The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria. Quasi Einstein Kähler Metrics

Transcription

1 ESI The Erin Schrödinger International Boltmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Quasi Einstein Kähler Metrics Henrik Pedersen Christina Tønnesen Friedman Galliano Valent Vienna, Preprint ESI 796 (1999) November 6, 1999 Supported by Federal Ministry of Science and Transport, Austria Available via anonymous ftp or gopher from FTP.ESI.AC.AT or via WWW, URL:

2 Quasi-Einstein Kähler Metrics Henrik Pedersen, Christina Tønnesen Friedman, Galliano Valent Abstract We rite an ansat for quasi-einstein Kähler metrics and construct ne complete examples. Moreover, e construct ne compact generalied quasi-einstein Kähler metrics on some ruled surfaces, including some of Guan s examples as special cases. 1 Introduction Let (M, J) be a complex manifold. In this paper e consider pairs (g, V ) consisting of a Kähler metric and a real holomorphic vector field V on M, such that JV is an isometry of g and ρ λω = L V Ω, (1) here ρ is the Ricci form, Ω is the Kähler form and λ is some constant. Such structures are called quasi-einstein Kähler metrics or Kähler Ricci solitons [4, 7, 9, 14]. Quasi-Einstein metrics are solitons for the Hamilton flo [9] d dt g t = r t + s t n g t, () here r t is the Ricci curvature tensor and s t is the average scalar curvature of g t. Indeed, if g 0 is quasi-einstein then (Φ t ) g 0 solves (), here Φ t = exp(tv ). Friedan [6] studied quasi-einstein metrics in connection ith bosonic σ-models. He shoed that the one-loop renormaliability of the model is ensured if and only if the metric of the target space is quasi-einstein. If M is a compact manifold, equation (1) implies that [ρ] = λ[ω] H (M, R). Thus a necessary condition for M to admit a quasi-einstein Kähler metric is that c 1 (M) = [ ρ ] is π either positive, negative, or ero. In short, c 1 must have a sign. If c 1 0, then Calabi, Yau Joint Odense, Aarhus, LPTHE, and ESI preprint Dept. of Mathematics and Computer Science, Odense University, Campusvej 55, DK 530 Odense M, Denmark Dept. of Mathematical Sciences, University of Aarhus, DK-8000 Århus C, Denmark Laboratoire de Physique Théorique et des Hautes Energies, Unité Associée au CNRS URA 80, Université Paris 7, Place Jussieu, 7551 Paris Cedex 05, France 1

3 and Aubin [3, 16, 1] shoed that there exist Kähler-Einstein metrics. Hoever, for c 1 > 0 e do not alays have Kähler-Einstein metrics and the quasi-einstein Kähler metrics serve as suitable generaliations. For any Kähler metric on a compact manifold, e have that ρ ρ H = 1 ϕ Ω, here ρ H is the harmonic part of ρ and ϕ Ω is called the Ricci potential. In fact, ϕ Ω = Gs, here G is the Green s operator of the Laplacian and s is the scalar curvature. In particular, if [ρ] = λ[ω], then ρ λω = i ϕ Ω = L ( ϕω ) Ω = L 1 gradϕ Ω Ω il J 1 gradϕ Ω Ω = L 1 gradϕ Ω Ω, here the last equality follos from the fact that ρ λω is real (e use for raising indices and for loering indices). Thus if gradϕ Ω is a holomorphic vector field and [ρ] = λ[ω], then (g, 1 gradϕ Ω) is a Kähler-Ricci soliton. Conversely, if (g, V ) is a Kähler-Ricci soliton on a compact manifold ith positive first Chern class, then V = 1 gradϕ Ω: From equation (1) e have that L V Ω = L 1 gradϕ Ω Ω. Let W = V ijv be the holomorphic (1, 0) vector field corresponding to V. Then or L W Ω = 1 ϕ Ω d(jw ) = 1 ϕ Ω. Since JW is holomorphic and g is Kähler, e have that Thus (JW ) = ( 1 ϕ Ω ). (JW ) = 1 ϕ Ω + α, here α is a -closed (0, 1)-form. Moreover, by the Hodge decomposition, (JW ) = f + H for some function f and a harmonic form H. But since e assumed that c 1 > 0, there are no non-trivial harmonic 1-forms (see page 34 in []) and hence f = 1 ϕ Ω + α.

4 This gives Therefore and α must vanish. Thus or hich is equivalent to f = 1 ϕ Ω. f = 1ϕ Ω + constant (JW ) = 1 ϕ Ω W = ( ϕ Ω ), V = 1 gradϕ Ω. In particular, gradϕ Ω is a holomorphic real vector field. The observation above tells us that the existence of quasi-einstein Kähler metrics ith non-trivial vector fields is an obstruction to the existence of Kähler-Einstein metrics: The Futaki invariant of the Kähler class on the vector field V is given as the L -norm of V [14]. In the compact case, Tian and Zhu [14] have proved uniqueness (modulo automorphisms) for Kähler-Ricci solitons ith a fixed vector field. In this paper, e construct quasi-einstein Kähler metrics on complex line bundles (or their compactifications P(O L)) over Kähler-Einstein base manifolds B. We impose no restriction on the sign of the scalar curvature s B of B and our results therefore extend knon constructions. Hoever, for s B 0, the first Chern class of P(O L) does not have a sign. This motivates our study of generalied quasi-einstein Kähler metrics in section 4. Previously, a family of extremal Kähler metrics as obtained [13] on P(O L) over Riemann surfaces of genus at least to. Contrary to the result in [13], our construction of generalied quasi-einstein Kähler metrics on these ruled surfaces exhausts the Kähler cone. The structure of this paper is as follos. In section, e rite an ansat for quasi- Einstein Kähler metrics ith a torus symmetry. In section 3, e find solutions in the case of S 1 symmetry and some additional assumptions. This gives ne complete (non-compact) quasi-einstein Kähler metrics, as ell as some already knon examples. In section 4, e consider Guan s generalied quasi-einstein Kähler metrics [7]. We then construct such metrics in every Kähler class on the compactification of holomorphic line bundles over compact Riemann surfaces. This construction includes ne compact metrics, as ell as some metrics already constructed by Guan and Koiso. An ansat for quasi-einstein Kähler metrics In this section, assuming the existence of a real torus acting through holomorphic isometries on a Kähler manifold, e construct an ansat for quasi-einstein Kähler metrics. 3

5 .1 The moment map construction of Kähler metrics Folloing [1], e consider the situation of a real torus T N acting freely on the Kähler manifold M m through holomorphic isometries. Proposition 1 [1] Let ( ij ), i, j = 1,..., N be a positive definite symmetric matrix and (h µν ), µ, ν = 1,..., m N a positive definite hermitian matrix of smooth functions on an open set U in C m N R N ith coordinates (ξ µ, i ). Assume that the -form Ω h := 1 h µνdξ µ dξ ν is a Kähler form on an open set in C m N ith corresponding Kähler metric h. Let M be a T N -bundle over U ith connection 1-form ω = (ω 1,..., ω N ). Suppose that h µν i + 4 ij = 0, (3) j ξ µ ξν ij k and assume the torus bundle has curvature = ik j (4) F i = 1 h µν i dξ µ dξ ν + 1 ij ξ µ dj dξ µ 1 ij ξ µ dj dξ µ. (5) Then g = h + ij d i d j + ij ω i ω j, (6) here ij = ( 1 ) ij, is a Kähler metric on M. Conversely any Kähler metric ith a torus acting freely through holomorphic isometries can locally be constructed as above. Proof: The proof is straightforard and e just make some remarks concerning the second part of the proposition. Suppose that M is a T N -symmetric Kähler manifold ith metric g, Kähler form Ω and complex structure J. Suppose further that (X 1,..., X N ) are Hamiltonian vector fields generated by the torus action. Let d j = i Xj Ω define the Hamiltonian functions j. Then the metric is given as in equation ( 6), here h is a Kähler metric in the quotient space of each level set of the Hamiltonians. Note that ij = g(x i, X j ) and ω i = ij X j so Jω i = ij d j and Ω = d i ω i + Ω h, here Ω h is the Kähler form of the of the Kähler quotient. As J is integrable, the exterior derivative dϕ i of the (1, 0) forms ϕ i = ij d j + 1ω i must have no (0, ) part. Also for g to be Kähler e need dω = 0. These conditions are captured by equation (4) and by the equation dω i = F i ith F i as in (5) 1. Then equation (3) is just the integrability condition df i = 0. 1 To be absolutely precise, the pull-back of F i ith respect to the bundle projection is given by dω i. 4

6 . Quasi-Einstein Kähler metrics No let M m be a T N -symmetric Kähler metric as above. Let V be the holomorphic vector field V = 1 N J (c i X i ), i=1 here c i are constants. We ill look for the condition under hich the metric is quasi- Einstein Kähler ith respect to V and some constant λ. Thus the equation to examine is ρ λω = L V Ω. (7) The Ricci form ρ is given by ρ = 1 u = 1 djdu, here u = log( det h ). The Lie derivative of Ω ith respect to V is given by det L V Ω = d(i V Ω) = 1 c id( ij ω j ). Using equation (5) ith dω i = F i, e find that equation (7) is equivalent to the pair of equations ( c j + u ) ij = λ ( i + B i ) j and here B i is some constant. We conclude ( 4 u ξ µ ξ = λ h µν ( i + B i ) h ) µν, ν i Proposition Let M m be a T N -symmetric Kähler manifold as in Proposition 1. Then the metric is a quasi-einstein Kähler metric, ρ λω = L V Ω, ith vector field V = 1 N i=1 J(c ix i ) if and only if the folloing equations are satisfied; here B i is some constant. ( c j + u ) ij = λ ( i + B i ) j (8) ( 4 u ξ µ ξ = λ h µν ( i + B i ) h ) µν, ν i (9) Notice that by setting c j = 0 in the above ansat, e get the Pedersen and Poon ansat for Kähler-Einstein metrics [1]. In fact, the difference beteen the to ansäte appears only in equation (8). 5

7 In the case m = and N = 1, e have that h = e u (dx + dy ) ith ξ = x + iy and the above equations can be reritten as and u + c = λ( + B) (10) u xx + u yy + λ(e u ( + B)(e u ) ) = 0. (11) By changing the moment map by a constant, e may assume that B = 0. Using equation (10), one can then replace equation (11) by an equation completely decoupled for the u function: u xx + u yy + ( (e u ) ) + c ( e u) = 0. (1) Equation (3), i.e., the constraint folloing from the Kähler property, is given by xx + yy + (e u ) = 0. Hoever, this equation follos from equations (10) and (1). Thus, as in the Kähler- Einstein case considered in [1], e are left ith the single partial differential equation (1). Once e have solved this equation, e easily find, by using equation (10). 3 A construction of ne complete quasi-einstein Kähler metrics In this section e consider the case N = 1. By solving the differential equations from the previous section in a special case, e find ne complete quasi-einstein Kähler metrics. First, e give the details of the special case in hich e solve the equations. Then, e apply the ansat from the previous section. 3.1 The assumptions Let (B, g B ) be a (m 1)-dimensional compact Kähler manifold ith scalar curvature s B. Assume that the Kähler form Ω B is such that the derham class [ Ω B π ] is contained in the image of H (B, Z) H (B, R). Let L be a holomorphic line bundle such that c 1 (L) = [ Ω B ]. On the total space M of (L 0) π B e can form an S 1 -symmetric Kähler π metric g = g B + d + 1 ω here, being the coordinate of (a, b) (0, ], becomes the moment map of g ith the obvious S 1 action on L, is a positive function depending only on, and ω is the connection one-form of the connection induced by g on the S 1 -bundle (L 0) (π,) B (a, b). 6

8 That is dω = Ω B. Clearly, equations (3) and (4) are satisfied. The complex structure J on M is given by the complex structure on B and Jω = d. The Kähler form is given by Ω = Ω B + d ω. If X is the Hamiltonian vector field generated by the S 1 action, then d = Ω( X, ) = ( JX), and The Ricci form ρ is given by ω = 1 = g(x, X), g(x, ) g(x, X) = X. ρ = ρ B i log( m 1 ) hich implies that the scalar curvature s is given by ) s = s B ( m 1. m 1 If 1 (by hich e mean 1/) is such that 1 (a) = 0 and ( 1 ) (a) =, then e can add a copy of B at = a and extend the Kähler metric g over the ero-section of the bundle L B. If moreover b <, 1 (b) = 0, and ( 1 ) (b) = then e can add another copy of B at = b and extend g to a Kähler metric on the total space of the CP 1 -bundle P(O L). We refer to [10, 11] for the details. 3. Applying the ansat In this special case, h is the local description of g B. That is, if locally Ω B = then h µν = q µν. Equation (8) is here 1 q µνdξ µ dξ ν, (c + u ) 1 = λ ( + B) (13) and equation (9) is 4 u ξ µ ξ ν = λbq µν. (14) 7

9 Since u = (m 1) log log + log(det q) and ρ B = 1 log(det q) dξ µ dξ ν ξ µ ξ ν e see from equation (14) that g B must be Kähler-Einstein. From no on, e assume that λ 0. Since, by the Lefschet decomposition, e have that B = or The general solution of this ODE is ρ B = s B. Equation (13) no becomes λ(m 1) s B (m 1) Ω B (15) c + u = λ + s B (m 1) ( m 1 ) + c m 1 = λm + s B (m 1) m 1. (16) ( m 1 ) = ke c + A m m + + A 1 + A 0, (17) here the A i s are fixed (given by λ, c, m and s B ) and k is chosen freely. For s B 0, e have from equation (15) that c 1 (L) = (m 1) s B [ ρ B (m 1) ] = c 1 (K 1 ), π s B here K 1 is the anti-canonical line bundle of B. Since c 1 (L) must be an integer class, there are some restrictions on the sie of s B. For s B = 0 e cannot use equation (15) to calculate c 1 (L). 3.3 The case s B 0: Given a > 0, suppose e chose (the unique) k such that 1 (a) = 0. Then from equation (16), e see that ( 1 ) (a) = if and only if λ = s B (m 1). a(m 1) Assume no that e have chosen k and λ such that the endpoint conditions at = a are met. For any b > a, 1 (b) = 0 implies, together ith equation (16), that ( 1 ) (b) = λb + s B (m 1) = b a + s B (m 1) (1 b a ) > 0. 8

10 Since ( 1 ) (a) > 0, e conclude that there can be no such b. Hence the function 1 is positive for all > a. Thus e have a Kähler quasi-einstein metric g on the total space of L B. If c is chosen to be positive, then This implies [1, 8] that g is complete. lim 1 = A m. Theorem 1 Let (B, g B ) be a non-positive compact Kähler-Einstein manifold of dimension (m 1). Assume that [ Ω B π ] is an integer cohomology class. Let L be a holomorphic line bundle on B such that c 1 (L) = [ Ω B ]. Let X denote the Hamiltonian vector field generating π the natural S 1 action on L and let J denote the complex structure on the total space of L B. Then, for a given a > 0 and a given c > 0, there exists a complete Kähler metric g on the total space of the bundle L B such that the pair (g, 1 cjx) is quasi-einstein, satisfying the equation ρ λω = L 1 cjxω, here λ = s B (m 1). a(m 1) Notice that there is no hope of producing quasi-einstein metrics on the compact manifold P(O L) B. This can be seen by the fact that c 1 has no sign for s B 0: If m = the argument is quite simple. Let C denote the fiber in P(O L) B and let E 0 denote the ero-section in P(O L) B. Then by the adjunction formula for complex surfaces e have that c 1 C = > 0 and c 1 E 0 = deg L (g B 1), here g B denotes the genus of B. Thus for g B > 0 (or equivalently s B 0) e see that c 1 cannot have a sign. In higher dimension, take any compact metric on M of the type described in subsection 3.1. Since C ρ > 0 and E 0 ρ Ω m B ] does not have a sign. In the next section e ill consider Guan s generaliation of quasi-einstein Kähler metrics and get existence results for all ruled surfaces of the form P(O L) B. 3.4 The case s B > 0: < 0 hen s B 0, e conclude that c 1 = [ ρ π This case has already been considered by others [9, 7, 5]. If L is such that c 1 of the compact complex manifold M = P(O L) B is positive, then there exist compact quasi-einstein Kähler metrics on M of the type described in this section. This follos as a special case of the results in [7], hich generalies Koiso s construction [9]. Moreover, Chave and Valent [5] considered the case B = CP m 1. They found the compact solutions on M = P(O K p m ) B for p = 1,..., m 1, here K is the canonical line bundle. For p = m + 1, m +,... they found complete non-compact quasi-einstein Kähler metrics on In a private communication Prof. Guan pointed out that, due to editorial error, Theorem and Corollary in [7] are both missing the assumption that ρ B (hence s B ) is non-negative. 9

11 the total space of K p m B. Note that s B = m(m 1). Using the same arguments that e p used in the case s B 0, it is easy to sho the folloing general result. Theorem Let (B, g B ) be a positive compact Kähler-Einstein manifold of dimension (m 1). Assume that s B = m(m 1) ith p = m + 1, m +,.... Let L be a holomorphic line p bundle on B such that c 1 (L) = p c m 1(K), here K is the canonical line bundle. (Such a line bundle L exists hen B = CP m 1. In general, e may, at orst, have to assume that p = m, 3m,....) Then for a given a > 0 and a given c > 0, there exists a complete Kähler metric g on the total space of the bundle L B such that the pair (g, 1 cjx) is quasi-einstein, satisfying the equation here ρ λω = L 1 cjxω, λ = m p pa. 4 Generalied quasi-einstein Kähler metrics In [7] Guan defines generalied quasi-einstein Kähler metrics in any Kähler class on any compact complex manifold. In this section, e introduce the defining equation for such metrics. Then e solve it under the same conditions as in subsection 3.1, ith the additional assumptions that m =, g B is Kähler-Einstein and that the final metric g is compactified to a Kähler metric on P(O L) B. In fact, our solutions exhaust the Kähler cone. 4.1 The equation Let g be a Kähler metric and V be a real holomorphic vector field on a compact manifold M. The pair (g, V ) is said to be generalied quasi-einstein if ρ ρ H = L V Ω. (18) A generalied quasi-einstein Kähler metric serves as a generaliation of constant scalar curvature Kähler metrics. Indeed, Guan proved that a generalied quasi-einstein Kähler metrics has constant scalar curvature if and only if the Futaki invariant of the Kähler class vanishes. Thus, these metrics behave very much like the extremal Kähler metrics. 4. The equation in a special case We no make the same assumptions as in subsection 3.1. Moreover e assume that m =, that is, B is a compact Riemann surface, and that g B is Kähler-Einstein, that is, ρ B = s B Ω B. Finally, e assume that g is such that it can be extended to a smooth Kähler metric on the compact manifold M = P(O L) B. This means that for a given pair 10

12 (a, b) (determining the Kähler class [13]), ith 0 < a < b, at the point = a (resp. b) the function 1 vanishes and has derivative equal to (resp. ). By rescaling, e may assume that a = 1. Since g is a positive definite metric, e have that 1 must be positive on the interval (1, b). If V = 1 cjx, then equation (18) implies that s s = L V Ω, Ω = d(i V Ω), Ω = c dx, Ω = c d( 1 ω), Ω = c djd, Ω = c 1, ω = c, here s denotes the average scalar curvature s = sdµ M dµ. M On the other hand, if then, by the Hodge decomposition, hich implies that Then gradϕ Ω = cgrad = V and, since s s = c, Gs = c, ϕ Ω = Gs = c + constant. ρ ρ H = L 1 gradϕ Ω Ω is satisfied for any compact Kähler metric, e conclude that (g, V ) is generalied quasi- Einstein. The above is summaried in the folloing lemma. Lemma 1 Let B be a compact Riemann surface ith Kähler-Einstein metric g B and let M be the ruled surface P(O L) B, here L is a holomorphic vector bundle such that c 1 (L) = [ Ω B ]. For each b > 1, that is, for each Kähler class on M [13], consider the π Kähler metric from subsection 3.1 g = g B + d + 1 ω, 11

13 here 1 is a smooth function, positive on the interval (1, b), such that it satisfies the boundary conditions 1 (1) = 1 (b) = 0 (19) and ( 1 ) (1) =, (0) ( 1 ) (b) =. (1) Then the pair (g, k 1 JX) is a generalied quasi-einstein Kähler metric if and only if 1 is such that s s = k. () In this case, kjx = gradϕ Ω. 4.3 Solving the equation in the special case Recall that s = s B ( ). Inserting equation (19), (0), and (1) into equation (), e get s B ( ) s B(b 1)+(b+1) (b 1) = k = k d( 1 ω), Ω or ( ) ( ) k = k(( 1 ) + 1 ) = ( s B(b 1) + (b + 1) ) + s (b B. (3) 1) Integrating equation (3) and inserting equations (19), (0) and (1) e get ( ) ( ) k = s B(b 1) + (b + 1) (b 1) + s B + b(b + 1) s Bb(b 1). (4) (b 1) Notice no, that if solves equation (4) and equation (19) is satisfied, then equations (0) and (1) follo. Thus e have to find a solution of (4) such that 1 vanishes at = 1 and at = b but is positive in the interval (1, b). The solution of (4), such that ( ) vanishes at = 1, is given by e k G(, k) = (Au + Bu + C)e ku du, 1 here P (u) = Au + Bu + C is the polynomial on the right hand side of equation (4). Notice that P (1) = and P (b) = b. Let u 0 be a ero of the polynomial P (u) hich lies 1

14 in (1, b) and u 1 the second possible ero. In the case s B b+1, elementary computations b 1 sho that P (u) = A(u u 1 )(u u 0 ), ith A(u u 1 ) < 0 for u (1, b). In the case s B = b+1, e have that b 1 Au + Bu + C = B(u u 0 ), ith B < 0. It follos that e can rite P (u) = p(u)(u u 0 ) ith p(u) negative for u (1, b). We ant to prove that, given a fixed b > 1, G(b, k) = 0 has a unique non-ero solution in k. Consider the auxiliary function F (k) = G(b, k)e u 0k = b 1 p(u)(u u 0 )e k(u u 0) du. Its derivative ith respect to the variable k is positive. Thus F (k) is monotonic and increasing. Furthermore, F ( ) = and F (+ ) = +. This proves the existence and uniqueness of k. Lastly, the possibility of k = 0 needs to be excluded. Toard this end, e compute G(b, 0) = (b 1) 3 ( sb ) b 1 b + 1 This vanishes only if s B = 4 b+1 > 4, hich can be excluded by the folloing reasoning. b 1 Recall that if s B 0 then c 1 (L) = s B c 1 (K), here K is the canonical line bundle on B. Thus if s B > 0 (equivalently, if the genus of B is equal to 0), then c 1 (L) = 4 s B. Hoever, since c 1 (L) must be an integer [15], the situation s B > 4 is not possible. Since G(b, 0) < 0, it follos that k is alays strictly positive. At this point, since equations (19), (0), and (1) are satisfied, and ( ) is nothing but the sum of an exponential function and a second degree polynomial, e can sho that the function ( ( ) and hence must be positive beteen = 1 and = b: clearly ) has at most three eroes (since the graph of an exponential function intersects a parabola at, at most, three points). We kno from (19) that it vanishes at = 1 and = b. Suppose that a third vanishing point, x, lies beteen 1 and b. Then one easily sees that ( ) is positive on the interval (1, x) and negative on the interval (x, b) - or vice versa. In either case, ( ) has the same sign at = 1 and at = b. By equations (0) and (1), this is not possible. Therefore ( ( ) 0 for 1 < < b. Since ) is positive in a small neighborhood to the right of = a (alternatively in a small neighborhood to the left of = b), it must be positive everyhere on the interval (1, b). By rescaling g B appropriately, e can obtain any (negative) holomorphic line bundle on B. We conclude ith the folloing theorem. Theorem 3 Let M = P(O L) B, here L is a non-trivial holomorphic line bundle on a compact Riemann surface B. Then any Kähler class on M admits a generalied quasi- Einstein Kähler metric (g, V ), here V = 1 gradϕ Ω and W := V ijv is a multiple of the holomorphic vector field generating the natural C action on L.. 13

15 Remark 1 If the genus of B is less than, then the metrics in the above Theorem ere constructed by Guan in [7]. In particular, if the genus of B is equal to 0, L = K 1 and b is such that the Kähler class is a multiple of c 1 (M), e have the quasi-einstein Kähler metric on CP CP constructed by Koiso in [9] (see also [5]). If the genus of B is at least, e notice that, as opposed to the family of extremal Kähler metrics constructed in [13], the metrics do exhaust the Kähler cone. Acknoledgements The authors ould like to thank Daniel Guan, Claude LeBrun and Gideon Maschler for very helpful conversations. References [1] T. Aubin, Equations du type de Monge-Ampère sur les variétés kähleriennes compactes, C.R. Acad. Sci. Paris 83, (1976), [] A.L. Besse, Einstein Manifolds, Springer, Berlin, [3] E. Calabi, On Kähler manifolds ith vanishing canonical class, Algebraic geometry and topology, a symposium in honor of S.Lefschet, Princeton Univ. Press (1955), [4] H.-D. Cao, Existence of gradient Kähler Ricci solitons, Elliptic and Parabolic Methods in Geometry, (Minneapolis, MN, 1994) B. Cho, R. Gulliver, S. Levy, J. Sullivan ed., AK Peters (1996), [5] T. Chave, G. Valent, On a class of compact and non-compact quasi-einstein metrics and their renormaliability properties, Nuclear Physics B 478 (1996), [6] D. Friedan, Nonlinear models in + ε dimension, Phys. Rev. Lett. 45 (1980), [7] Z.-D. Guan, Quasi-Einstein metrics, Int. J. Math. 6 (1995), [8] A.D. Hang, M.A. Singer, A Momentum Construction for Circle-Invariant Kähler Metrics, preprint, math.dg/ [9] N. Koiso, On rotationally symmetric Hamilton s equation for Kähler-Einstein metrics, in Recent topics in differential and analytic geometry, Advanced Studies in Pure Math 18-I (1990), [10] C. LeBrun, Scalar-flat Kähler metrics on blon-up ruled surfaces, J. Reine Ange. Math. 40 (1991),

16 [11] C. LeBrun, S. Simanca, Extremal Kähler metrics and complex deformation theory, Geom. Func. Anal. 4 (1994), [1] H. Pedersen, Y.S. Poon, Hamiltonian construction of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature, Comm. Math. Phys. 136 (1991), [13] C. Tønnesen-Friedman, Extremal Kähler Metrics on Minimal Ruled Surfaces, J. Reine Ange. Math. 50 (1998), [14] G. Tian, X.H. Zhu, Uniqueness of Kähler-Ricci solotions, pre-print [15] R.O. Wells, Differential Analysis on Complex Manifolds, Springer, Ne York, [16] S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure and Appl. Math 31 (1978),