one side of he momen esimae is nohing bu an improved version of he volume elemen esimaes obained by Shi in [Sh1-2]. The derivaion for he linear case here also suggess a simple proof for he oher side of he momen esimae for he nonlinear Kähler-Ricci flow, which was discovered in [6] very recenly. This unified reamen cerainly clarifies and simplifies he previous resuls on Kähler-Ricci

3 The Poisson Equaion and Hermiian-Einsein Merics 681 flow. Using he resuls on he hea equaion in Secion 3, we sudy he Hermiian- Einsein hea equaion in Secion 4. The main resul is conained in Theorem 4.1 and Theorem 4.6. In Theorem 4.1 we prove a long ime exisence of he Hermiian-Einsein hea equaion on any complee Kähler manifold, under he assumpion ha he iniial meric has bounded mean curvaure. In Theorem 4.6 we prove he convergence on a complee Ricci nonnegaive manifold under some average decay assumpions of he iniial meric s mean curvaure, which gives a hea flow proof of Theorem Under an assumpion similar o ha of Theorem 4.6, in [24] and [25], W.X. Shi proved he long ime exisence of he Ricci flow equaion on a complee Kähler manifold wih nonnegaive bisecional curvaure. Our assumpion here is slighly sronger han his. However, we should poin ou ha he sronger assumpion is used o prove he convergence of he Hermiian-Einsein flow, and we have shown he long ime exisence under much more flexible condiion in Theorem 4.1. On he oher hand, a simple use of H. D Cao s Li-Yau-Hamilon inequaliy (cf. [4] shows ha, for Kähler-Ricci flow equaion, he soluion never converges excep ha he iniial meric is fla (cf. [22, Corollary 2.3]. Finally we should poin ou ha he mehod developed here can be modified o consruc harmonic maps from various complee noncompac Riemannian manifolds o Riemannian manifolds wih nonposiive secional curvaure or a domain in a manifold wih a convex defining funcion. 2. HERMITIAN-EINSTEIN EQUATION In his secion we prove he exisence of Hermiian-Einsein merics on a holomorphic vecor bundle E over a complee Kähler manifold M, assuming ha E admis a Hermiian meric H 0 such ha he mean curvaure of H 0 saisfies cerain decay condiions. This assumpion has appeared in various ocassions dealing wih he nonlinear ellipic or parabolic equaions, such as he harmonic map equaion and Ricci flow on complee noncompac manifolds. We focus on he ellipic mehod here and deal wih he corresponding parabolic equaion laer. The main resul is ha we can solve he Hermiian-Einsein equaions, a nonlinear sysem, when he corresponding linear equaions on he deerminan line bundle have a nonnegaive soluion. In [20], he exisence resul has been proved for he manifold wih λ 1 (M > 0, where λ 1 (M denoes he lower bound of he specrum of he Laplacian operaor. Here we are going o prove i for a class of Kähler manifolds, including ones wih nonnegaive Ricci curvaure. The mehod we use is much simpler han he previous one. I reduces he global nonlinear problem o some local a priori esimaes and solving a global linear problem wih nonnegaive soluion. Firs le us fix he noaion. Le M m be a Kähler manifold of complex dimension m, andle (E, H be a holomorphic vecor bundle wih a Hermiian meric H. Denoe by ω = 1g βdz α α d z β he Kähler form, where g = g βdz α α d z β is he Kähler meric. Then we can define he operaor Λ

4 682 LEI NI as he conracion wih 1ω, i.e., for any (1, 1-form a α βdz α d z β Λ a α βdz α d z β = g α βa α β. A connecion A onhevecorbundle E is called Hermiian-Einsein (cf. [9] if he corresponding curvaure form F A saisfies he following equaion: (2.1 ΛF A = λi, where λ is some consan. In he case when M is a compac Kähler manifold λ is a opological invarian (cf. [27]. If for a Hermiian meric H on E, he corresponding Hermiian connecion is Hermiian-Einsein, hen he meric H is called a Hermiian-Einsein meric (cf. [9]. Recall ha under a holomorphic local frame {e i },ifh =(h i j, where h i j = e i,e j, hen he Hermiian connecion A H and he corresponding curvaure form F H are given by he following formulae: (2.2 (2.3 A H = HH 1, F H = ( HH 1. If we wrie F H = F j iα β dzα d z β,wedefine ΛF H = g α βf j iα β α,β he mean curvaure of H. We also denoe R(x = r ΛF H,hescalar curvaure of H. In his paper we will focus on he case when λ = 0. The general cases do no differ from his special case. In [20] we proved ha, for he complee Kähler manifold wih λ 1 (M > 1, if here is a Hermiian meric H 0 such ha he mean curvaure of H 0 saisfies ha ΛF H0 L p (M for a p>1, hen here exiss a meric H on E such ha ΛF H 0. A similar resul was proved for Kähler-Einsein merics before in [7]. In [20] we applied a hea flow mehod. Namely, we firs proved ha he Hermiian-Einsein flow has a global soluion. Then we showed ha i converges o a soluion o he ellipic problem. In he following we will adap he direc ellipic mehod. The ellipic mehod reduces he problem essenially o a linear problem. This mehod can also esablish some exisence resuls, for he Hermiian- Einsein merics on holomorphic vecor bundles over complee Kähler manifolds wih nonnegaive Ricci curvaure, or general nonparabolic Kähler manifolds (see Theorems 2.9 and 2.10 for he precise saemen. Le us firs sar wih some simple lemmas on he soluions of he Poisson equaion. Lemma 2.1. Le M be a complee Riemannian manifold. Assume ha λ 1 (M > 0. Then for a nonnegaive coninuous funcion f he Poisson equaion u = f

5 The Poisson Equaion and Hermiian-Einsein Merics 683 has a nonnegaive soluion u W 2,n loc (M C 1,α loc (M (0 <α<1 if f L p (M for some p 1. Proof. Le Ω i be a compac exhausion of M. We firs solve he following Dirichle problem on Ω i : u i = f, (2.4 u i Ωi =0. Firs, by he maximum principle, we know ha u i 0. Now muliplying u p 1 i on boh sides of he equaion and inegraing by pars we have ha (p 1 u i 2 u p 2 i dv = fu p 1 i dv. Ω i Ω i Rewrie i as 4(p 1 p 2 (u i p/2 2 dv = Ω i fu p 1 i dv. Ω i Using he fac ha λ 1 (M > 0, we have Therefore we have Ωi u pi dv p 2 from which we have λ 1 (M u p i dv (u i p/2 2 dv. Ω i Ω i 4λ 1 (M(p 1 p 2 ( 4λ 1 (M(p 1 fu p 1 i dv Ω i M 1/p ( (p 1/p f p dv u p i dv Ω i ( 1/p u p i dv p 2 ( 1/p f dv p. Ω i 4λ 1 (M(p 1 M Using he inerior L p esimaes for he linear ellipic PDE (cf. [11, Theorem 9.11] we know ha, over a compac subse Ω, here will be a uniform upper bound for u i W 2,p (Ω. Therefore, using Rellich s compacness heorem, by passing o a subsequence we know ha u i will converge o a soluion u W 2,p loc (M on he manifold M. Clearly u 0, a.e. Since f L q loc(m for any q n, he regulariy heory for he linear ellipic PDE (cf. [1, Theorem 3.54]. implies ha u W 2,q loc (M. This complees he proof of he lemma.

6 684 LEI NI The proof above does no work for he case p = 1, which needs a differen reamen, which we will give a lile laer. If on M we have he L 2 -Sobolev ( (n 2/n (2.5 ϕ 2 dv S(M ϕ dv 2n/(n 2, M M an argumen similar o he proof of Lemma 2.1 shows a similar lemma. Lemma 2.2. Le M be a complee Riemannian manifold saisfying he L 2 -Sobolev inequaliy (2.5. Then for a nonnegaive coninuous funcion f he Poisson equaion u = f has a nonnegaive soluion u W 2,n loc (M C 1,α loc (M (0 <α<1 if f L p (M for some n/2 >p 1. Proof. We only need o replace he Poincaré inequaliy by he Sobolev inequaliy in he proof of Lemma 2.1. Remark. For p>1, Lemmas 2.1 and 2.2 were essenially proved before in auhor s hesis [19] (cf. Theorems 2.8 and 2.9 herein. Now we prove Lemma 2.1 and Lemma 2.2 for he case p = 1. The following saemen was essenially proved in he proof of Theorem 3.2 in [21]. Lemma 2.3. Le M be a complee nonparabolic manifold (in he sense ha M suppors a posiive Green s funcion. Then for a nonnegaive coninuous funcion f he Poisson equaion u = f has a nonnegaive soluion u W 2,n loc (M C 1,α loc (M (0 <α<1 if f L 1 (M. Proof. Since M is nonparabolic, here exiss a posiive Green s funcion on M. Iissufficien if we can show ha M G(x,yf(ydy is locally bounded. For any x, here are R such ha x B o (R. Here o isafixedpoin. Thushereare consans C, C, C, depending only on R and he manifold M, such ha ( G(x,yf(ydy = + G(x,yf(ydy M Bo(2R M\Bo(2R C + G(x,yf(ydy M\Bo(2R C + C G(o,yf(ydy M\Bo(2R C + C f(ydy C. M\Bo(2R Here in he second las inequaliy we used he Harnack inequaliy, and in he las inequaliy we used he fac ha sup M\B G(o, y < and he fac ha f is o(2r inegrable. Thus we have proved Lemma 2.3.

7 The Poisson Equaion and Hermiian-Einsein Merics 685 On he oher hand, i is no hard o prove ha M is nonparabolic under he assumpion of Lemma 2.1 or Lemma 2.2. For example, he exisence of he posiive Green s funcion was proved in [23] under he assumpion ha λ 1 (M > 0 (cf. [23, Theorem A.3]. Under he assumpion of Lemma 2.2, i was firs proved by J. Nash in [18] (see also [5] ha H(x,x,, where H(x,y, is he hea kernel of M, has an upper bound H(x,x, C(M n. By [8] (See also [12] one has a Gaussian bound H(x,y, C(M n ( exp Cr2 (x, y. Inegraing he above inequaliy along he ime direcion gives a posiive Green s funcion on M. Therefore he proof for he case p = 1 of Lemma 2.1 and Lemma 2.2 follows from Lemma 2.3. The nex lemma deals wih he case when M is a complee Riemannian manifold wih nonnegaive Ricci curvaure. Lemma 2.4. Le M be a complee Riemannian manifold wih nonnegaive Ricci curvaure. For any nonnegaive coninuous funcion f 0, he Poisson equaion u = f has a nonnegaive soluion (which can be chosen o be W 2,n loc (M C 1,α loc (M if and only if, for every x M, 0 f (y dy d <. V x ( Bx( Proof. This lemma was essenially proved in [21]. For he sake of compleeness, we include a proof here. Suppose ha f (y dy d <. V x ( Bx( 0 If f 0, hen u 0 is a soluion. Oherwise he condiion above means ha M is non-parabolic, by a heorem of Li-Yau in [17]. On he oher hand, le G(x, y be he minimal posiive Green s funcion; by Li-Yau s esimae (cf. [17, Theorem

8 686 LEI NI 5.2] and Fubini s heorem we have ( G(x,yf(y C M 0 r V x ( d fdadr Bx(r ( ( =C fdadr d 0 0 V x ( Bx(r = C f (y dy d <. 0 V x ( Bx( From here i is clear o see ha M G(x,yf(ydy is a nonnegaive soluion o u = f. Conversely, suppose ha here is a nonnegaive soluion o u = f ; we need o derive he desired average decay for f.ifmis a parabolic, hen u is a consan. Therefore f = 0 and he esimae holds rivially. Oherwise we assume ha M is nonparabolic. For R>0, le G R (x, y be he Dirichle Green s funcion on B x (R. Then we have u(x = G R (x, y u(y dy Bx(R = G R (x, yf (y dy Bx(R G R (x, yf (y dy. Bx(R u G R Bx(R ν u G R Bx(R ν Leing R we have M G(x,yf(ydy u(x. Then for any R>0wehave u(x G(x,yf(ydy Bx(R R ( s C(n 0 V x (s ds (( = C(n V x ( d R + f(yda Bx( Bx(R R 0 f(ydy ( V x ( d f(ydy Bx( d

9 The Poisson Equaion and Hermiian-Einsein Merics 687 C(n R 0 ( f(ydy d. V x ( Bx( Since R is arbirary we finish he proof of he lemma. Now we use he mehod of compac exhausion o prove he exisence of he Hermiian-Einsein merics. The argumen shown below gives anoher proof of he resul in [20], using Lemma 2.1. By [10], we know ha he following Dirichle problem is solvable on Ω i. Namely, here exiss a Hermiian meric H i (x such ha ΛF Hi = 0, for x Ω i, (D i H i (x Ωi = H 0 (x. In order o prove ha we can pass o limi and evenually obain a soluion on he whole manifold M, we need o esablish some a priori esimaes. The key is he so-called zeroh order esimae. In order o do ha we have o recall some funcions defined by Donaldson. The firs one is τ i (x = τ(h i,h 0 =r(h i H 1 0. Since H i H0 1 is a secion of Hom(E, E, i is a well-defined funcion. The second one is σ i (x = σ(h i,h 0 =r(h i H0 1 + r(h 0H 1 i 2k, where k is he rank of E. σ(x measures he disance of wo merics in he C 0 opology, in he sense ha H i H 0 if σ(h i,h 0 0 uniformly. A Bochner ype inequaliy is he following one. The proof is similar o ha of Proposiion 3.7 in [20]. Lemma 2.5. Le τ i (x, H 0 and H i as above. Then log τ i ΛF H0 for x Ω i, (2.6 log τ i Ωi = log k. Le f i = log τ i log k. We hen have log f i ΛF H0 for x Ω i, (2.7 log f i Ωi = 0. Lemma 2.6. Assume ha ΛF H0 L p (M for some p 1 and g is he soluion o he Poisson equaion g = ΛF H0 consruced by Lemma 2.1. Then we have f i g or τ i ke g.

10 688 LEI NI Proof. Noicing ha f i g is subharmonic and f i g Ωi 0, he proof is jus he maximum principle (cf. [11, Theorem 9.1]. Since he same Bochner ype inequaliy holds for log r(h 0 H 1 i, he same argumen as he above gives he esimae for r(h 0 H 1 i. Combining hem we have he C 0 -esimaes. Proposiion 2.7. Le M be a complee Kähler manifold such ha λ 1 (M > 0, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0 such ha ΛF H0 L p (M for some p 1. LeH i be he soluions o he Dirichle problem for he Hermiian-Einsein merics. Le g be he soluion o he Poisson equaion g = ΛF H0 consruced by Lemma 2.1. Then σ i 2ke g 2k. Here σ i (x = r(h i H0 1 + r(h 0H 1 i 2k. One can see easily ha he corresponding resul holds if M saisfies he assumpions in Lemmas 2.2, 2.3. When M is a complee manifold wih nonnegaive Ricci curvaure, using Lemma 2.4 and he exac same proof of he above proposiion we can have he following resul. Proposiion 2.8. Le M be a complee Kähler manifold wih nonnegaive Ricci curvaure, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0 such ha ΛF H0 saisfies, for every x M, 0 ΛF H0 (y dy <. V x ( Bx( Le H i be he soluions o he Dirichle problem for he Hermiian-Einsein merics. Le g be he soluion o he Poisson equaion g = ΛF H0 consruced by Lemma 2.4. Then σ i 2ke g 2k. Here σ i (x = r(h i H0 1 + r(h 0H 1 i 2k. Once we have esablished he C 0 -esimae, we can prove wo exisence heorems. The key ingredien of he proof is o obain a C 1 -esimae, which we will esablish laer in he proof. The case when λ 1 (M > 0andp>1has been proved in [20] by he hea equaion mehod. Theorem 2.9. Le M be a complee Kähler manifold, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0 such ha ΛF H0 L p (M. Assume ha eiher (i λ 1 (M > 0 and p 1, or (ii M saisfies L 2 -Sobolev and n/2 >p 1,or (iii M is nonparabolic and p = 1.

11 The Poisson Equaion and Hermiian-Einsein Merics 689 Then here exiss a meric H on E such ha Furhermore, ΛF H 0. σ(h(x,h 0 (x 2ke g 2k, where g is he soluion o he Poisson equaion g = ΛF H0 consruced by Lemmas We shall only give a complee proof for he case when M has nonnegaive Ricci curvaure, since he argumen is robus enough o apply o he oher cases. Theorem Le M be a complee Kähler manifold wih nonnegaive Ricci curvaure, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0 such ha ΛF H0 saisfies, for every x M, Then here exiss a meric H on E such ha Furhermore, 0 ΛF H0 (y dy <. V x ( Bx( ΛF H 0. σ(h(x,h 0 (x 2ke g 2k, where g is he soluion o he Poisson equaion g = ΛF H0 consruced by Lemma 2.4. Proof. In general for any wo merics H and K, direc calculaion using (2.3 gives (F H F K h = ( K hh 1 h = 1 K h + K hh h. Here h = HK 1. If we denoe h = HH 1 0, we hen can wrie ΛF H 0as (2.8 h = Λ( H0 hh 1 h + ΛF H0 h. I is a semi-linear ellipic sysem. Solving H for he Hermiian-Einsein equaion is equivalen o solving h for he above nonlinear sysem, where h is a smooh secion of he endomorphism bundle Hom(E, E. Fixing H 0 is jus fixing a gauge on he vecor bundle. In he following discussion we will also use H i0,forsomei 0, consruced before as he fixed base meric. Le Ω i be a compac exhausion, and le H i be he soluion o he Dirichle problem over Ω i. We denoe h i = H i H0 1. The C 0 -esimaes and he proof show ha for any compac subse Ω M here exis posiive consans C 1 and C 2 such ha (2.9 C 2 I h i C 1 I,

12 690 LEI NI where I is he ideniy on E. Here C 1 and C 2 depend only on M, Ω and k. Choose a i 0 such ha Ω Ω i0. In he following, we firs show ha, by passing o a subsequence, H i uniformly converges o a meric H, which is a leas coninuous, over Ω. Once we have his, he gradien esimae will follow easily. Le σ ij (x = σ(h i (x, H j (x. We are going o show ha, by passing o a subsequence, for any ε>0, (2.10 σ 2 jk (x dx ε, Bo(1 for sufficienly large j and k. On he oher hand, since log r(h i H 1 j 0, i is no hard o verify ha σ ij 0. Then he uniform convergence over a compac domain follows from he he uniform L 2 convergence and he meanvalue inequaliy. Le h j = H j H 1 i 0 and τ j = r(h j ; a direc calculaion shows ha (2.11 τ j = r(λ( Hi0 h j (h j 1 ( h j. Now we define e j (x = r(λ( Hi0 h j ( h j. Clearly i conains all he squares of he firs order derivaives of h j. Using (2.9 and (2.11 we have τ j C 3 e j (x. Le ϕ be a cu-off funcion over B o (2. Muliplying ϕ 2 τ j on boh sides of he above inequaliy and inegraing by pars we have e j (xτ j dx Bo(1 1 C 3 = 1 C 3 2 C 3 e j (xτ j ϕ 2 dx Bo(2 Bo(2 Bo(2 Bo(2 Combining wih (2.9 we know ha Bo(1 ( τ j τ j ϕ 2 dx ( τ j 2 ϕ 2 + ϕ τ j τ j ϕ dx ϕ 2 τ 2 j dx 2 τ 2 j C dx. 3 e j (x dx 2 τ 2 j C 1 C dx. 3 Bo(2 Bo(2 Since τ j are uniformly bounded, we know ha h j is bounded in W 1,2 (B o (1. By he Sobolev embedding heorem we know ha by passing o a subsequence i converges in L 2 (B o (1. This implies (2.10. Now using he subharmoniciy of σ jk we have ha, over a compac subdomain Ω, H j uniformly converges o a meric H (x, which is a leas coninuous. In order o prove ha one can exrac a convergen subsequence ou of H i,we need o esablish he esimaes for he higher order derivaives. The equaion (2.8

13 The Poisson Equaion and Hermiian-Einsein Merics 691 and he sandard boosrapping for he ellipic PDE make i sufficien o only prove a uniform esimae for he C 1 bounds. Namely, we only need o show ha for a compac subse Ω M, here exiss a consan C 4, independen of j, such ha (2.12 sup H j C 4. Ω I will be enough if we can prove i for small balls. Therefore, le Ω = B o ( 1 2. By he esimae in he las paragraph we know ha here exiss a consan C, independen of j,suchha: H j 2 dx C. B 0 (1 By our C 0 esimae (2.9, i implies ha: ( H j H 1 j 2 C B 0 (1 for some consan C > 0 independen of j. Also we can assume ha B o ( 1 2 is small enough such ha he vecor bundle is rivial over B o ( 1 2. We also know ha ( H j H 1 j 2 is a subharmonic funcion by [10] (see [10, Secion 2.3]. By he mean-value inequaliy of Li-Schoen [15], or he local version, [11, Theorem 8.17], we have ha ( H j H 1 j 2 C ( H j H 1 j 2. B 0 (1 sup Bo(1/2 Here C only depends on he volume of B o ( 1 2 and he lower bound of curvaure of he Kähler meric ω on B o (1. Combining hem ogeher we have (2.12. Once we have he C 1 -esimae (2.12, he sandard ellipic heory shows ha, by passing o a subsequence, H j converges uniformly over any compac subse of M o a meric H which is a soluion of ΛF H 0 on he whole manifold. Therefore we complee he proof of Theorem MOMENT TYPE ESTIMATES In his secion we esablish some momen ype esimaes for he soluion o he hea equaion. The mehod we use here is differen from Nash s mehod in he sense ha we do no need o inroduce he enropy quaniy, and derive he esimae direcly from he Guassian ype upper bound of he hea kernel proved by Li-Yau [17]. The proof is also simpler. A by-produc resul, which parially generalizes a heorem of Li on he long ime behavior of he soluion o he hea equaion on Ricci nonnegaive manifold, follows aferwards. Le us sar wih he momen ype esimae firs.

14 692 LEI NI Theorem 3.1. Le M be a Riemannian manifold wih nonnegaive Ricci curvaure. Le u(x, be he nonnegaive soluion o he hea equaion ( u(x, = 0, u(x, 0 = f(x. Here f is a nonnegaive funcion. As in [21] we define k(x, r = 1 f(ydv y. V x (r Bx(r Then for any d ( n 2 <d< here exiss a consan C(n,d such ha u(x, C(n, da d/2, for 1andsomeA>0 if and only if k(x, r Ar d,forr 1.Here nis he dimension of he manifold. Moreover, if A 1 r d k(x, r Ar d, we have he momen esimae: (3.1 C 1 (n, da 1 d/2 u(x, C(n, da d/2. For he case k(x, r A log(1 + r we have he similar esimae: u(x, C(nAlog(1 +, and u(x, A log(1 + also implies ha k(x, r C(nAlog(1 + r. Proof. Firs, by he uniqueness of he nonnegaive soluion, we can wrie u(x, in erms of he hea kernel on M as u(x, = M H(x,y,f(ydv y. Now recall he hea kernel esimae of Li-Yau, which says ha, for some C = C(n, H(x,y, C(n V x ( e r2 (x,y/((4+ε.

15 The Poisson Equaion and Hermiian-Einsein Merics 693 Then we have u(x, C(n ( V x ( e r2 /((4+ε 0 C(n V x ( 0 + C(n V x ( C(nA d/2 + C(n Bx(r f(yda y dr Bx(r ( e r2 /((4+ε ( 1 V x ( f(yda y dr Bx(r f(yda y dr f(ydv y Bx(r 2r (4+ε e r 2 /((4+ε dr. The second erm above follows afer inegraion by pars. Using he assumpion and volume comparison we hen have ( ( u(x, C(nA d/2 Vx (r + V x ( r d 2r (4 + ε e r 2 /((4+ε dr ( ( C(nA d/2 r n + r d 2r (4 + ε e r 2 /((4+ε dr C(n, da d/2. Here we have used he fac ha ( r n+d e r 2 /((4+ε n+d 2rdr=Γ +1 ((4 + ε 0 2 (n+d/2+1. The condiion d> n 2is used here o have he inegrabiliy of he lef hand side above near r = 0. To prove he necessary par we use he lower bound esimae of Li-Yau on he hea kernel. A d/2 u(x, C 1 1 (n V x ( e r2 /((4+ε 0 C 1 1 (n V x ( e r2 /((4+ε 0 C 1 1 (n V x ( Bx( f(ydv y. f(yda y Bx(r f(yda y Bx(r The proof for he logarihmic case needs a lile more care on he calculaion. As before, by Li-Tau s upper bound on he hea kernel we have ha u(x, C(nk(x, + C(n V x ( e r2 /(5 f(yda. Bx(r

16 694 LEI NI The firs erm is conrolled by C(nAlog(1+ by he assumpion ha k(x, r log(1+r. Doing he inegraion by pars as above, we have he following esimae for he second par. C(n V x ( e r2 /(5 ( Bx(r f(yda Bx(r f(ydy e r2 /(5 C(n V x ( C(nA n/2+1 log(1 + rr n+1 e r2 /(5 dr C(nA log(1 + ττ n/2 e τ/(5 dτ ( 2r dr 5 n/2+1 0 C(nA log(1 + 5ss n/2 e s ds 0 C(nAlog(1 + 0 sn/2+1 e s ds C(nAlog(1 +. The oher direcion is simpler and very similar o he previous case. Noe ha, if in he above esimae (of Theorem 3.1 we choose f(xo be r(x, he disance funcion o a fixed poin o M, we obain he momen esimae on complee Riemannian manifolds wih nonnegaive Ricci curvaure. This was firs proved by Nash for he Euclidean case in [18]. In [16], Li-Tam proved an upper bound esimae on he soluion o he hea equaion on general Riemannian manifolds wih lower bound on he Ricci curvaure. As a corollary we have he following resul on he asympoic behavior of he soluion o he hea equaion. Corollary 3.2. In he case d = 0, here exiss a consan C = C(n such ha C 1 (nk(x, u(x, C(n sup k(x, r. r In paricular, lim u(x, = 0 if and only if lim r k(x, r = 0. Corollary 3.3. Le M be a complee Riemannian manifold wih nonnegaive Ricci curvaure. Suppose ha u(x, is he soluion of he hea equaion ( u(x, = 0, u(x, 0 = f(x. Assume ha, for some consan a, 1 (3.2 lim f(y a dy = 0. r V x (r Bx(r Then lim u(x, = a.

17 The Poisson Equaion and Hermiian-Einsein Merics 695 Proof. By he assumpion we know ha boh (f a + = max{0,(f a} and (f a = min{0,(f a} saisfy (2.1. Le u +(x, (u (x, behe soluion wih he iniial daa (f a + ((f a. By Corollary 3.2 we have ha ( 1 0 u + (x, C(n sup (f a + (y dy 0, as. r V x (r Bx(r Therefore lim u +(x, = 0. Similarly we have lim u (x, = 0. This also holds for u (x, = u + + u, which solves he hea equaion wih he iniial daa f a. Our saemen is an easy consequence of i. This parially generalizes Li s heorem (cf. [14, Theorem 3] since we need no assume eiher ha M is of maximum volume growh or ha f is bounded. Bu our assumpion (3.1 on he iniial daa is sronger han he assumpion in Theorem 3 of [14]. In order o sudy he Hermiian-Einsein hea equaion we need he following resul, which is he parabolic version of Lemma 2.4. Proposiion 3.4. Le M be a complee Riemannian manifold wih nonnegaive Ricci curvaure. Le f be a nonnegaive coninuous funcion on M. Consider he hea equaion ( v(x, = f(x, v(x,0 = 0. Then i has a nonnegaive soluion v(x, such ha sup 0 v(x, < if and only if r f (y dy dr <. V x (r Bx(r 0 Proof. The case when M is parabolic is he same as in Lemma 2.4. We jus focus on he nonparabolic case. We sar wih he necessary par. Suppose ha here is a soluion v(x, such ha sup 0 v(x, <. v (x, isanonnegaive soluion of he hea equaion such ha 0 (v d < by he assumpion. Therefore here will be a sequence i such ha v(x, i converges over a compac subse of M. By passing o a subsequence we hen obain a nonnegaive soluion of he Poisson equaion by aking v (x = lim i v(x, i. Applying Lemma 2.4 we prove he necessary par. To prove he sufficien par we do he following esimaes. v(x, H(x,y,τf(ydv y dτ 0 M 1 C(n 0 0 V x ( e r2 /((4+ε f(yda y dr d. B x (r

18 696 LEI NI Here we have used he hea kernel esimae of Li-Yau. Using Fubini heorem we have ha ( ( 1 v(x, C(n 0 0 V x ( e r2 /((4+ε d f(yda dr B x (r ( r 2 ( 1 = C(n 0 0 V x ( e r2 /((4+ε d f(yda dr B x (r ( ( 1 + C(n 0 r 2 V x ( e r2 /((4+ε d f(yda dr B x (r = I + II. Applying he esimae in he proof of Theorem 5.2 of [17], we have ha ( ( 1 I + II C(n 0 r 2 V x ( d f(yda dr B x (r ( ( = C(n V x ( d f(yda dr. 0 r B x (r Applying Fubini s heorem one more ime as in he proof of Lemma 2.4, we have r v(x, C(n f(ydv y dr <. 0 V x (r Bx(r Leing u(x, = v (x, we can wrie he following corollary. Corollary 3.5. Le M be a complee Riemannian manifold wih nonnegaive Ricci curvaure. Le f be a nonnegaive coninuous funcion on M. Consider he hea equaion ( u(x, = 0, u(x, 0 = f(x. Then i has a nonnegaive soluion u(x, such ha 0 u(x, s ds < if and only if r f (y dy dr <. V x (r Bx(r 0 We should poin ou ha, if we replace he non-negaiviy of he Ricci curvaure by he assumpion ha M saisfies he L 2 -Sobolev inequaliy, he similar resul holds. In fac we have he following resul. Theorem 3.6. Le M be a complee Riemannian manifold of dimension n,which suppors he L 2 -Sobolev inequaliy. Le f be a nonnegaive coninuous funcion on M. Denoe k(x, r = 1 r n f(ydy. Bx(r

19 The Poisson Equaion and Hermiian-Einsein Merics 697 Consider he soluion u(x, o he scalar hea equaion wih he iniial daa u(x, 0 = f(x.wehaveha (i if k(x, r Ar 2d for some n/2 < d <, hen here exiss a consan C(n,d such ha u(x, C(n, da d/2 ; (ii if lim r k(x, r = 0,henlim u(x, = 0. The proof of he above resul is very similar o ha of Theorem 3.1. When f(x=r(o,x, he disance funcion o a poin o M, Nash s momen esimae in [18] is a wo-sides esimae, which is similar o Theorem 3.1 and beer han he above resul. Even his argumen is applicable o general manifolds, i does no seem o work if one replace r(o,xby a general funcion f(x. In his sense, our resul is more general. 4. HERMITIAN-EINSTEIN HEAT EQUATION In his secion we are going o prove a long ime exisence for he Hermiian- Einsein flow under a very relaxed condiion on he iniial meric on any complee Kähler manifold. Using he resuls from las secion we shall also give a hea flow proof of Theorem Theorem 4.1. Le M be a complee Kähler manifold, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0. If here is a consan λ such ha he mean curvaure ΛF H0 saisfies ΛF H0 λi Afor some posiive number A, hen he Hermiian-Einsein flow h h 1 = ΛF H +λi, h(0 = I has a long ime soluion on M [0,. Hereh(x, = H(x,H0 1 (x. Noeha we do no assume ha M has nonnegaive Ricci curvaure. As before, we use he compac exhausion consrucion o prove he long ime exisence. Also, we will only give he proof for he case λ = 0. Le Ω i be a compac exhausion of M. By Donaldson s heorem we know ha he following Dirichle problem is solvable: (D i h h 1 = ΛF H, H(x,0 = H 0 (x, H(x, Ωi = H 0 (x. Denoe by H i (x, he soluion o he above Dirichle problem. To prove he heorem we need o show ha here exiss a subsequence of H i ha converges o a soluion defined on he whole manifold M.

20 698 LEI NI Before we prove he heorem we need a couple of lemmas. The following lemmas were proved in [9]. Le v(x, = log de(hh0 1 (x,, R(x, = r ΛF H,andR(x = r ΛF H0. The firs lemma is on he equaion of ΛF H (4.1 (4.2 Lemma 4.2. ( ( ΛF H = 0, R(x, = 0. An easy consequence of above lemma is he following resul. Lemma 4.3. Le H( be a soluion o he Hermiian-Einsein flow. ê(x, = ΛF H( saisfies ê ê 0. and Then From he maximum principle we have ha ê i (x, = ΛF Hi (, where H i (x, are soluions o (D i, saisfies (4.3 sup ê i A. Ω i [0, Anoher consequence of Lemma 4.2 is he following corollary. Corollary 4.4. ( (4.4 v(x, = R(x. Proof. Firs, from he definiion of he so-called Bo-Chern class [3], i is easy o see ha ( r F H r F H0 = r h 0 h 1 (x, s ds. Using he fac we can conclude ha ( log de(h = r h h 1, (4.5 log de(h = r FH r F H0. Nowiiseasyoseeha (4.6 v(x, = R(x R(x,.

21 The Poisson Equaion and Hermiian-Einsein Merics 699 Direc calculaion also shows ha ( h v = r h 1 = R(x,, from which our claim follows. Le H and K o be wo Hermiian merics on E. Then HK 1 is a secion of Hom(E, E. Recall ha Donaldson s disance funcions τ(h,k and σ(h,kare defined as τ(h,k = r(hk 1 and σ(h,k = τ(h,k+τ(k,h 2rank(E. The funcion τ(h(x,,h 0 (x saisfies he following lemma. Lemma 4.5. Le H(x, be a soluion o Hermiian-Einsein flow wih iniial meric H 0 (x. Then ( log τ(h,h 0 ΛF H0. As a corollary, if we denoe ρ i (x, = log τ(h i,h 0 log(rank(e, hen we have he following (4.7 ( ρ i (x, ê 0 (x, ρ i (x, 0 = 0, ρ i (x, Ωi = 0. Here ê 0 (x = ΛF H0. Now applying he maximum principle o e ρ i, noicing ha ( / ρ i A, we can have he following (4.8 sup ρ i (x, e T A. Ω i [0,T ] Proof of Theorem 4.1. Denoe v i (x, = log de(h i H0 1. Direc calculaion shows ha v i = r( ΛF H i. Togeher wih Lemma 4.3, his implies ha v i A. Therefore de(h i H0 1 (x, e A. Using he above inequaliy and (4.8, we have ha sup Ω i [0,T ] r(h 0 H 1 i C 1 (T, A.

22 700 LEI NI This would imply ha (4.9 sup σ(h i,h 0 C 2 (T, A. Ω i [0,T ] In paricular, over any compac subse K, we have he so-called C 0 -esimae (4.10 sup σ(h i,h 0 C 2 (T, A. K [0,T ] Once we have esablished he C 0 -esimaes, we can adap he argumen in [20] o conclude ha here exiss a subsequence of H i (x, such ha i converges o a soluion o he Hermiian-Einsein flow (cf. [20, Theorem 3.8]. The argumen is basically he following. From (4.3, ogeher wih a similar differenial inequaliy on τ i, one can esablish he inegral bound for he gradien. Then he Sobolev embedding implies he L 2 -convergence, hen he C 0 -convergence of H i, afer applying he mean value inequaliy of Li-Tam [16] for he nonnegaive subsoluions o he hea equaion. Then one can use Donaldson s scaling argumen o obain he C 0 -bounds for he gradiens. The argumen in he las par of he proof of Theorem 2.10 also works here. This finishes he proof of Theorem 4.1. Theorem 4.6. Le M be a complee Kähler manifold wih nonnegaive Ricci curvaure, and le E be a holomorphic vecor bundle of rank k over M wih meric H 0 such ha ΛF H0 saisfies, for every x M, Then he Hermiian-Einsein flow 0 ΛF H0 (y dy <. V x ( Bx( h h 1 = ΛF H, h(0 = I. has a long ime soluion on M [0,, andh(x, converges o a soluion o he Hermiian-Einsein equaion. Proof. The key poin is o derive an upper bound esimae on σ(h(x,,h 0 (x which is independen of. I is a his poin we need o use some sharper esimaes han he one used in he proof of Theorem 4.1. Le v (x, be he soluion o he equaion ( v (x, = ê 0 (x, v (x, = 0.

23 The Poisson Equaion and Hermiian-Einsein Merics 701 By Proposiion 3.4 we know ha sup 0 v (x, <. From Lemma 4.5 and he comparison principle we know ha τ(h(x,,h 0 (x ke v (x,.wegive anoher derivaion of he esimae for σ(h(x,,h 0 (x, which makes use of he funcion v(x, = log de(hh0 1 (x,. Using he fac v(x,0 = 0and he comparison principle, we have ha v(x, v (x,. Thisgivesalower bound on de(hh0 1 (x,. Togeher wih he esimae on τ(h(x,,h 0(x we have a upper bound esimae for τ(h 0 (X, H(x, by simple algebraic consideraion, and hen he upper bound esimae of σ(h(x,,h 0 (x. Ifweuse he fac ha τ(h 0 (x, H(x, also saisfies Lemma 4.5 we can, as in he ellipic case, have a more precise esimae σ(h(x,,h 0 (x 2k(e v (x, 1. Applying Proposiion 3.4 we have ha on any compac subse K M here is a uniform upper bound for σ(h(x,,h 0 (x. Togeher wih he gradien esimae we derived in he proof of Theorem 2.10 we have a leas i such ha H(x, i converges o a meric H (x. Using Lemma 4.3 and he fac ê(x, 0 = ΛF H0, by Corollary 3.3 and he comparison principle we have ha lim ê(x, = 0. Therefore H (x is a soluion o he Hermiian-Einsein equaion. The convergence of H(x, o H (x follows from a more careful esimae of σ(h(x,,h 0 (x as shown below. Le 1 2 and le ĥ(x, = H(x,H 1 (x, 1. I is an easy maer of checking o see ha From here i is easy o see ha Then we have ha ĥ ĥ 1 = ΛF H. log r(ĥ = r ( H (x, H 1 (x, 1 r(h(x, H 1 (x, 1 = r( ΛF HH(x,H 1 (x, 1 r (H(x,H 1 ΛF H. (x, 1 (4.11 τ(h(x,,h(x, 1 ke 1 ΛF H (x,s ds. We have a similar esimae for τ(h(x, 1, H(x,. Combining hem we have σ(h(x,,h(x, 1 2k(e 1 ΛF H (x,s ds 1.

24 702 LEI NI Le u(x, be a soluion of he hea equaion wih u(x, 0 = ê 0 (x. Then by Corollary 3.5 and Lemma 4.2 we know ha 2 σ(h(x, 2, H(x, 1 2k(e 1 u(x,s ds 1 2k(e 1 u(x,s ds 1 0, as 1. Thus we have proved ha H(x, converges o H (x. Remark. When M has λ 1 (M > 0, he hea flow proof of he Hermiian- Einsein merics was given in [20]. If M saisfies he L 2 -Sobolev inequaliy, one can provide a hea flow proof of he exisence of he Hermiian-Einsein merics if he mean curvaure of he iniial meric has cerain decay condiions in erms of k(x, r, which is defined in Theorem 3.6 of he las secion. The resul will be slighly differen from he saed resul in Theorem 2.9. We leave he deails o he ineresed readers. Acknowledgmens. The auhor wishes o hank Professor L-F. Tam for suggesing a simple proof of Theorem 3.1 afer reading he firs draf. The auhor would also like o hank Professor P. Li, Professor R. Schoen and Professor L. Simon for heir coninuous encouragemens. Research was parially suppored by NSF gran DMS and DMS REFERENCES [1] T. AUBIN, Nonlinear analysis on manifold. Monge-Amperé equaions, Fundamenal Principles of Mahemaical Sciences Volume 252, Springer-Verlag, [2] S. BANDO & R. KOBAYASHI, Ricci-fla Kähler merics on affine algebraic manifolds. II, Mah. Ann. 287 (1990, [3] R. BOTT & S.S. CHERN, Hermiian vecor bundles and he equidisribuion of he zero of heir holomorphic secions, Aca Mah. 114 (1965, [4] H.D. CAO, On Harnack s inequaliies for he Kähler-Ricci flow, Inven. Mah. 109 (1992, [5] S.Y. CHENG & P. L I, Hea kernel esimaes and lower bounds of eigenvalues, Commen. Mah. Helv. 56 (1981, [6] B.L. CHEN,S.H.TANG & X.P. ZHU, A uniformizaion heorem of complee noncompac Kähler surfaces wih posiive bisecional curvaure, preprin. [7] S.Y. CHENG & S.T. YAU, On he exisence of a complee Kähler manifolds and he regulariy of Fefferman s equaion, Comm. Pure Appl. Mah. 33 (1980, [8] E.B. DAVIES, Explici consans for Guassian upper-bounds on hea kernels, Amer. J. Mah. 109 (1987, [9] S.K. DONALDSON, Ani self-dual Yang-Mills connecions over complex algebraic surfaces and sable vecor bundle, Proc.LondonMah.Soc.50 (1985, [10], Boundary value problems for Yang-Mills fields, Journal of Geomery and Physics 8 (1992,

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