Riemannian geometry of Hartogs domains
aa r X i v : . [ m a t h . DG ] M a r Riemannian Geometry of Hartogs domains ∗ Antonio J. Di Scala, Andrea Loi, Fabio Zuddas
Abstract
Let D F = { ( z , z ) ∈ C n | | z | < b, k z k < F ( | z | ) } be a stronglypseudoconvex Hartogs domain endowed with the K¨ahler metric g F associated to the K¨ahler form ω F = − i ∂∂ log (cid:0) F ( | z | ) − k z k (cid:1) .This paper contains several results on the Riemannian geometry ofthese domains. These are summarized in Theorem 1.1, Theorem 1.2and Theorem 1.3. In the first one we prove that if D F admits a nonspecial geodesic (see definition below) through the origin whose trace isa straight line then D F is holomorphically isometric to an open subsetof the complex hyperbolic space. In the second theorem we prove thatall the geodesics through the origin of D F do not self-intersect, wefind necessary and sufficient conditions on F for D F to be geodesicallycomplete and we prove that D F is locally irreducible as a Riemannianmanifold. Finally, in Theorem 1.3, we compare the Bergman metric g B and the metric g F in a bounded Hartogs domain and we prove that if g B is a multiple of g F , namely g B = λg F , for some λ ∈ R + , then D F isholomorphically isometric to an open subset of the complex hyperbolicspace. Keywords : K¨ahler metrics; Hartogs domain; geodesics.
Subj.Class : 53C55, 32Q15, 32T15.
Let b ∈ R + ∪ { + ∞} and let F : [0 , b ) → (0 , + ∞ ) be a non increasingsmooth function on [0 , b ). The n -dimensional Hartogs domain D F ⊂ C n associated to the function F is defined by D F = { ( z , z ) ∈ C n | | z | < b, k z k < F ( | z | ) } , (1)where z = ( z , . . . , z n − ) and k z k = | z | + · · · + | z n − | . ∗ This work was supported by the MIUR Project “Riemannian metrics and differentiablemanifolds” . nder the assumption that D F is strongly pseudoconvex one canprove (see Proposition 2.1 in [11]) that the natural (1 , D F given by ω F = − i ∂∂ log (cid:0) F ( | z | ) − k z k (cid:1) (2)is a K¨ahler form on D F and this is equivalent to the requirement that F satisfies the condition (cid:18) xF ′ F (cid:19) ′ < , x ∈ [0 , b ) , (3)where the derivatives are taken with respect to the variable x = | z | .Notice that (3) (and hence the strongly pseudoconvexity of D F ) turnsout to be equivalent to the strongly pseudoconvexity of the bound-ary of D F at all z = ( z , z , . . . , z n − ) with | z | < x (see Propo-sition 2.1 in [11] for a proof). Denote by g F the K¨ahler metric as-sociated to the K¨ahler form ω F . Throughout all this paper we as-sume that D F is equipped with this K¨ahler metric. Notice that when F ( x ) = 1 − x, ≤ x < D F equals the n -dimensional complex hy-perbolic space C H n , namely the unit ball B n in C n equipped with thehyperbolic metric g hyp = g F . Hartogs domains are interesting bothfrom the mathematical and the physical point of view (see for exam-ple [4], [9] and [11] for the study of some Riemannian properties of g F and the Berezin quantization of ( D F , g F ), [1] and [10] for the construc-tion of global symplectic coordinates on these domains and [12] for theconstruction of K¨ahler-Einstein metrics on Hartogs type domains onsymmetric spaces).In this paper we are interested in the Riemannian geometry of Har-togs domains. In particular we study the geodesics, the completenessand local irreducibility of such domains with respect to the metric g F .We denote by G the set of those geodesics passing through the originwhose traces are straight lines of C n intersected with D F . Since theisometry group of D F contains U (1) × U ( n − S of geodesics of D F passing through the origin and containedin the plane z = 0 or in the complex line z = · · · = z n − = 0 isincluded in G . A geodesic ℓ ∈ G will be called a special geodesic if itbelongs to S .Our first result is the following interesting characterization of thecomplex hyperbolic space amongst Hartogs domains. Theorem 1.1
Let ( D F , g F ) be a Hartogs domain. If there exists ℓ ∈ G such that ℓ / ∈ S , then D F is holomorphically isometric to an open subsetof C H n . In other words the previous theorem asserts that if there exists one nonspecial geodesic ℓ through the origin of D F whose trace is a straight ine, then D F ⊂ C H n (and hence, in this case, G coincide with the setof all the geodesic through the origin).Our second result is the following: Theorem 1.2
Let ( D F , g F ) be a Hartogs domain. Then the followingfacts hold true.(i) All the geodesics through the origin of D F do not self-intersect;(ii) D F is geodesically complete with respect to the K¨ahler metric g F if and only if Z √ b s − (cid:18) xF ′ F (cid:19) ′ | x = u du = + ∞ , (4) where we define √ b = + ∞ for b = + ∞ ;(iii) ( D F , g F ) is locally irreducible around any of its points. The first part of the previous theorem should be compared with themain result of D’Atri and Zhao [2] asserting that in a bounded homo-geneous domain equipped with its Bergman metric all the geodesics donot intersect. (Notice that the homogeneous assumption for an Har-togs domain implies that D F is holomorphically equivalent to B n , theunit ball in C n (see e.g. Theorem 6.11 in [7]).Other properties of the geodesics of the Bergman metric can befound in [5] and [6]. In [5] Fefferman deeply studied the geodesics ofthe Bergman metric at the boundary points of a bounded domain D while [6] is concerned with the existence of a closed geodesic in any nontrivial homotopy class of a (non simply-connected) bounded domain.Regarding the completeness of the Bergman metric on a bounded do-main, the reader is referred to the classical paper of S. Kobayashi [8].By the previous discussion it is natural to compare the Bergmanmetric g B and the metric g F on a bounded Hartogs domain. Our thirdand last result is the following:
Theorem 1.3
Let D F be a bounded Hartogs domain. Assume that g B is a multiple of g F , namely g B = λg F , for some λ ∈ R + . Then g F isK¨ahler–Einstein and therefore D F is holomorphically isometric to anopen subset of C H n . The first part of the proof of the previous theorem is an adaptationof the proof of the following (unpublished) proposition communicatedby Miroslav Engliˇs to the second author and which deals with the moregeneral class of generalized
Hartogs domains.
Proposition 1.4 (Engliˇs) Let e Ω = { ( z, w ) ∈ Ω × C k : k w k < F ( z ) } e a bounded and simply-connected generalized Hartogs domain, where Ω is a pseudoconvex domain in C n and − log F is a smooth strictly-PSH function on Ω . Let g B be the Bergman metric and let g F be theK¨ahler metric on e Ω whose K¨ahler potential is − log( F ( z ) − k w k ) . If g B = λg F , for some λ ∈ R + , then g F is K¨ahler-Einstein. The next section is dedicated to the proof of our theorems.
The following lemma is the main tool in the proofs of Theorem 1.1 andTheorem 1.2.
Lemma 2.1
Let ( D F , g F ) be a Hartogs domain. Let M ⊂ D F be thereal (plane) surface given by: M = D F ∩ { Im ( z ) = Im ( z ) = 0 , z j = 0 , j = 2 , . . . , n − } , (5) and denote by g the metric induced on M by g F . Then ( M, g ) istotally geodesic, has constant Gaussian curvature equal to − and isgeodesically complete if and only if condition (4) above is satisfied. Proof:
The surface M is the fixed point set of the isometry of D F given by ( z , z , z , . . . , z n − ) (¯ z , ¯ z , − z , . . . , − z n − ) and hence itis totally geodesic in D F . By setting u = Re ( z ) and v = Re ( z ), thissurface can be described as M = { ( u, v ) ∈ R | v < F ( u ) , u < b } . (6)Furthermore, it is not difficult to see that the metric g induced by g F on M is given by g = (cid:18) g g g g (cid:19) = 2( F − v ) (cid:18) C − F ′ uv − F ′ uv F (cid:19) , (7)where C = F ′ · u − ( F ′ + F ′′ · u )( F − v ) and F and its derivatives areevaluated at u . By a straightforward, but long computation, one canverify that the Gaussian curvature of g equals − /
2. Hence (
M, g ) isisometric to an open subset, say U , of R H ( − ), namely the unit disk { ( x, y ) | x + y < } in R endowed with the Beltrami-Klein metric g BK = 2(1 − x − y ) (cid:2) (1 − y ) dx + 2 xydxdy + (1 − x ) dy (cid:3) . (8)An isometry between ( M, g ) and U can be described explicitly. Indeed,let ψ : ( −√ b, √ b ) → R be the strictly increasing real valued functiondefined by ψ ( u ) = Z u s − (cid:18) xF ′ F (cid:19) ′ | x = s ds. hen, it is not hard to see that the mapΨ : M → R H ( −
12 ) , ( u, v ) Tanh( ψ ( u )) , v Cosh( ψ ( u )) p F ( u ) ! is an injective local diffeomorphism satisfying Ψ ∗ ( g BK ) = g . Therefore,the completeness of M is equivalent to Ψ( M ) = R H , which is easilyseen to be equivalent to condition (4), and we are done. (cid:3) Remark 2.2
The fact that the surface M in Lemma 2.1 is totallygeodesic and that the isometry group of D F contains U (1) × U ( n − D F , fixing the origin and takingany given geodesic passing through the origin of D F to a geodesic lyingin M . This will be a key point in the proofs of both Theorem 1.1 andTheorem 1.2. Remark 2.3
All the n -dimensional Hartogs domain D F contains thecomplex totally geodesic surface B = { z j = 0 , j = 1 , . . . , n − } ∩ D F , (9)which in the literature of complex analysis is called the base of theHartogs domain D F . In view of the previous lemma, it is natural toconsider the Hartogs domains where the Gaussian curvature of B isconstant, , say equal to K ,. It is not hard to see that such domainscan be classified as follows:(a) if K = 0 then F ( t ) = ce − kt , c, k > t ∈ [0 , + ∞ ), (complexanalysts often refer to these domains as Spring domains );(b) If K > F ( t ) = ( c + c t ) − K , with c > c > t ∈ [0 , + ∞ );(c) If K < F ( t ) = ( c + c t ) − K , with c > c < t ∈ [0 , − c c ).Notice that in the case (b), the corresponding Hartogs domain D F cannot be geodesically complete. In fact in this case also its base B would be complete and hence biholomorphic to C P , yielding thecontradiction B ∼ = C P ⊂ D F ⊂ C n (cfr. Example 2.6 at the end ofthe paper). Proof of Theorem 1.1.
Let ℓ be a geodesic as in the statement of the theorem. Since ℓ / ∈ S ,by Remark 2.2, we can assume ℓ ⊂ M and that ℓ = { v = ku, k = 0 } ∩ M, here u and v are the parameters introduced in the proof of Lemma2.1. Hence ℓ can be parametrized as t ( u ( t ) , v ( t ) = ku ( t )), where t varies in a real interval containing the origin and the following geodesicequations have to be satisfied u ′′ +Γ u ′ +2Γ u ′ v ′ +Γ v ′ = 0 , v ′′ +Γ u ′ +2Γ u ′ v ′ +Γ v ′ = 0 , namely u ′′ + Γ u ′ + 2 k Γ u ′ + k Γ u ′ = 0 (10) ku ′′ + Γ u ′ + 2 k Γ u ′ + k Γ u ′ = 0 , (11)where Γ ijk , i, j, k = 1 , Γ = 12 D „ g ∂g ∂u − g „ ∂g ∂u − ∂g ∂v «« == − uD ( v − F ) [ u (2 F ′ + v F ′′ ) − F ( v − F )(2 F ′′ + u F ′′′ ) − F F ′ (2 F ′ +3 u F ′′ )] , (12)Γ = 12 D „ − g ∂g ∂u + g „ ∂g ∂u − ∂g ∂v «« == 4 u vD ( v − F ) [ − u F ′′ + F ′ ( F ′′ + u F ′′′ )] , (13)Γ = 12 D „ g ∂g ∂v − g ∂g ∂u « == − vD ( v − F ) [ − u F ′ + F ( F ′ + u F ′′ )] , (14)Γ = 12 D „ g ∂g ∂u − g ∂g ∂v « == 4 uF ′ D ( v − F ) [ − u F ′ + F ( F ′ + u F ′′ )] , (15)Γ = 12 D „ − g ∂g ∂v + g „ ∂g ∂v − ∂g ∂u «« = 0 , (16)Γ = 12 D „ g ∂g ∂v − g „ ∂g ∂v − ∂g ∂u «« == − vD ( v − F ) [ − u F ′ + F ( F ′ + u F ′′ )] , (17) here D = g g − g = 4 CF − F ′ u v ( F − v ) . By solving (10) with respect to u ′′ and substituting into (11) we get u ′ [Γ + k (2Γ − Γ ) + k (Γ − ) − k Γ ] = 0 (18)Since u ′ = 0 we getΓ + k (2Γ − Γ ) + k (Γ − ) − k Γ = 0 (19)(where Γ kij = Γ kij ( u, ku )).By using (12) - (17), after a very long but straightforward calcula-tion, the previous equation becomes8 ku (cid:0) u F ′′ + F (2 F ′′ + u F ′′′ ) − F ′ (2 u F ′′ + u F ′′′ ) (cid:1) D ( k u − F ) = 0 , (20)which, by setting u = t, ≤ t < b , is equivalent to the following ODE t F ′′ + F (2 F ′′ + tF ′′′ ) − F ′ (2 tF ′′ + t F ′′′ ) = 0 . (21)Notice that for t = 0 this equation can be written as t F ′′ + (cid:18) Ft − F ′ (cid:19) ( t F ′′ ) ′ = 0 . (22)By setting G = t F ′′ equation (22) becomes G ′ = − F ′′ tF − F ′ t G (23)(notice that F − F ′ t > < t < b since F is not decreasing)and hence G ( t ) = c e R − F ′′ tF − F ′ t dt = c ( F − F ′ t ) , (24)for some c ∈ R . For t → cF (0) = 0, i.e. c = 0. Therefore G = t F ′′ = 0, which implies F ( t ) = c − c t for some c , c >
0. Thenthe map φ : D F → C H n , ( z , z , . . . , z n − ) z p c /c , z √ c , . . . , z n − √ c ! is a holomorphic isometry of D F into an open subset of C H n and thisconcludes the proof of Theorem 1.1. (cid:3) emark 2.4 In the very definition of a Hartogs domain D F we haveassumed (cfr. the introduction) that F is non increasing in the inter-val [0 , b ). The statement of Theorem 1.1 holds true also without thisassumption. Indeed, it follows by condition (3) that F ′ ( t ) < ≤ t < ǫ < b , for some ǫ , and the proof works also inthis case. Proof of Theorem 1.2
Let ℓ ⊂ D F be a geodesic passing through the origin. By Remark 2.2we can assume ℓ ⊂ M . On the other hand by Lemma 2.1, ( M, g ) isisometric to an open subset of R H ( − ) where it is well-known thatall the geodesics do not self intersect. This proves (i) of Theorem 1.2.Notice that, again by Remark 2.2 and by Hopf–Rinow’s theoremthe completeness of g F is equivalent to that of g , which by Lemma 2.1is equivalent to (4) and we this proves (ii).In order to prove (iii), assume by contradiction that D F is locallyreducible around some point, say p ∈ D F . Since the group U (1) × U ( n −
1) acts by isometries on ( D F , g F ) we can assume that p ∈ M where M is given by (5). So p = ( u, v, , · · · ,
0) and we can assumethat both u, v are (real numbers) different from zero (indeed if oneof them is zero, say p = 0, then D F is locally reducible around thepoint p ′ = ( p ′ , p , , . . .
0) with p ′ sufficiently close to zero). Thereforethere exists an neighborhood D ⊂ D F of p ∈ D F such that ( D, g F )splits as a Riemannian product i.e. D = A × B , where A and B areK¨ahler manifolds. So the Lie algebra g of Killing vector fields of D alsosplits into two (or more) factors. Since u (1) × u ( n − ⊂ g it followsthat g has at most two factors. Moreover since p = ( u, v, , · · · , u, v = 0 we can recover the tangent space to the Riemannianfactors A and B . Thus, the factor A is an open subset A ⊂ C , with u ∈ A , and B is an open subset B ⊂ C n − , with ( v, , · · · , ∈ B .In particular such Riemannian factors must be orthogonal w.r. to g F .Then the coefficient g of the metric g on M induced by g F has to bezero for u and v different from zero. On the other hand, by (7) above, g = − F ′ uv = 0, a contradiction. This concludes the proof of (iii). (cid:3) With the aid of (ii) in Theorem 1.2 we now study the completeness oftwo specific Hartogs domains.
Example 2.5 If F ( t ) = ce − kt , c, k > t ∈ [0 , + ∞ ) then condition(4) is easily seen to be satisfied, so we get that the the Spring domainsare complete (cfr. (a) of Remark 2.3). Example 2.6 If F ( t ) = c + c t ) p ( p ∈ N + ), t ∈ [0 , + ∞ ) , then Z √ b s − (cid:18) xF ′ F (cid:19) ′ | x = u du = Z + ∞ √ c c pc + c u du = π √ p < ∞ hich proves that, for such F , the domain D F is not complete (cfr.(b) of Remark 2.3).We now prove the last result of this paper. Proof of Theorem 1.3
Recall that the Bergman metric g B on D F is, by definition, the onegiven by the K¨ahler potential log e K ( z , z ; z , z ), where e K ( z , z ; z ′ , z ′ )is the Bergman kernel of D F . Let˜ F ( z , z ) := F ( | z | ) − k z k . (25)Note that this is a local defining function (positively signed) for D F at any boundary point ( z , z ) with z ∈ B , and such boundary pointsare strictly pseudoconvex. The hypothesis of the theorem and the factthat D F is contractible means thatlog e K ( z , z ) = − λ log e F ( z , z ) + 2 Re G ( z , z )for some holomorphic function G on D F ; here and below we will writejust e K ( z , z ) for e K ( z , z ; z , z ). By rotational symmetry of e K and F ,the pluriharmonic function 2 Re G must depend only on | z | and k z k ,hence must be a positive constant, say µ . Thus e K ( z , z ) = µ e F ( z , z ) λ . (26)On the other hand, by Fefferman’s formula [5] for the boundary singu-larity of the Bergman kernel, e K ( z , z ) = a ( z , z ) e F ( z , z ) n +1 + b ( z , z ) log e F ( z , z ) , ( z , z ) ∈ D F , (27)where a, b ∈ C ∞ ( B × C n − ) and a ( z , z ) = n ! π n J [ ˜ F ]( z , z ) , (28)for z ∈ B and k z k = F ( | z | ) and where J [ e F ] is the Monge-Amperedeterminant J [ e F ] = ( − n det e F ∂ e F∂z ∂ z e F ∂ e F∂ ¯ z ∂ e F∂z ∂ ¯ z ∂ z ( ∂ e F∂ ¯ z ) ∂ ¯ z e F ∂ ¯ z ( ∂ e F∂ ¯ z ) ∂ ¯ z ∂ z e F . A direct computation gives J [ e F ] = − F ∂ log F∂z ∂ ¯ z . (29) which depends only on | z | ). By comparing (26) with (27) one gets: µ = a ( z , z ) e F ( z , z ) λ e F ( z , z ) n +1 + b ( z , z ) e F ( z , z ) λ log e F ( z , z ) , ( z , z ) ∈ D F , which evaluated at k z k = F ( | z | ), forces λ = n + 1. Further, by (28)and (29), the last expression gives − F ∂ log F∂z ¯ ∂z = c, for all z ∈ B and k z k = F ( | z | ), where c is the negative constantgiven by c = − µπ n n ! (notice that the condition k z k = F ( | z | ) is su-perfluous, since nothing there depends on z ). Feeding this back intoformula (29) one gets J [ F ]( z , z ) = c for all ( z , z ) ∈ D F , i.e. g F isK¨ahler-Einstein.Let us recall now Lemma 3.1. of [11]. Lemma 2.7
Let ( D F , g F ) be an n -dimensional Hartogs domain. As-sume that one of its generalized scalar curvatures is constant. Then ( D F , g F ) is holomorphically isometric to an open subset of the n -dimensional complex hyperbolic space. Since the scalar curvature is one of the generalized scalar curvaturesthe proof of Theorem 1.3 is complete. (cid:3)
References [1] F. Cuccu and A. Loi,
Global symplectic coordinates on complexdomains , J. Geom. and Phys. 56 (2006), 247-259.[2] J. E. D’Atri and Y. D. Zhao,
Geodesics and Jacobi fields inbounded homogeneous domains , Proc. Amer. Math. Soc. 89 no.. 1(1983), 55-61.[3] M. P. Do Carmo,
Riemannian Geometry , Birkh¨auser 1992.[4] M. Engliˇs,
Berezin Quantization and Reproducing Kernels onComplex Domains , Trans. Amer. Math. Soc. vol. 348 (1996), 411-479.[5] C. Fefferman,
The Bergman kernel and biholomorphic mappingsof pseudoconvex domains , Invent. Math. 26 (1974), 1–65.[6] G. Herbort,
On the geodesics of the Bergman metric , Math. Ann.264 no. 1 (1983), 39–51.[7] A. V. Isaev and S. G. Krantz,
Domains with non-compact auto-morphism group: a survey , Adv. Math. 146 no. 1 (1999), 1–38.
8] S. Kobayashi,
Geometry of bounded domains , Trans. Amer. Math.Soc. vol. 92 (1959), pp. 267-290.[9] A. Loi,
Regular quantizations of K¨ahler manifolds and constantscalar curvature metrics , J. Geom. and Phys. 53 (2005), 354-364.[10] A. Loi and F. Zuddas,
Symplectic maps of complex domains intocomplex space forms , preprint (2007).[11] A. Loi and F. Zuddas,
Extremal metrics on Hartogs domains ,arXiv:0705.2124.[12] G. Roos, A. Wang, W. Yin and L. Zhang,
The K¨ahler-Einsteinmetric for some Hartogs domains over symmetric domains , Sci.China Ser. A 49 no. 9 (2006), 1175-1210.Dipartimento di Matematica, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Milano, Italy,
E-mail: [email protected]
Dipartimento di Matematica e Informatica Universit`a di Cagliari,Via Ospedale 72, 09124 Cagliari, Italy,
E-mail: [email protected]
Dipartimento di Matematica e Informatica Universit`a di Cagliari,Via Ospedale 72, 09124 Cagliari, Italy,
E-mail: [email protected]@unica.it