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1 ASIAN J. MATH. c 2004 International Press Vol. 8, No., pp , March THE EINSTEIN-KÄHLER METRIC WITH EXPLICIT FORMULAS ON SOME NON-HOMOGENEOUS DOMAINS AN WANG, WEIPING YIN, LIYOU ZHANG, AND WENJUAN ZHANG Dedicated to Yum-Tong Siu on the occasion of his 60th birthday. Abstract. In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit forms of the complete Einstein-Kähler metrics on Cartan-Hartogs domains which are the non-homogeneous domains. Let M be a complex manifold. Then a Hermitian metric i,j g i,j dzi dz j defined on M is said to be Kähler if the Kähler form Ω = i,j g i,j dzi dz j is closed. The Ricci form is given by log det(g i,j ). If the Ricci form of the Kähler metric is proportional to the Kähler form, the metric is called Einstein-Kähler. If the manifold is not compact, we require the metric to be complete. Clearly for a noncompact complex manifold to admit such a metric, it is necessary that there exists a volume form, the negative of whose Ricci tensor defines a complete Kähler metric. The volume form of this Kähler metric must be equivalent to the original volume form. If we normalize the metric by requiring the scalar curvature to be minus one, then the Einstein-Kähler metric is unique. Cheng and Yau[CY] proved that any bounded domain D which is the intersection of domain with C 2 boundary admits a complete Einstein-Kähler. Without any regularity assumption on the domain D, Mok and Yau[MY] proved that the complete Einstein-Kähler metric always exists. This Einstein-Kähler metric is given by E D (z) := z i z j dz i dz j, where g is an unique solution to the boundary problem of the Monge-Ampère equation: ( 2 ) g det = e (n+)g z D, z i z j g = z D, We call g the generating function of E D (z). It is obvious that if one determines g explicitly, then the Einstein-Kähler metric is also determined explicitly. Therefore if one would like to compute the Einstein-Kähler metric explicitly, it suffices to compute the generating function g in explicit formula. The explicit formulas for the Einstein-Kähler metric, however, are only known on homogeneous domains. In his famous paper [Wu], H.Wu points out that among the four classical invariant metrics(i.e. the Bergman metric, Carathéodory metric, Received April 3, 2003; accepted for publication September 8, Research supported in part by NSF of China (Grant No and 07068) and NSF of Beijing (Grant No and 02004). Department of Mathematics, Capital Normal University, Beijing 00037, P. R. China (wangancn@sina.com). wyin@mail.cnu.edu.cn or wpyin@263.net 39

2 40 A. WANG, W. YIN, L. ZHANG AND W. ZHANG Kobayashi metric and Einstein-Kähler metric), the Einstein-Kähler metric is the hardest to compute because its existence is proved by complicated nonconstructive methods. The purpose of this paper is to compute the explicit formulas of complete Einstein-Kähler metrics on Cartan-Hartogs domains of the following four types: Y I (,m,n; K) := {w C,Z R I (m, n) : w 2K < det(i ZZ T ),K >0} := Y I, Y II (,p; K) := {w C,Z R II (p) : w 2K < det(i ZZ T ),K >0} := Y II, Y III (,q; K) := {w C,Z R III (q) : w 2K < det(i ZZ T ),K >0} := Y III, Y IV (,n; K) := {w C,Z R IV (n) : w 2K < 2ZZ T + ZZ T 2,K >0, } := Y IV. Where R I (m, n),r II (p),r III (q) andr IV (n) are the first, second, third and fourth Cartan domains respectively in the sense of Loo-Keng HUA[Hu], Z T indicates the conjugate and transpose of Z, det indicates the determinant. The Cartan-Hartogs domains are introduced in 998. The Bergman kernel functions on Cartan-Hartogs domains are obtained in explicit formulas in [Yin,Yin2,Yin3,GY]. And these Bergman kernel functions are Bergman exhaustions, therefore all of the Cartan-Hartogs domains are bounded pseudoconvex domains. Some results on Hua domains can be found in [Yin-Yin3,GY,YW,YWZ]. This paper is organized as follows. Section presents some background material and known results from domain Y I needed for us. In section 2, by using the noncompact automorphism group of Y I and the biholomorphic invariance of the Einstein- Kähler metric, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z,w) = w 2 [det(i ZZ T )] K, this differential equation can be solved to give an implicit function in X. In section 3, we give explicit forms of complete Einstein-Kähler metric on Y I. And the explicit formulas of complete Einstein-Kähler metrics on Y II,Y III and Y IV are given in section 4. At that time the Y I,Y II,Y III,andY IV are the non-homogeneous domains. The authors would like to express their gratitude to the referee for his/her review of this paper and much useful suggestions.. Preliminaries. Ω C can first of all be interpreted as the total space of the anticanonical line bundle over Ω. The latter is endowed a Hermitian metric unique up to a global multiplicative constant. Aut(Ω) acts canonically on Ω C preserving the unit disk bundle D Ω C, i.e., the set of all vectors of length <. The action of Aut(Ω) on Ω extends naturally to an action on Ω C and hence on D. Write D = {(z,w) : w <ϕ o (z)}. The Cartan-Hartogs domains are of the form D α = {(z,w) : w <ϕ α o (z)} for an arbitrary positive real constant α. We may now interpret D α as the unit ball bundle of a positive power of the anticanonical line bundle represented by (Ω C,h α ), where the exponent is real. For an arbitrary α>0 we do not have a canonical extension of the action of of Aut(Ω) on Ω to an action on D α. Instead, there is a semi-direct product H = Aut(Ω) S (an extension of Aut(Ω) by S ), acting on Ω C, such that S is a normal subgroup acting on Ω C by scalar mutliplication (z,w) (z,e iθ w),θ R, in the second factor, and such that H acts as holomorphic bundle isomorphisms of (Ω C,h α ) and hence as automorphisms of D α. When α is rational, α = p/q, p, q coprime, H contains a proper subgroup Aut(Ω) µ q, where µ q S denotes the group of q-th roots of unity, µ q = Z/qZ. Dα with α rational are thus particularly interesting geometrically. In fact, if Γ Aut(Ω) is a torsion-free cocompact discrete subgroup, then we have an action of Γ µ q on D α,

3 EINSTEIN-KÄHLER METRIC ON NON-HOMOGENEOUS DOMAINS 4 and the normal quotient space D α /Γ carries the structure of a strongly pseudoconvex manifold whose exceptional set is the compact Hermitian locally symmetric manifold Ω/Γ =N. If it is furthermore possible to define a q-th root of the canonical line bundle on N, then Γ µ q is a direct product, so that Γ acts on D α without any fixed point, and the Einstein-Kähler metric on D α descends to an Einstein-Kähler metric on the strongly pseudoconvex manifold D α /Γ. The exceptional set N D α /Γ is then a totally geodesic complex submanifold with respect to the Einstein-Kähler metric. Lemma. The automorphism group Aut(Y I )ofy I consists of the following mappings F (z,w; z 0,θ 0 ): { w = e iθ0 w det(i Z 0 Z T 0 ) 2K det(i ZZ T 0 ) K, Z = A(Z Z 0 )(I Z T 0 Z) D, where A T A =(I Z 0 Z T 0 ), D T D =(I Z T 0 Z 0 ), Z 0,Z R I (m, n),θ 0 R. Proof. See [Yi]. Obviously, the F (z,w; z 0,θ 0 ) maps points (Z 0,w) Y I on to points (0,w )and Z = A(Z Z 0 )(I Z T 0 Z) D is holomorphic automorphism of R I (m, n). Lemma 2. Let X = X(Z, w) = w 2 [det(i ZZ T )] K. under the mapping of Aut(Y I ). That is X(Z,w )=X(Z, w). Proof. See [Yi]. Then X is invariant Lemma 3. If F = F (z,w; z 0,θ 0 ) Aut(Y I ). Let J F be the Jacobi matrix of F (z,w; z 0,θ 0 ), i.e. z w J F = z z w, 0 w where z =(z,,z n,z 2,,z 2n,,z m,,z mn ) is a vector, z jk is the element of Z and Z =(z jk ) m n R I (m, n). Then one has z z w z w w =(A T D T ) z0=z z0=z z0=z z0=z, = K eiθ0 det(i ZZ T ) 2K E(Z) T w, = e iθ0 det(i ZZ T ) 2K, where E(Z) = (tr[(i ZZ T ) I Z T ], tr[(i Z T ) I 2 Z T ],, tr[(i ZZ T ) I mn Z T ]) is a column vector with mn entries. I pq is defined as a m n matrix, the (p,q)-th entry of I pq, i.e. the entry located at the junction of the p-th row and q-th column of I pq, is, and others entries of I pq are zero. The meaning of is following(see [Lu]):

4 42 A. WANG, W. YIN, L. ZHANG AND W. ZHANG Let A = (a ij ) p q, B = (b kl ) r s, then A B = (c αβ ) pr qs is defined as c αβ = a ik b jl, where α = q(i ) + j, β = s(k ) + l. Proof. It can be obtained by direct computation. Lemma 4. If F = F (z,w; z 0,θ 0 ) Aut(Y I )andt = T [(z,w), (z,w)] is the metric matrix of the Einstein-Kähler metric of Y I, one has T [(z,w), (z,w)] = J F z0=zt [(0,w ), (0,w )]J F T z0=z, and J F 2 z 0=z = det(i ZZ T ) (m+n+ K ), where J F = det J F. Proof. It can be proved by using the invariance of the Einstein-Kähler metric under the holomorphic automorphism of Y I. 2. Reduction of the Monge-Ampère equation to an ordinary differential equation. Let Z =(z jk ) m n R I (m, n). We denote (z,w) =(z,,z n,z 2,,z 2n,,z m,,z mn,w)=(z,z 2,,z N ), where N = mn +,and 2 g g αβ (z,w) = z α z β α, β =, 2,,N. Note that z N = w. Suppose g(z,w) generates the Einstein-Kähler metric of Y I. Then g(z,w) isa solution to the boundary problem of the Monge-Ampère equation: { det(gαβ (z,w)) = e (N+)g(z,w) (z,w) Y I, () g = (z,w) Y I. Let F :(z,w) (z,w ), F = F (z,w; z 0,θ 0 ) Aut(Y I ). Because of the invariance of the metric, it is easy to show that det(g αβ (z,w)) = J F 2 det(g αβ (z,w )). So Thus e (N+)g(z,w) = J F 2 e (N+)g(z,w ). e g(z,w ) = J F N+ e g(z,w). For arbitrary (z, w) Y I, especially take z 0 = z, θ 0 = arg w, that is F 0 = F (z,w; z, arg w). We have e g(0,w ) 2 = J F0 N+ e g(z,w) = det(i ZZ T ) (m+n+/k) N+ e g(z,w), where w = w = X 2. If λ = K(m + n) +, then J F0 2 = X λ w 2λ. Let h(x) =e g(0,x 2 ) = e g(0,w ). We obtain 2 h(x) = J F0 N+ e g(z,w) = X λ N+ w 2 λ N+ e g(z,w). 2 Hence h w = h (X) w = J F 2 N+ e g(z,w) ( g) w.

5 EINSTEIN-KÄHLER METRIC ON NON-HOMOGENEOUS DOMAINS 43 It is obvious that Where Then { X w = w = X w, w = X w. w det(i ZZ T ) /K w 0, 0 w =0. g w = X w h (X) h(x). (2) Since h(x) =X λ N+ w 2 λ N+ e g(z,w),wehave h = h (X) = h(x) g + λ X h(x), z α z α z α N + z α where α =, 2,,N. Then g ( λ = z α N + X h (X) ). h(x) z α Let then Y (X) = λ N + X h (X) h(x), (3) g = YX, α =, 2,,N. z α z α g w = ( Y λ ) N + w. So, we have w w = Y X w 2, w z β = w Y z β, z α w = Y X w + YX 2 X z α z α w YX 2 w z α, (4) = Y X + YX z α z β z β z α 2 X YX z α z β 2 z β z α,

6 44 A. WANG, W. YIN, L. ZHANG AND W. ZHANG where α, β =, 2,,N. Because = z pq K Xtr[(I ZZT ) I pq Z T ], z st = K Xtr[(I ZZT ) ZI T st]. From (2) and (5), we have = w, w = =0, z α z pq where α = n(p ) + q, β = n(s ) + t. And from (5), we have 2 X z pq z st = K = w, w = =0, z β z st z st tr[(i ZZ T ) I pq Z T ]+ X K tr[(i ZZT ) I pq I T st] + X K tr[(i ZZT ) ZI T st(i ZZ T ) I pq Z T ]. From (2), (5), (6), (7), by computation, we obtain 2 X z pq z st = X K δ psδ qt, where δ ps = 2 X z pq w =0, 2 X w z st =0, { p = s, 0 p s. From (6), (8) and (4), we obtain w w = Y, w z β =0, z α w =0, δ qt = z α z β = YK δ αβ, { q = t, 0 q t. (5) (6) (7) (8)

7 EINSTEIN-KÄHLER METRIC ON NON-HOMOGENEOUS DOMAINS 45 where α, β =, 2,,N. Therefore det(g αβ (0,w )) = det Y K 0... Y K 0 Y = ( ) N Y Y. K Since det(g αβ (z,w)) = det(g αβ (0,w )) J F0 2 = ( ) N Y Y J F0 K 2, and e (N+)g(z,w) = J F0 2 e (N+)g(0,w ) = J F0 2 h (N+), we reduce the Monge-Ampère equation to an ordinary differential equation: ( ) N Y Y = h (N+). K It is equivalent to Differentiating with respect to X log(y N Y )+(N + ) log h log K N =0. (Y N Y ) (Y N Y ) By computation and (3), we obtain +(N +)h h =0. (Y N Y ) (Y N Y ) + λ X (N +)Y X =0. So we have It is equivalent to [X(Y N Y )] =(Y N+ ) λ N (Y N ). XY N Y = Y N+ λ N Y N + C, (9) where C is a constant. Because g w =(Y λ N + ) w. So Y = w g w + λ N +,

8 46 A. WANG, W. YIN, L. ZHANG AND W. ZHANG it holds for (z,w) Y I. If take (z,0) Y I, then X =0,thusw g w w=0 = 0, therefore Y (0) = λ (λ N )λn N+. We obtain C = from (9). N(N+) N+ Suppose g is a solution of Monge-Ampère equation. Then from the above we have g = N + log[( Y K )N Y det(i ZZ T ) (m+n+ K ) ], where Y are the solutions to the following problem: XY N Y = Y N+ λ N Y N + Y (0) = (λ N )λn N(N +) N+, λ (0) N +. The solutions of above problem are not unique. If g is the generating function of the complete Einstein-Kähler metric on Y I, then g is the unique solution of the problem () and has the form as follows: g = N + log[( Y K )N Y det(i ZZ T ) (m+n+ K ) ], where Y is a solution of problem (0). The unique solution g of (), which includes the condition that g(x) as x Y I, determines a particular solution to (0). For the problem (0), we obtain where ( ) C X =(Y Y (0)) Y N λ N N Y (0) k Y N k e ϕ(y ), N(N +) Y ϕ(y )= Y (0) Y N λ N N(N+) k= (N +)Y N N k= Y dy. (0)k Y N k C is a positive constant, and Y is the function in X. Therefore the problem (0) can be solved to give an implicit function in X. There is a unique choise of the positive constant corresponding to the unique solution g of problem (). 3. Complete Einstein-Kähler metric with explicit formula on Y I. Let K = mn+ m+n, m iny I.Wehaveλ = N +,C = 0. The equation (0) becomes { XY = Y 2 Y, () Y (0) =. We obtained solution Y = C X, where C is a constant. Let C =. Then Y = X. Therefore g = N+ log[( Y K )N Y det(i ZZ T ) (m+n+ K ) ] = log[ X det(i ZZT ) K K N +N ]. Now, we will prove g is the solution of problem (). Obviously, det(g αβ (z,w)) = e (N+)g(z,w) for any (z,w) Y I.

9 EINSTEIN-KÄHLER METRIC ON NON-HOMOGENEOUS DOMAINS 47 If point ( z, w) Y I and w 0, when (z,w) Y I, (z,w) ( z, w), we have X. So X + and det(i ZZT ) w 2K > 0. Therefore, g(z,w) + as (z,w) Y I. If point ( z, w) Y I,andw= 0, when (z,w) Y I, (z,w) ( z, 0), we have X >, and det(i ZZT ) 0, det(i ZZ T ) K +. That is g(z,w) + as (z,w) Y I. Therefore g generates the complete Einstein-Kähler metric of Y I in the case of, m, and is given by the following explicit formula: K = mn+ m+n g = log[ X det(i ZZT ) N K K +N ]. wherex = X(Z, w) = w 2 [det(i ZZ T )] K,N = mn +and(w, Z) Y I. The Y I in the case of K = mn+ m+n, m is nonhomogeneous domain. 4. Complete Einstein-Kähler metric with explicit formulas on Y II, Y III, Y IV. Follow the above idea and similar procedure, we get the explicit formulas of complete Einstein-Kähler metrics on Y II,Y III and Y IV as follows.. For Y II. Suppose g is a solution of Monge-Ampère equation on Y II. Then we have g = N + log[( Y K )N Y det(i ZZ T ) (p++ K ) ], where Y are the solutions to the following problem: XY N Y = Y N+ λ N Y N + Y (0) = (λ N )λn N(N +) N+, λ (2) N +. Where N = 2p(p +)+,λ= K(p +)+. In the case of K =/(p +)+p/2,p >, if g generates the complete Einstein- Kähler metric of Y II then g is given by the following explicit formula: g = log[ X det(i ZZT ) N K K +N ]. wherex = X(Z, w) = w 2 [det(i ZZ T )] K and (w, Z) Y II. 2. For Y III. Suppose g is a solution of Monge-Ampère equation on Y III. Then we have g = N + log[( Y K )N Y det(i ZZ T ) (q + K ) ], where Y are the solutions to the following problem: XY N Y = Y N+ λ N Y N + Y (0) = (λ N )λn N(N +) N+, λ (3) N +.

10 48 A. WANG, W. YIN, L. ZHANG AND W. ZHANG Where N = 2q(q ) +,λ= K(q ) +. In the case of K =/(q ) + q/2,q >, If g generates the complete Einstein- Kähler metric of Y III then g is given by the following explicit formula: g = log[ X det(i ZZT ) K K ( 2 ) N +N ]. wherex = X(Z, w) = w 2 [det(i ZZ T )] K and (w, Z) Y III. 3. For Y IV. Suppose g is a solution of Monge-Ampère equation on Y IV. Then we have g = N + log[(2y K )N Y β(z, Z) (n+ K ) ], where Y are the solutions to the following problem: XY N Y = Y N+ λ N Y N + Y (0) = (λ N )λn N(N +) N+, λ (4) N +. Where N = n +,λ= Kn +. In the case of K =/n +, if g generates the complete Einstein-Kähler metric of Y IV then g is given by the following explicit formula: g = log[ X β(z, Z) N K (K/2) +N ]. wherex = X(Z, w) = w 2 [β(z, Z)] K, (w, Z) Y IV, and β(z, Z) =+ZZ T ZZ T 2ZZ T. In the above cases, Y II,Y III and Y IV are non-homogeneous domains. REFERENCES [CY] S.Y. Cheng and S.T. Yau, On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman s equation, Comm Pure Appl Math, 33 (980), pp [GY] Bingxin Guan and Weiping Yin, The Bergman kernel function on Cartan-Hartogs domain of the fourth type and a biholomorphic invariant(in Chinese), J.of Xiaman University (Nature Science Edition), 39:5 (2000), pp [He] M. Heins, On a class of conformal metrics, Nagoya Math. J., 2 (962), pp. 60. [Hu] Loo-Keng Hua, Harmonic Analysis of Function of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, RI, 963. [Lu] Qi-keng Lu, The Classical Manifolds and the Classical domains (in chinese), Shanghai Scientific and Technical Publisher, Shanghai, 963. [MY] N. Mok and S.T. Yau, Completeness of the Kähler-Einstein metric on bounded domain and the characterization of domain of holomorphy by curvature conditions, ProcSymposia Pure Math, 39 (983), pp [Wu] H. Wu, Old and new invariants metrics on complex manifolds, Several Complex Variables: Proceedings of the Mittag-Leffler Institute, (J.E.Fornaess, ed.), Math. Notes, Vol. 38, Princeton Univ. Press, Princeton, NJ, 993, pp [Yin] Weiping Yin, The Bergman Kernel function on super-cartan domains of the first type, Science in China(Series A), 43: (2000), pp [Yin2] Weiping Yin, The Bergman kernel function on Cartan-Hartogs domain of the second type (in Chinese), Chinese Annals of Mathematics, 2A:3 (2000), pp

11 EINSTEIN-KÄHLER METRIC ON NON-HOMOGENEOUS DOMAINS 49 [Yin3] [YW] [YWZ] Weiping Yin, The Bergman kernel function on Cartan-Hartogs domains of the third type (in Chinese), Advances in Mathematics(CHINA), 29:5 (2000), pp Weiping Yin, An Wang, Zhengang Zhao, Xiaoxia Zhao, Bingxin Guan, The Bergman kernel functions on Hua Domains, Science in China (Series A), 44:6 (200), pp Weiping Yin, An Wang and Xiaoxia Zhao, Comparison theorem on Cartan-Hartogs domain of the first type, Science in China (Series A), 44:5 (200), pp

12 50 A. WANG, W. YIN, L. ZHANG AND W. ZHANG

ENGG1410-F Tutorial 6

ENGG1410-F Tutorial 6 Jianwen Zhao Department of Computer Science and Engineering The Chinese University of Hong Kong 1/16 Problem 1. Matrix Diagonalization Diagonalize the following matrix: A = [ ] 1 2 4 3 2/16 Solution The