A certain Kähler potential of the Poincaré metric and its characterization
aa r X i v : . [ m a t h . C V ] A ug A CERTAIN K ¨AHLER POTENTIAL OF THE POINCAR´EMETRIC AND ITS CHARACTERIZATION
YOUNG-JUN CHOI, KANG-HYURK LEE AND SUNGMIN YOO
Abstract.
We will show a rigidity of a K¨ahler potential of the Poincar´e metricwith a constant length differential. Introduction
From the fundamental result of Donnelly-Fefferman [4], the vanishing of the spaceof L harmonic ( p, q ) forms has been an important research theme in the theory ofcomplex domains. Since M. Gromov ([6], see also [2]) suggested the concept of theK¨ahler hyperbolicity and gave a connection to the vanishing theorem, there havebeen many studies on the K¨ahler hyperbolicity of the Bergman metric, which isa fundamental K¨ahler structure of bounded pseudoconvex domains. The K¨ahlerstructure ω is K¨ahler hyperbolic if there is a global 1-form η with dη = ω andsup k η k ω < ∞ .In [3], H. Donnelly showed the K¨ahler hyperbolicity of Bergman metric on someclass of weakly pseudoconvex domains. For bounded homogeneous domain D in C n and its Bergman metric ω D especially, he used a classical result of Gindikin [5]to show that sup k d log K D k ω D < ∞ . Here K D is the Bergman kernel function of D so log K D is a canonical potential of ω D .In their paper [7], S. Kai and T. Ohsawa gave another approach. They provedthat every bounded homogeneous domain has a K¨ahler potential of the Bergmanmetric whose differential has a constant length. Theorem 1.1 (Kai-Ohsawa [7]) . For a bounded homogeneous domain D in C n ,there exists a positive real valued function ϕ on D such that log ϕ is a K¨ahlerpotential of the Bergman metric ω D and k d log ϕ k ω D is constant. It can be obtained by the facts that each homogeneous domain is biholomorphicto a Siegel domain (see [10]) and a homogeneous Siegel domain is affine homoge-neous (see [8]).More precisely, let us consider a bounded homogeneous domain D in C n anda biholomorphism F : D → S for a Siegel domain S . For the Bergman kernelfunction K S of S which is a canonical potential of the Bergman metric ω S , it iseasy to show that d log K S has a constant length with respect to ω S from the affinehomogeneity of S (the group of affine holomorphic automorphisms acts transitivelyon S ). Since log K S is a K¨ahler potential of ω S , the transformation formula of the Mathematics Subject Classification.
Key words and phrases.
The Poincar´e metric, K¨ahler hyperbolicity.The research of first and second named authors was supported by the National ResearchFoundation of Korea (NRF) grant funded by the Korea government (No. 2018R1C1B3005963,No. NRF-2019R1F1A1060891). The last named author was supported by IBS-R003-D1.
Bergman kernel implies that the pullback F ∗ log K S = log K S ◦ F is also a K¨ahlerpotential of ω D . Using the fact that F : ( D, ω D ) → ( S, ω S ) is an isometry, we have k d ( F ∗ log K S ) k ω D = k d log K S k ω S ◦ F . As a function ϕ in Theorem 1.1, we canchoose the pullback K S ◦ F of the Bergman kernel function of the Siegel domain.At this junction, it is natural to ask: If there is a K¨ahler potential log ϕ with a constant k d log ϕ k ω D , isit always obtained by the pullback of the Bergman kernel functionof the Siegel domain? The aim of this paper is to discuss of this question in the 1-dimensional case.The only bounded homogeneous domain in C is the unit disc ∆ = { z ∈ C : | z | < } up to the biholomorphic equivalence and the 1-dimensional correspondence of theBergman metric, namely a holomorphically invariant hermitian structure, is onlythe Poincar´e metric. Hence the main theorem as follows gives a positive answer tothe question. Theorem 1.2.
Let ω ∆ be the Poincar´e metric of the unit disc ∆ . Suppose thatthere exists a positive real valued function ϕ : ∆ → R such that log ϕ is a K¨ahlerpotential of the Poincar´e metric and k d log ϕ k ω ∆ is constant on ∆ . Then ϕ is thepullback of the canonical potential on the half-plane H = { z ∈ C : Re z < } . Note that 1-dimensional Siegel domain is just the half-plane. We will introducethe Poincar´e metric and related notions in Section 2. As an application of the maintheorem, we can characterize the half-plane by the canonical potential.
Corollary 1.3.
Let D be a simply connected, proper domain in C with a Poincar´emetric ω D = iλdz ∧ d ¯ z . If k d log λ k ω D is constant on D , then D is affine equivalentto the half-plane H = { z ∈ C : Re z < } . In Section 2, we will introduce notions and concrete version of the main theorem.Then we will study the existence of a nowhere vanishing complete holomorphicvector field which is tangent to a potential whose differential is of constant length(Section 3). Using relations between complete holomorphic vector fields and modelpotentials in Section 4, we will prove theorems.2.
Background materials
Let X be a Riemann surface. The Poincar´e metric of X is a complete hermitianmetric with a constant Gaussian curvature, −
4. The Poincar´e metric exists on X if and only if X is a quotient of the unit disc. If X is covered by ∆, thePoincar´e metric can be induced by the covering map π : ∆ → X and it is uniquelydetermined. Throughout of this paper, the K¨ahler form of the Poincar´e metric of X , denoted by ω X , stands for the metric also. When ω X = iλdz ∧ d ¯ z in the localholomorphic coordinate function z , the curvature can be written by κ = − λ ∂ ∂z∂ ¯ z log λ . So the curvature condition κ ≡ − ∂ ∂z∂ ¯ z log λ = 2 λ , equivalently dd c log λ = 2 ω X , CERTAIN K¨AHLER POTENTIAL OF THE POINCAR´E METRIC 3 where d c = i ( ∂ − ∂ ). That means the function log λ is a local K¨ahler potentialof ω X . Any other local potential of ω X is always of the form log λ + log | f | where f is a local holomorphic function on the domain of z . We call log λ the canonical potential with respect to the coordinate function z . For a domain D in C , the canonical potential of D means the canonical potential with respect to thestandard coordinate function of C .Let us consider the Poincar´e metric ω ∆ of the unit disc ∆: ω ∆ = i (cid:16) − | z | (cid:17) dz ∧ d ¯ z = iλ ∆ dz ∧ d ¯ z . The canonical potential λ ∆ satisfies k d log λ ∆ k ω ∆ = (cid:13)(cid:13)(cid:13)(cid:13) ∂ log λ ∆ ∂z dz + ∂ log λ ∆ ∂ ¯ z d ¯ z (cid:13)(cid:13)(cid:13)(cid:13) ω ∆ = ∂ log λ ∆ ∂z ∂ log λ ∆ ∂ ¯ z λ ∆ = 4 | z | , so does not have a constant length. By the same way of Kai-Ohsawa [7], we canget a model for ϕ in Theorem 1.1 for the unit disc,(2.1) ϕ θ ( z ) = (cid:12)(cid:12) e iθ z (cid:12)(cid:12) (cid:16) − | z | (cid:17) for θ ∈ R as a pullback of the canonical potential λ H = 1 / | Re w | on the left-half plane H = { w : Re w < } by the Cayley transforms (see (4.3) for instance). Theterm θ depends on the choice of the Cayley transform. Since log ϕ θ = log λ ∆ +log (cid:12)(cid:12) e iθ z (cid:12)(cid:12) , the function log ϕ θ is a K¨ahler potential. Moreover k d log ϕ θ k ω ∆ ≡ . At this moment, we introduce a significant result of Kai-Ohsawa.
Theorem 2.1 (Kai-Ohsawa [7]) . For a bounded homogeneous domain D in C n ,suppose that there is a K¨ahler potential log ψ of the Bergman metric ω D with aconstant k d log ψ k ω D , then k d log ψ k ω D = k d log ϕ k ω D where ϕ is as in Theorem 1.1. Suppose that a positively real valued ϕ on ∆ satisfies that dd c log ϕ = 2 ω ∆ and k d log ϕ k ω ∆ ≡ c for some constant c . Theorem 2.1 implies that c must be 4.Therefore, we can rewrite Theorem 1.2 by Theorem 2.2.
If there exists a function ϕ : ∆ → R satisfying (2.2) dd c log ϕ = 2 ω ∆ and k d log ϕ k ω ∆ ≡ . Then ϕ = rϕ θ as in (2.1) for some r > and θ ∈ R . Corollary 1.3 can be also written by
Corollary 2.3.
Let D be a simply connected, proper domain in C with a Poincar´emetric ω D = iλdz ∧ d ¯ z . If k d log λ k ω D ≡ , then D is affine equivalent to thehalf-plane H = { z ∈ C : Re z < } . YOUNG-JUN CHOI, KANG-HYURK LEE AND SUNGMIN YOO Existence of nowhere vanishing complete holomorphic vectorfield
In this section, we will study an existence of a complete holomorphic tangentvector field on a Riemann surface X which admits a K¨ahler potential of the Poincar´emetric with a constant length differential.By a holomorphic tangent vector field of a Riemann surface X , we means a holo-morphic section W to the holomorphic tangent bundle T , X . If the correspondingreal tangent vector field Re W = W + W is complete, we also say W is complete.Thus the complete holomorphic tangent vector field generates a 1-parameter familyof holomorphic transformations.In this section, we will show that Theorem 3.1.
Let X be a Riemann surface with the Poincar´e metric ω X . If thereis a function ϕ : X → R with (3.1) dd c log ϕ = 2 ω X and k d log ϕ k ω X ≡ then there is a nowhere vanishing complete holomorphic vector field W such that (Re W ) ϕ ≡ .Proof. Take a local holomorphic coordinate function z and let ω X = iλdz ∧ d ¯ z . Theequation (3.1) can be written by(log ϕ ) z ¯ z = 2 λ and (log ϕ ) z (log ϕ ) ¯ z = 4 λ Here, (log ϕ ) z = ∂∂z log ϕ , (log ϕ ) ¯ z = ∂∂ ¯ z log ϕ and (log ϕ ) z ¯ z = ∂ ∂z∂ ¯ z log ϕ . Thisimplies that (cid:16) ϕ − / (cid:17) z = ∂∂z ϕ − / = − ϕ − / (log ϕ ) z ; (cid:16) ϕ − / (cid:17) z ¯ z = ∂ ∂z∂ ¯ z ϕ − / = − ϕ − / (log ϕ ) z ¯ z + 14 ϕ − / (log ϕ ) z (log ϕ ) ¯ z = − ϕ − / (cid:18) (log ϕ ) z ¯ z −
12 (log ϕ ) z (log ϕ ) ¯ z (cid:19) = 0 . Thus we have that the function ϕ − / is harmonic so (cid:0) ϕ − / (cid:1) z is holomorphic.Let us consider a local holomorphic vector field, W = i (cid:0) ϕ − / (cid:1) z ∂∂z = − iϕ / ϕ z ∂∂z = − iϕ / (log ϕ ) z ∂∂z . In any other local holomorphic coordinate function w , we have W = i (cid:0) ϕ − / (cid:1) z ∂∂z = i (cid:0) ϕ − / (cid:1) w ∂w∂z ∂w∂z ∂∂w = i (cid:0) ϕ − / (cid:1) w ∂∂w . so W is globally defined on X . Now we will show that W satisfies conditions in thetheorem.Since (cid:13)(cid:13)(cid:13) ϕ − / W (cid:13)(cid:13)(cid:13) ω X = (cid:13)(cid:13)(cid:13)(cid:13) − i (log ϕ ) z ∂∂z (cid:13)(cid:13)(cid:13)(cid:13) ω X = 4 λ (log ϕ ) z (log ϕ ) ¯ z = 1 , CERTAIN K¨AHLER POTENTIAL OF THE POINCAR´E METRIC 5 the vector field ϕ − / W has a unit length with respect to the complete metric ω X ,so the corresponding real vector field Re ϕ − / W = ϕ − / ( W + W ) is complete.Moreover (Re W ) ϕ = − iϕ / ϕ z ϕ z + 2 iϕ / ϕ ¯ z ϕ ¯ z = 0 . Hence it remains to show the completeness of W . Take any integral curve γ : R → X of ϕ − / Re W . It satisfies (cid:16) ϕ − / (Re W ) (cid:17) ◦ γ = ˙ γ equivalently (Re W ) ◦ γ = (cid:16) ϕ / ◦ γ (cid:17) ˙ γ The condition (Re W ) ϕ ≡
0, equivalently ϕ − / (Re W ) ϕ ≡
0, implies that thecurve γ is on a level set of ϕ so ϕ / ◦ γ ≡ C for some constant C . The curve σ : R → X defined by σ ( t ) = γ ( Ct ) satisfies(Re W ) ◦ σ ( t ) = (Re W )( γ ( Ct )) = C ˙ γ ( Ct ) = ˙ σ ( t )This means that σ : R → X is the integral curve of Re W ; therefore Re W iscomplete. This completes the proof. (cid:3) Complete holomorphic vector fields on the unit disc
In this section, we introduce parabolic and hyperbolic vector fields on the unitdisc and discuss their relation to the model potential,(4.1) ϕ = | z | (cid:16) − | z | (cid:17) where it is ϕ θ in (2.1) with θ = 0.4.1. Nowhere vanishing complete holomorphic vector fields from the left-half plane.
On the left-half plane H = { w ∈ C : Re w < } , there are two kindsof affine transformations: D s ( w ) = e s w and T s ( w ) = w + 2 is for s ∈ R . Their infinitesimal generators are D = 2 w ∂∂w and T = 2 i ∂∂w which are nowhere vanishing complete holomorphic vector fields of H . Note that(4.2) ( T s ) ∗ D = 2( w − is ) ∂∂w = D − s T and ( T s ) ∗ T = 2 i ∂∂w = T for any s .For the Cayley transform F : H → ∆ defined by(4.3) F : H −→ ∆ w z = 1 + w − w , we can take two nowhere vanishing complete holomorphic vector fields of ∆: H = F ∗ ( D ) = ( z − ∂∂z YOUNG-JUN CHOI, KANG-HYURK LEE AND SUNGMIN YOO and P = F ∗ ( T ) = i ( z + 1) ∂∂z . When we define H s = F ◦ D s ◦ F − and P s = F ◦ T s ◦ F − , vector fields H and P are infinitesimal generators of H s and P s , respectively. Moreover Equation (4.2)can be written by(4.4) ( P s ) ∗ H = H − s P and ( P s ) ∗ P = P . There is another complete holomorphic vector field R = iz∂/∂z generating therotational symmetry(4.5) R s ( z ) = e is z . The holomorphic automorphism group of ∆ is a real 3-dimension connected Liegroup (cf. see [1, 9]), we can conclude that any complete holomorphic vector fieldcan be a real linear combination of H , P and R . Since H ( −
1) = P ( −
1) = 0 and R ( −
1) = − i∂/∂z , we have Lemma 4.1. If W is a complete holomorphic vector field of ∆ satisfying W ( −
1) =0 , then there exist a, b ∈ R with W = a H + b P . Hyperbolic vector fields.
In this subsection, we will show that the hyper-bolic vector field H can not be tangent to a K¨ahler potential with a constant lengthdifferential.By the simple computation, H (log ϕ ) = ( z −
1) 2(1 + ¯ z )(1 + z )(1 − | z | ) = 2 | z | + z − ¯ z − − | z | ) , we get (Re H ) log ϕ ≡ − . That means Re H is nowhere tangent to ϕ . Moreover Lemma 4.2.
Let ϕ : ∆ → R with dd c log ϕ = 2 ω ∆ and k d log ϕ k ω ∆ ≡ . If (Re H ) log ϕ ≡ c for some c , then c = ± .Proof. Since dd c log ϕ = 2 ω ∆ also, the function log ϕ − log ϕ is harmonic; hencewe may let log ϕ = log ϕ + f + ¯ f for some holomorphic function f : ∆ → C . Thenthe condition (Re H ) log ϕ ≡ c can be written by(4.6) (Re H ) log ϕ = − z − f ′ + (¯ z −
1) ¯ f ′ ≡ c . This implies that ( z − f ′ is constant. Thus we can let(4.7) f ′ = Cz − C ∈ C . Since ∂∂z log ϕ = f ′ + ∂∂z log ϕ = f ′ + 2(1 + ¯ z )(1 + z )(1 − | z | ) , we have k d log ϕ k ω ∆ = (cid:18) ∂∂z log ϕ (cid:19) (cid:18) ∂∂ ¯ z log ϕ (cid:19) λ ∆ = | f ′ | (1 − | z | ) + 2(1 + ¯ z )(1 − | z | )(1 + z ) ¯ f ′ + 2(1 + z )(1 − | z | )(1 + ¯ z ) f ′ + k d log ϕ k ω ∆ . CERTAIN K¨AHLER POTENTIAL OF THE POINCAR´E METRIC 7
From the condition k d log ϕ k ω ∆ ≡ ≡ k d log ϕ k ω ∆ , it follows | f ′ | (1 − | z | ) = − z )(1 − | z | )(1 + z ) ¯ f ′ − z )(1 − | z | )(1 + ¯ z ) f ′ , equivalently(4.8) 12 | f ′ | (1 − | z | ) = − (1 + ¯ z )(1 + z ) ¯ f ′ − (1 + z )(1 + ¯ z ) f ′ . Applying (4.7) to the right side above, − (1 + ¯ z )(1 + z ) ¯ f ′ − (1 + z )(1 + ¯ z ) f ′ = (1 + ¯ z )(1 + z ) ¯ C − ¯ z + (1 + z )(1 + ¯ z ) C − z = (1 + ¯ z − z − | z | ) ¯ C + (1 − ¯ z + z − | z | ) C | − z | . Let C = a + bi for a, b ∈ R , then(1 + ¯ z − z − | z | ) ¯ C + (1 − ¯ z + z − | z | ) C = 2 a (1 − | z | ) + 2 bi ( z − ¯ z ) . Now Equation (4.8) can be written by12 | C | | z − | (1 − | z | ) = 2 a (1 − | z | ) + 2 bi ( z − ¯ z ) | − z | , so we have ( | C | − a )(1 − | z | ) = 4 bi ( z − ¯ z )on ∆. Take ∂ ¯ ∂ to above, we have | C | − a = 0 . Simultaneously b = 0 so C = a . Now we have a = 4 a . Such a is 0 or 4. If f ′ = 4 / ( z − c = 4 from (4.6). If f ′ = 0, then c = − (cid:3) Parabolic vector fields.
Since P (log ϕ ) = i ( z + 1) z )(1 + z )(1 − | z | ) = 2 i | z | − | z | , we have (Re P ) log ϕ ≡ . That means that the parabolic vector field P is tangent to ϕ . The vector field P isindeed the nowhere vanishing complete holomorphic vector field as constructed inTheorem 3.1 corresponding to ϕ . The main result of this section is the following. Lemma 4.3.
Let ϕ : ∆ → R with dd c log ϕ = 2 ω ∆ and k d log ϕ k ω ∆ ≡ . If (Re P ) log ϕ ≡ c for some c , then c = 0 and ϕ = rϕ for some r > .Proof. By the same way in the proof of Lemma 4.2, we let log ϕ = log ϕ + f + ¯ f for some holomorphic f : ∆ → C . Since(4.9) (Re P ) log ϕ = i ( z + 1) f ′ − i (¯ z + 1) ¯ f ′ ≡ c it follows that ( z + 1) f ′ is constant. Thus we have(4.10) f ′ = C ( z + 1) for some C ∈ C . Since (4.8) also holds, we can apply (4.10) to the right side of(4.8) to get − (1 + ¯ z )(1 + z ) ¯ f ′ − (1 + z )(1 + ¯ z ) f ′ = − (1 + ¯ z )(1 + z ) ¯ C (¯ z + 1) − (1 + z )(1 + ¯ z ) C ( z + 1) = − ¯ C | z | + − C | z | = − ¯ C − C | z | Now Equation (4.8) is can be written by | C | | z + 1 | (1 − | z | ) = 2 − ¯ C − C | z | equivalently | C | (1 − | z | ) = − (cid:0) C + 2 C (cid:1) | z | . Evaluating z = 0, we have | C | = − C − C . And taking ∂ ¯ ∂ to above, we have − | C | = − C − C . It follows that C = 0 so f is constant. Moreover Equation(4.9) implies that c = 0. (cid:3) Proof of the main theorem
Now we prove Theorem 2.2 and Corollary 2.3
Proof of Theorem 2.2.
Let ϕ : ∆ → R be a function with dd c log ϕ = 2 ω ∆ and k d log ϕ k ω ∆ ≡ . By Theorem 3.1, we can take a nowhere vanishing complete holomorphic vector field W with (Re W ) ϕ ≡
0. Since every automorphism of ∆ has at least one fixed pointon ∆ and W is nowhere vanishing on ∆, any nontrivial automorphism generatedby Re W has no fixed point in ∆ and should have a common fixed point p at theboundary ∂ ∆. This means p is a vanishing point of W . Consider a rotationalsymmetry R θ in (4.5) satisfying R θ ( −
1) = p . We will show that ϕ ◦ R θ = rϕ where ϕ is as in (4.1) and r >
0. This implies that ϕ = rϕ − θ .Now we can simply denote by ϕ = ϕ ◦ R θ and W = ( R − θ ) ∗ W . Since − W , Lemma 4.1 implies W = a H + b P for some real numbers a , b .Suppose that a = 0. Equation (4.4) implies that( P s ) ∗ W = ( P s ) ∗ ( a H + b P ) = a H − as P + b P = a H + ( b − as ) P . Take s = b/ a , then f W = ( P s ) ∗ W = a H . Let ˜ ϕ = ϕ ◦ P − s for this s . Then˜ ϕ satisfies conditions in Theorem 2.2 and (Re f W ) ˜ ϕ ≡
0. But Lemma 4.2 saidthat (Re f W ) ˜ ϕ = a (Re H ) ˜ ϕ ≡ ± a ˜ ϕ . It contradicts to (Re W ) ϕ ≡ f W ) ˜ ϕ ≡
0. Thus a = 0.Now W = b P . Since W is nowhere vanishing already, b = 0. The condition(Re W ) ϕ ≡ P ) ϕ ≡
0. Lemma 4.3 says that ϕ = rϕ for some positive r . This completes the proof. (cid:3) Proof of Corollary 2.3.
Let D be a simply connected proper domain in C and let ω D = iλ D dz ∧ d ¯ z be its Poincar´e metric with k d log λ D k ω D ≡
4. By Theorem 3.1,
CERTAIN K¨AHLER POTENTIAL OF THE POINCAR´E METRIC 9 there is a nowhere vanishing complete holomorphic vector field W with (Re W ) λ D ≡
0. Take a biholomorphism G : ∆ → D and let ϕ = λ D ◦ G and Z = ( G − ) ∗ W . Note that (Re Z ) ϕ ≡ R θ of ∆which is also affine, we may assume that Z ( −
1) = 0 and we will prove that G is aCayley transform.Since G : (∆ , ω ∆ ) → ( D, ω D ) is an isometry, we have G ∗ ω D = ω ∆ , equivalently ϕ = λ ∆ | G ′ | . Moreover d log ϕ = d ( G ∗ log λ D ) implies that k d log ϕ k ω D = k d ( G ∗ log λ D ) k ω D ≡ λ ∆ | G ′ | = ϕ = rϕ = rλ ∆ | z | for some positive r . This means that G ′ = e iθ ′ / √ r (1 + z ) for some θ ′ ∈ R so that G = e iθ ′ √ r z − z + 1 + C Since the function z ( z − / ( z +1) is the inverse mapping of the Cayley transform F : H → ∆ in (4.3), we have G ◦ F : H → Dz e iθ ′ √ r z + C .
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Department of Mathematics, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, Republic of Korea
E-mail address : [email protected] Department of Mathematics and Research Institute of Natural Science, GyeongsangNational University, Jinju, Gyeongnam, 52828, Korea
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