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A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature
Differential Geometry and its Applications 43 (2015) 1–20
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature Hongchuan Xia, Chunping Zhong ∗ School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
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Article history: Received 21 March 2015 Received in revised form 20 July 2015 Available online xxxx Communicated by Z. Shen
In this paper, we give a characterization of strongly pseudoconvex complex Finsler metric F which is unitary invariant. A necessary and sufficient condition for F to be a weakly complex Berwald metric and a necessary and sufficient condition for F to be a weakly Kähler Finsler metric are given, respectively. We also give a classification of unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures. © 2015 Elsevier B.V. All rights reserved.
MSC: 53C60 53C40 Keywords: Unitary invariant Weakly complex Berwald metric Weakly Kähler Finsler metric Constant holomorphic curvature
1. Introduction and main results Let ·, · be the canonical complex Euclidean inner product in the complex n-dimensional linear space C , · be the norm induced by ·, ·, that is, n
z, v =
n
zi vi ,
z =
z, z,
∀z = (z 1 , · · · , z n )T , v = (v 1 , · · · , v n )T ∈ Cn ,
i=1
where and in the following bars and overlines denote conjugations of complex numbers, and (· · · )T denotes the transpose of vectors in Cn . Denote U (n) the set of n-by-n unitary matrices over the complex number field C. A domain D ⊂ Cn is called unitary invariant if Az ∈ D whenever z ∈ D and A ∈ U (n). * Corresponding author. E-mail addresses: [email protected] (H. Xia), [email protected] (C. Zhong). http://dx.doi.org/10.1016/j.difgeo.2015.08.001 0926-2245/© 2015 Elsevier B.V. All rights reserved.
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
Definition 1.1. Let F be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . F is called unitary invariant if F (Az, Av) = F (z, v), ∀A ∈ U (n), z ∈ D, v ∈ Tz1,0 D.
(1.1)
It is well known that the canonical complex Euclidean inner product ·, · in Cn is unitary invariant. For strongly pseudoconvex complex Finsler metrics in the sense of Abate and Patrizio [2], we have the following characterization. Theorem 1.2. Let F be a strongly pseudoconvex complex Finsler metric on a unitary invariant domain D ⊂ Cn . Then F is unitary invariant if and only if there exists a smooth function φ(t, s): [0, +∞) × [0, +∞) → (0, +∞) such that F (z, v) =
rφ(t, s),
r := v2 ,
t := z2 ,
s :=
|z, v|2 r
(1.2)
for every z ∈ D and every nonzero holomorphic tangent vector v ∈ Tz1,0 D. In [3], a strongly pseudoconvex complex Finsler metric is called a complex Berwald metric if the horiγ zontal Chern–Finsler connection coefficients Γν;μ (z, v) associated with F depend only on the base manifold 1 n coordinates z = (z , · · · , z ). In [7], a strongly pseudoconvex complex Finsler metric is called a weakly complex Berwald metric if the complex Berwald connection coefficients Gγνμ (z, v) associated with F depend only on the base manifold coordinates z = (z 1 , · · · , z n ). It was also shown in [7] that the complex Wrona metric is a weakly complex Berwald metric, but not a complex Berwald metric. In [8], unitary invariant strongly pseudoconvex complex Finsler metrics of the form (1.2) were systematically studied. It was proved that Theorem 1.3. (See [8].) A strongly pseudoconvex complex Finsler metric F = D ⊂ Cn is a weakly complex Berwald metric if and only if
rφ(t, s) defined on a domain
φ(φst + φss ) − φs (φt + φs ) = g(t) (φ − sφs ) φ + (t − s)φs + s(t − s)φφss
(1.3)
for a real-valued smooth function g(t). In this paper we are able to simplify the second-order nonlinear PDE (1.3) to a first-order linear PDE and solve it completely. We obtain Theorem 1.4. Let F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly complex Berwald metric if and only if φt + 1 − tg(t)s + g(t)s2 φs = f (t) + g(t)s φ
(1.4)
holds for some real-valued smooth functions f (t) and g(t). The solution of (1.4) satisfying φ(0, s) = ϕ(s)
(1.5)
for a given smooth positive function ϕ(s) of s ∈ R is necessary of the form
t φ(t, s) = exp 0
s−t h(t, s), f (τ )dτ ϕ h(t, s)
(1.6)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
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where h(t, s) is
t h(t, s) = exp
τ g(τ )dτ
σ
t + (s − t)
0
g(σ) exp 0
τ g(τ )dτ dσ.
(1.7)
0
In Section 3, we also obtain a sufficient condition for the solution (1.6) of (1.4) to be a weakly complex Berwald metric on a unitary invariant domain D ⊂ Cn . A fundamental result of Lempert [4] states that the Kobayashi and Carathéodory metrics on strongly convex bounded domains with smooth boundaries in Cn are weakly Kähler Finsler metrics, whether they are Kähler Finsler metrics is still open [1]. These metrics, however, don’t have explicit formulae. So far there is no explicit weakly Kähler Finsler metric found in literature. It was prove in [8] that among unitary invariant strongly pseudoconvex complex Finsler metrics, there is neither non-Hermitian complex Berwald metric nor Kähler Finsler metric. For weakly Kähler Finsler metrics, a necessary and sufficient condition for F = rφ(t, s) to be a weakly Kähelr Finsler metric was obtained, which is characterized by a second-order nonlinear PDE (cf. Theorem 4.3 in [8]). Among weakly complex Berwald metrics, however, the author in [8] also obtained a negative result under the assumption that φ satisfying φ(φst + φss ) − φs (φs + φt ) = 0.
(1.8)
In this paper, we are able to get rid of the assumption (1.8) and prove that Theorem 1.5. Let F = rφ(t, s) be a weakly complex Berwald metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric if and only if it satisfies the following system of PDEs
φss = 0, φs − φt + sφts = 0.
(1.9)
The solution of (1.9) is given by φ(t, s) = a0 (t) + a0 (t)s,
(1.10)
where a0 (t) is a positive smooth function satisfying a0 (t) + ta 0 (t) > 0. Remark 1.6. Theorem 1.5 shows that among unitary invariant weakly complex Berwald metrics there is also no non-Hermitian weakly Kähler Finsler metric. Note that Theorem 1.4 implies that there are lots of non-Hermitian weakly complex Berwald metrics which are unitary invariant. One may wonder among them whether there are metrics of constant holomorphic curvatures. In this paper, we obtain the following classification theorem. Theorem 1.7. Let F = rφ(t, s) be a weakly complex Berwald metric defined on a unitary invariant domain D ⊂ Cn with constant holomorphic curvature KF along a nonzero vector v ∈ Tz1,0 D . Then (i) KF = constant = 0 if and only if φ(t, s) is given by φ(t, s) = ϕ(s − t), where ϕ(w) is a positive smooth function satisfying Proposition 2.2. (ii) KF = constant = 0 if and only if φ is given by
(1.11)
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
φ(t, s) =
B 2 A(s − t) + B , B − At
(1.12)
where A, B are constants with A = 0, B > 0, B − At > 0. In this case KF = −4AB −2 .
(1.13)
2. Characterization of unitary invariant strongly pseudoconvex complex Finsler metric In this section we shall give a characterization of unitary invariant strongly pseduconvex complex Finsler metrics. We have Theorem 2.1. Let F be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is unitary invariant if and only if there exists a smooth function φ(t, s) : [0, +∞) × [0, +∞) → (0, +∞) such that F (z, v) =
rφ(t, s),
r := v2 ,
t := z2 ,
s :=
|z, v|2 r
(2.1)
for every z ∈ D and every nonzero holomorphic tangent vector v ∈ Tz1,0 D. Proof. Since the complex Euclidean inner product ·, · in Cn is unitary invariant, the sufficiency is clear. Next we prove the necessity. Suppose that F is a unitary invariant strongly pseudoconvex complex Finsler metric, i.e., F satisfies (1.1) and is smooth outside of the zero section of T 1,0 D. Denote the unit orthogonal basis of Cn by e1 , · · · , en , where (j th)
ej = (0, · · · , 0, 1 , 0, · · · , 0),
j = 1, · · · , n.
Let z be an arbitrary fixed point in D and v ∈ Tz1,0 D with v = 0. Case 1: If z, v = 0 and v = λz for any λ ∈ C. Then ε1 =
z, vv , |z, v|v
ε2 =
z − z, ε1 ε1 z − z, ε1 ε1
(2.2)
are two unit orthogonal vectors in Cn since v = λz implies that z − z, ε1 ε1 2 = z2 − |z, ε1 |2 = z2 −
|z, v|2
= 0. v2
(2.3)
Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
Aε2 = e2 .
It follows from (2.2)–(2.4) that Az = A(z, ε1 ε1 + z − z, ε1 ε1 ε2 ) |z, v|2 Aε2 = z, ε1 Aε1 + z2 − v2 |z, v| |z, v|2 e1 + z2 − = e2 . v v2
(2.4)
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Using (1.1) and the homogeneity of F , i.e., F (z, λv) = |λ|F (z, v) for any λ ∈ C, we get
z, vv = vF Az, A |z, v|v
|z, v|2 |z, v| 2 e1 + z − e 2 , e1 = vF (Az, Aε1 ) = vF v v2 √ √ √ = rF ( se1 + t − se2 , e1 ).
z, vv F (z, v) = vF z, |z, v|v
Thus one need only taking √ √ φ(t, s) = F 2 ( se1 + t − se2 , e1 ).
(2.5)
Case 2: If z, v = 0 and z = 0, then ε1 =
v , v
ε2 =
z z
(2.6)
are two unit orthogonal vectors in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
Aε2 = e2 .
(2.7)
Using (1.1) and the homogeneity of F , we get √ √ v F (z, v) = vF z, = vF (Az, Aε1 ) = vF (ze2 , e1 ) = rF ( te2 , e1 ). v Thus one need only taking √ φ(t, s) = F 2 ( te2 , e1 ).
(2.8)
Case 3: If z, v = 0 and z = 0, then ε1 =
v v
(2.9)
is a unit vector in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
(2.10)
so that √ v = vF (A0, Aε1 ) = vF (0, e1 ) = rF (0, e1 ). F (z, v) = vF 0, v Thus one need only taking φ(t, s) = F 2 (0, e1 ) = constant > 0. Case 4: If z = λv for some λ ∈ C. If λ = 0, then it reduces to case 3. If λ = 0, then ε1 =
|λ|v λv
(2.11)
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is a unit vector in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
(2.12)
from which we get Av =
λv e1 . |λ|
(2.13)
So that Az = λAv = |λ|ve1 = ze1 .
(2.14)
Using (1.1) and the homogeneity of F , we get F (z, v) = vF (z, ε1 ) = vF (Az, Aε1 ) = vF (ze1 , e1 ) =
√
√ rF ( te1 , e1 ).
Thus one need only taking √ φ(t, s) = F 2 ( te1 , e1 ). Since F is smooth outside of the zero section of T 1,0 D, it follows from the discussion of case 1 to 4 that φ(t, s) is a positive smooth function of t and s. This completes the proof. 2 In [8], the author gave the following characterization for a complex Finsler metric F to be a strongly pseudoconvex complex Finsler metric in the sense of Abate and Patrizio [2]. Proposition 2.2. (See [8].) A complex Finsler metric F = pseudoconvex if and only if φ satisfies φ − sφs > 0,
rφ(t, s) defined on a domain D ⊂ Cn is strongly
(φ − sφs ) φ + (t − s)φs + s(t − s)φφss > 0
(2.15)
when n ≥ 3 or (φ − sφs ) φ + (t − s)φs + s(t − s)φφss > 0 when n = 2. Remark 2.3. In fact, the condition φ(0, 0) > 0 and suitable smoothness of φ(t, s) is enough to ensure that F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric on Bn (ε) ⊂ Cn for some ε > 0. 3. Characterization of unitary invariant weakly complex Berwald metric In this section, we shall give a characterization of unitary invariant weakly complex Berwald metric, which is characterized by a first-order linear PDE with its solution explicitly given. Theorem 3.1. Let F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly complex Berwald metric if and only if φt + 1 − tg(t)s + g(t)s2 φs = f (t) + g(t)s φ
(3.1)
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holds for some real-valued smooth functions f (t) and g(t). The solution of (3.1) satisfying φ(0, s) = ϕ(s) for a given smooth positive function ϕ(s) of s ∈ R is necessary of the form
t φ(t, s) = exp
s−t h(t, s), f (τ )dτ ϕ h(t, s)
(3.2)
0
where h(t, s) is
t h(t, s) = exp
τ g(τ )dτ
0
σ
t + (s − t)
g(σ) exp 0
τ g(τ )dτ dσ.
(3.3)
0
Proof. We shall simplify the second-order nonlinear PDE (1.3) to a first-order linear PDE. Dividing by φ2 on both sides of (1.3), we have φ2 + (t − 2s)φφs − s(t − s)φ2s + s(t − s)φφss φ(φt + φs )s − φs (φt + φs ) = g(t) φ2 φ2 φφss − φ2s φs = g(t) 1 + (t − 2s) + s(t − s) φ φ2 φs , = g(t) 1 + (st − s2 ) φ s
(3.4)
where and in the following subscript s denotes the derivative with respect to this variable. Since the left hand side of (3.4) can be rewritten as φ(φt + φs )s − φs (φt + φs ) = φ2
φt + φs φ
. s
Thus integrating (3.4) with respect to s, we get φt + φs φs = g(t)s + g(t)(st − s2 ) + f (t) φ φ
(3.5)
for some real-valued smooth function f (t). Rearranging (3.5), we get (3.1), which is a first-order linear PDE. Next we shall solve the equation (3.1) with suitable initial-value. We consider the following Cauchy problem
φt + [1 − s(t − s)g(t)]φs = [f (t) + g(t)s]φ, φ(X(a), Y (a)) = Z(a), a ∈ I,
(3.6)
where I is an closed interval such that 0 is an interior point in I, and Γ : t = X(a), s = Y (a), b = φ(X(a), Y (a)) =: Z(a) is a given noncharacteristic curve. Note that since s ≤ t and t = 0 implies that s = 0. It is natural to choose Γ passing through the point (X(0), Y (0), Z(0)) = (0, 0, 1) on the surface Σ : b = φ(t, s). In the following we choose (X(a), Y (a), Z(a)) = (0, a, ϕ(a)) such that ϕ(a) is a positive function satisfying ϕ(0) = 1. First, let solve the following initial problem:
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
⎧ dt ⎪ t(0, a) = 0 ⎪ ⎨ dl = 1, ds dl = 1 − s(t − s)g(t), s(0, a) = a ⎪ ⎪ ⎩ db = [f (t) + g(t)s]b, b(0, a) = ϕ(a). dl
(3.7)
The above initial-value problem can be solved by using the method of characteristics. Denote the solution to the above equation by t = t(l, a), s = s(l, a), b = b(l, a). Note that by (3.7), we have ∂(t, s) ∂t ∂l (0, a) = ∂s ∂(l, a) ∂l (0, a)
∂t ∂a (0, a) ∂s ∂a (0, a)
1 = 1 + a2 g(0)
0
= 0 1
at least at a = 0. Thus by implicit function theorem we can obtain l = l(t, s), a = a(t, s) so that φ(t, s) = b(l(t, s), a(t, s)) is the unique solution in a neighborhood of the initial curve Γ :
t = 0, s = a, b = ϕ(a).
By (3.7), it follows that t = l is a solution to dt dl = 0 satisfies the initial condition t(0, a) = 0. Substituting 2 t = l into ds = 1 − tg(t)s + g(t)s and then setting u = s − l, we get dl du − lg(l)u = g(l)u2 , dl
(3.8)
which is a Bernoulli equation. 2 If a = 0, then it is clear that s = l is a solution to the equation ds dl = 1 − tg(t)s + g(t)s satisfies the db initial condition s(0, a) = a. In this case, substituting t = l, s = l into dl = [f (t) + g(t)s]b and integrating from 0 to t, we get
t φ(t, s) = φ(t, t) = exp
[f (τ ) + τ g(τ )]dτ .
(3.9)
0
If a = 0, then by (3.7) and (3.8), it follows that s = l. In this case it is necessary that a < 0 since 1 a = s(0, a) ≤ t(0, a) = 0. Setting y = u1 = s−l , we are able to obtain a first-order linear ODE with respect to y as follows: dy + lg(l)y = −g(l). dl
(3.10)
Solving equation (3.10) with the initial-value y|l=0 = a1 , we get
l
y = exp −
τ g(τ )dτ
1 a
−
0
l
σ g(σ) exp
0
τ g(τ )dτ dσ ,
0
from which we obtain exp s=l+ where we denote
l 0
1 a
τ g(τ )dτ
− c(l)
,
(3.11)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
l c(l) =
σ
0
τ g(τ )dτ
g(σ) exp
9
dσ.
0
Substituting t = l and (3.11) into the third equation in (3.7), and solving it with the initial-value b(0, a) = φ(t(0, a), s(0, a)) = φ(0, a) = ϕ(a), we get
l b(l, a) = ϕ(a) exp
exp
f (τ ) + g(τ ) τ +
τ 0
σg(σ)dσ
dτ − c(τ )
l τ g(τ ) exp 0 σg(σ)dσ f (τ ) + τ g(τ ) + = ϕ(a) exp dτ 1 a − c(τ ) 0
l l d 1 − c(τ ) a = ϕ(a) exp f (τ ) + τ g(τ ) dτ − 1 a − c(τ ) 1 a
0
0
0
l = ϕ(a) exp
f (τ ) + τ g(τ ) dτ
0
1 . |1 − ac(l)|
(3.12)
By (3.11), we get s−l s−l 1 = , = l l 1 − ac(l) a exp 0 τ g(τ )dτ a exp 0 τ g(τ )dτ
(3.13)
or equivalently, a= exp
l 0
s−l . τ g(τ )dτ + (s − l)c(l)
(3.14)
Since l = t and s < t, it follows by (3.12) and (3.14) that
t φ(t, s) = exp
s−t f (τ )dτ ϕ h(t, s), h(t, s)
(3.15)
0
where we denote t h(t, s) =: exp
τ g(τ )dτ + (s − t)c(t).
(3.16)
0
By (3.14) it is necessary that h(t, s) > 0 since a < 0 and s < t. We also note that the solution (3.9) which is corresponding the case a = 0 can be obtained by allowing s = t in (3.15). This completes the proof. 2 In the following, let’s denote φ(t, s) = L(t)ϕ(w)h(t, s), where
(3.17)
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t L(t) = exp
f (τ )dτ ,
w=
s−t , h(t, s)
(3.18)
0
h(t, s) = M (t) + (s − t)N (t),
t M (t) = exp
(3.19)
τ g(τ )dτ ,
(3.20)
0
σ
t g(σ) exp
N (t) = 0
t τ g(τ )dτ dσ = g(σ)M (σ)dσ.
0
(3.21)
0
The following Proposition is a sufficient condition for the solution (3.2) of (3.1) to be a unitary invariant strongly pseudoconvex complex Finsler metric. Proposition 3.2. Let φ(t, s) be defined by (3.17) with L(t) and h(t, s) defined by (3.18) and (3.19) on a unitary invariant domain D ⊂ Cn , respectively. If g(t) and ϕ(w) are smooth functions such that
g(t) ≤ 0
Then F =
and
⎧ ⎪ ⎨ ϕ (w) ≤ 0, ϕ(w) − wϕ (w) > 0, ⎪ ⎩ ϕ (w) ≥ 0.
(3.22)
rφ(t, s) is a weakly complex Berwald metric on D ⊂ Cn .
Proof. By a direct calculation, we have φ + (t − s)φs = L(t)M (t)(ϕ(w) − wϕ (w)),
sϕ (w)M (t) φ − sφs = L(t) ϕ(w) M (t) − tN (t) − , h(t, s) 1 (t − s)φss = φ + (t − s)φs = − 2 L(t)M 2 (t)wϕ (w). h (t, s) s Using our assumption (3.22), it is easy to check that φ − sφs > 0 Thus by Proposition 2.2, F =
and
(φ − sφs )[φ + (t − s)φs ] + s(t − s)φφss > 0.
rφ(t, s) is a weakly complex Berwald metric on D ⊂ Cn . 2
4. Condition for weakly complex Berwald metric to be weakly Kähler Finsler metric Since there are lots of weakly complex Berwald metrics which are unitary invariant, one may wonder among them whether there are weakly Kähler Finsler metrics? In [8], it was proved that Theorem 4.1. (See [8].) A weakly complex Berwald metric F = domain D ⊂ Cn is a weakly Kähler Finsler metric if and only if
rφ(t, s) defined on a unitary invariant
φs − φt + s(φst + φss ) − s2 (t − s)φss g(t) = 0 for a real-valued smooth function g(t).
(4.1)
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Under the assumption that φ(φst + φss ) − φs (φt + φs ) = 0,
(4.2)
the author in [8] proved that among weakly complex Berwald metrics which are unitary invariant, there is no non-Hermitian weakly Kähler Finsler metrics. In this paper, we are able to get rid of the assumption (4.2), thus in an equivalent way, solving the equation (4.1) completely. More precisely, we prove that Theorem 4.2. Let F = rφ(t, s) be a unitary invariant weakly complex Berwald metric defined on a domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric if and only if it satisfies the following system of PDEs
φss = 0, φs − φt + sφst = 0,
(4.3)
The solution of (4.3) is given by φ(t, s) = a0 (t) + a0 (t)s,
(4.4)
where a0 (t) is a positive smooth function satisfying a0 (t) + ta 0 (t) > 0. Proof. Assume that F = rφ(t, s) is a weakly complex Berwald metric, then by Theorem 3.1, φ is necessary of the form (3.2) for some smooth functions f (t) and g(t). By the proof of Theorem 3.10 in [8], a weakly complex Berwald metric F = rφ(t, s) is a weakly Kähler Finsler metric if and only if the following equation holds: 2φs + sφ
∂k2 − (φ − sφs )k2 = 0, ∂s
(4.5)
where k2 =
1 (φt + φs ) − s φ + (t − s)φs g(t) . φ
(4.6)
On the other hand, by the proof Theorem 3.4 in [8], a strongly pseudoconvex complex Finsler metric F = rφ(t, s) is a weakly complex Berwald metric if and only if ∂k2 = 0. ∂s
(4.7)
2φs − (φ − sφs )k2 = 0.
(4.8)
2φφs − (φ − sφs ) (φt + φs ) − s φ + (t − s)φs g(t) = 0.
(4.9)
Thus (4.5) reduces to
Substituting (4.6) into (4.8), we get
Again by Theorem 3.1, φ satisfies the following PDE: φt + φs = s φ + (t − s)φs g(t) + f (t)φ. Substituting (4.10) into (4.9), we get
(4.10)
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2φφs − (φ − sφs )f (t)φ = 0,
(4.11)
which implies that f (t) =
2φs φ − sφs
(4.12)
since φ > 0, φ − sφs > 0. Differentiating (4.12) with respect to s yields 0=
2φφss , (φ − sφs )2
(4.13)
φss = 0.
(4.14)
which is equivalent to
(4.14) together with (4.1) gives (4.3). Solving (4.3) and notice that φ have to satisfy (2.15), we get (4.4). This completes the proof. 2 Remark 4.3. By the proof of Theorem 3.4 in [7], a unitary invariant complex Finsler metric F is a weakly complex Berwald metric if and only if k3 = g(t) for some smooth function g(t). It was also shown in the proof of Theorem 3.4 in [7] that the condition k3 = g(t) implies that k2 is independent of the variable s, 2 i.e., ∂k ∂s = 0. Here k2 and k3 are respectively defined by (3.12) and (3.13) in [7]. Note that (4.6) and (4.10) implies that for a weakly complex Berwald metric, it is necessary that k2 = f (t).
(4.15)
Conversely, if k2 = f (t) and k3 = g(t) for some functions f (t) and g(t), then by (3.14) in [7], we have 2Gγ = k2 z, vv γ + k3 (z, v)2 z γ , which implies that the complex geodesic coefficients Gγ is quadratic with respect to the fiber coordinates v = (v 1 , · · · , v n ). Thus a unitary invariant complex Finsler metric F is a weakly complex Berwald metric if and only if k2 = f (t),
k3 = g(t).
(4.16)
5. Holomorphic curvature of unitary invariant weakly complex Berwald metric In this section, we shall investigate the unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures. Proposition 5.1. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F has vanishing holomorphic curvature, that is, KF = 0 if and only if f (t) = g(t) = 0,
(5.1)
φ(t, s) = ϕ(s − t)
(5.2)
and in this case
for a smooth positive function ϕ satisfying Proposition 2.2.
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
13
Proof. By Proposition 3.6 in [8], the holomorphic curvature KF of a weakly complex Berwald metric F = rφ(t, s) is given by dg(t) dg(t) dk2 2 + k2 + sφ s + 2g(t) + s(t − s)φs s + 2g(t) , KF (v) = − 2 φ s φ dt dt dt
(5.3)
where k2 is given by (4.15). Substituting (4.15) into (5.3) yields 2 KF (v) = − 2 φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) . φ
(5.4)
It is clear that f (t) = g(t) = 0 implies KF = 0. Conversely, if KF = 0, then by (5.4) φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) = 0.
(5.5)
Taking s = 0 in (5.5) we get f (t) = 0.
(5.6)
φ + (t − s)φs sg (t) + 2g(t) = 0.
(5.7)
Substituting (5.6) into (5.5) gives
Since F is a strongly pseudoconvex complex Finsler metric, taking s = 0 in (2.15) shows φ + tφs > 0.
(5.8)
g(t) = 0.
(5.9)
Thus again taking s = 0 in (5.7) yields
Finally, substituting (5.6) and (5.9) into (3.3), we get (5.2). This completes the proof. 2 By Theorem 4.2 and Proposition 5.1, we have the following corollary. Corollary 5.2. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric with vanishing holomorphic curvature if and only if φ(t, s) = constant > 0.
(5.10)
Proof. The sufficiency is obvious. We need only prove the necessity. Assume that the weakly complex Berwald metric F is a weakly Kähler Finsler metric satisfying KF = 0. By (5.2), we have φt + φs = 0,
(5.11)
φt = φs = 0.
(5.12)
which together with (4.3) implies
14
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
Thus we get (5.10), which completes the proof. 2 Definition 5.3. A complex Finsler metric F is said to be of isotropic holomorphic curvature if its holomorphic curvature KF is independent of the fiber coordinates v, that is KF (z, v) = KF (z). Now assume that F is of isotropic holomorphic curvature KF = k(t). Since the case k(t) = 0 has already been considered in Theorem 5.1, we need only investigate the case KF = k(t) = 0, and we shall discuss it in two cases: g(t) = 0 or g(t) = 0. Theorem 5.4. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn satisfying g(t) = 0. Then F has nonzero isotropic holomorphic curvature if and only if φ is either given by φ(t, s) = Be C0 t ;
(5.13)
or φ(t, s) =
B B − At
C1 A
A(s − t) + B
(5.14)
for real constants A, B, C0 , C1 satisfying C0 = 0, C1 = 0, B > 0, B − At > 0. The holomorphic curvatures of F are respectively given by KF = −
2C0 Be C0 t
(5.15)
and KF = −2C1 B −
C1 A
(B − At)
C1 A
−2
.
(5.16)
2 sf (t) + f (t) , φ
(5.17)
Proof. Substituting g(t) = 0 into (5.4), we get KF = k(t) = − or equivalently φ=−
2 sf (t) + f (t) . k(t)
(5.18)
On the other hand, (3.3) together with g(t) = 0 implies h(t, s) = 1.
(5.19)
Substituting (5.19) into (3.2), we have
t φ = exp
f (τ )dτ ϕ(s − t).
0
Note that by (5.18), φ is linear with respect to s, hence by (5.20) we have
(5.20)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
t φ = exp
f (τ )dτ
A(s − t) + B
15
(5.21)
0
for some real constant A and B satisfying B > 0,
B − At > 0,
(5.22)
since φ have to satisfy the strongly pseudoconvex condition (2.15) for F . It follows from (5.21) and (5.18) that ⎧ t 2 ⎨exp f (τ )dτ A = − k(t) f (t), 0 (5.23) t 2 ⎩exp f (τ )dτ (B − At) = − k(t) f (t). 0 If A = 0, then the first equality in (5.23) implies that f (t) = C0 ,
(5.24)
where C0 is nonzero constant. Substituting (5.24) into (5.21) and (5.17), we get (5.13) and (5.15). If A = 0, dividing the first equality of (5.23) by the second one yields A f (t) = . f (t) B − At
(5.25)
Solving equation (5.25), we obtain C1 , f (t) = B − At
t exp
f (τ )dτ
=
0
B B − At
CA1 ,
(5.26)
where C1 is nonzero constant. Substituting (5.26) into (5.21) and (5.23) yields (5.14) and (5.16). This completes the proof. 2 Theorem 5.5. There is no weakly complex Berwald metric of the form F = rφ(t, s) with φ given by (3.2) on a unitary invariant domain D ⊂ Cn and satisfying g(t) = 0 such that F has nonzero isotropic holomorphic curvature KF = k(t) = 0. Proof. Suppose that F = rφ(t, s) is a weakly complex Berwald metric. Denote by ψ(t) = φ(t, 0), here φ(t, s) is given by (3.2). In the following we shall use this ψ(t) as the initial value of the PDE of F = rφ(t, s) which is of isotropic holomorphic curvature. By (4.16), we have k2 = f (t). If KF (z, v) = k(t), then by (5.4), we have 2 (5.27) k(t) = − 2 φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) . φ Taking s = 0 in (5.27) yields k(t) = −
2 f (t). φ(t, 0)
(5.28)
Therefore there exists some t0 = 0 such that k(t0 ) = 0 and f (t0 ) = 0. We still denote t0 by t if it causes no confusion. For this fixed t, we can choose a positive number ε small enough such that s(t −s)(sg (t) +2g(t)) = 0 for each s ∈ (0, ε]. Thus (5.27) can be rewritten as
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
φs − p(t, s)φ = −q(t, s)φ2 ,
(5.29)
where p(t, s) =
s2 g (t) + 2sg(t) + sf (t) + f (t) , s(s − t)(sg (t) + 2g(t))
q(t, s) = −
k(t) . 2s(s − t)(sg (t) + 2g(t))
(5.30) (5.31)
Now we consider the following ODE:
φs − p(t, s)φ = −q(t, s)φ2 ,
s ∈ (0, ε],
φ(t, 0) = ψ(t).
(5.32)
By our assumption, the function φ given by (3.2) is a solution to equation (5.32). Furthermore, by the local existence and uniqueness theorem (Picard Theorem) for ODE, the solution to (5.32) is unique and is also given by (3.2). On the other hand, the equation (5.32) is a Bernoulli equation. By a routine exercise, it is easy to check that s exp 0 p(t, τ )dτ φ(t, s) = s τ (ψ(t))−1 + 0 q(t, τ ) exp 0 p(t, σ)dσ dτ
(5.33)
is a solution to (5.32). In the following we shall analyze the numerator of the right side of (5.33) to get a contradiction. In fact, by (5.30) we can view p(t, s) as a rational function with respect to s, then it can be decomposed as p(t, s) =
E(t)s + J(t) D(t) + s (s − t)(sg (t) + 2g(t))
(5.34)
for some functions D(t), E(t) and J(t) to be determined. It easy to see that D(t) is given by D(t) = −
f (t) . 2tg(t)
We conclude that D(t) = 0,
(5.35)
otherwise by (5.27), f (t) = 0 implies that k(t) = − φ22 s[φ + (t − s)φs ][sg (t) + 2g(t)], it contradicts the hypothesis that F has isotropic holomorphic curvature. s By (5.34) and (5.35), the integration 0 p(t, τ )dτ diverges, therefore (5.33) is not the solution to equation (5.32). It is a contradiction. 2 Theorem 5.5 shows that there is also no weakly complex Berwald metric F = rφ(t, s) satisfying g(t) = 0 such that F is of nonzero constant holomorphic curvature. Thus by Theorem 5.4 and Theorem 5.5, taking C1 A = 2 in Theorem 5.4 we get Corollary 5.6. Let F = rφ(t, s) be a weakly complex Berwald metric given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F has nonzero constant holomorphic curvature if and only if φ is given by
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
φ(t, s) =
B 2 A(s − t) + B B − At
17
(5.36)
for real constants A, B satisfying A = 0, B > 0, B − At > 0. In this case the holomorphic curvature of F is given by KF = −4AB −2 .
(5.37)
Remark 5.7. The canonical complex Euclidean metric in D = Cn is obtained by taking A = 0, B = 1 and φ(t, s) = 1 with KF = 0. The Bergman metric on the unit ball D = Bn is obtained by taking A = B = 1 and φ(t, s) =
s 1 + , 1 − t (1 − t)2
KF = −4.
(5.38)
The Fubini–Study metric on a local coordinate on CPn is obtained by taking A = −1, B = 1 and φ(t, s) =
s 1 − , 1 + t (1 + t)2
KF = 4.
(5.39)
Theorem 5.8. Let F = rφ(t, s) be a weakly complex Berwald metric on a unitary invariant domain D ⊂ Cn with holomorphic curvature KF . Then (i) KF = constant = 0 if and only if φ(t, s) is given by φ(t, s) = ϕ(s − t),
(5.40)
where ϕ(w) is a positive smooth function satisfying Proposition 2.2. (ii) KF = constant = 0 if and only if φ is given by φ(t, s) =
B 2 A(s − t) + B , B − At
(5.41)
where A, B are real constants satisfying A = 0, B > 0, B − At > 0. In this case KF = −4AB −2 . Proof. It follows from Proposition 5.1 and Corollary 5.6.
(5.42)
2
6. Ricci scalar curvature of weakly complex Berwald metrics In this section, we shall investigate Ricci scalar curvature of unitary invariant weakly complex Berwald metrics. There are two notions of Ricci scalar curvature in literature, we refer to [5–7] for more details. Definition 6.1. (See [6].) Let F be a strongly pseudoconvex complex Finsler metric defined on a domain D ⊂ Cn . Then the Ricci scalar curvature Ric of the Chern–Finsler connection associated with F is defined by Ric(z, v) = −δα (Γμ;μ )v α ,
(6.1)
˘ of the complex Berwald connection associated with F is defined by and the Ricci scalar curvature Ric ˘ Ric(z, v) = −Xα (Gμμ )v α ,
(6.2)
18
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
α where Γα ;μ and Gμ are the complex nonlinear connection coefficients of the Chern–Finsler connection and the complex Berwald connection, respectively, δα = ∂α − Γβ;α ∂˙β , Xα = ∂α − Gβα ∂˙β and the index μ is summed from 1 to n.
Theorem 6.2. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric of the Chern–Finsler connection associated with F is ∂ρ ∂ρ s− g(t)s2 (t − s) , Ric(z, v) = −r ρ + ∂t ∂s
(6.3)
where ρ = ρ(t, s) = (n − 1)d0 + f (t) + tg(t) + (s − t)d1 d4 , d0 =
φs , φ − sφs
d 1 d4 =
φ2 φss . (φ − sφs ) (φ − sφs )[φ + (t − s)φs ] + s(t − s)φss
Proof. By Proposition 3.1 in [8], we know that the complex nonlinear connection coefficients of a strongly pseudoconvex complex Finsler metric is given by 1 1 Γγ;μ = d0 z, vδμ¯γ + d1 d2 z, vz μ z γ + d3 z μ v γ + d4 (z, v)2 v μ z γ + d5 z, vv μ v γ , r r
(6.4)
where d0 =
φs , φ − sφs
d1 =
1 , (φ − sφs ){(φ − sφs )[φ + (t − s)φs ] + s(t − s)φss }
d2 = φ2 φst − sφφs (φst + φss ) − φs (φ − sφs )(φt + φs ), d3 = (φ − sφs ) φ + (t − s)φs (φt − sφst ) + s(t − s)φt φss + s2 φφs φss , d4 = φ2 φss , d5 = −sφφss [φ + (t − s)φs ]. Since F is a weakly complex Berwald metric, by (4.15) in this paper and Proposition 3.2 in [8], we get d0 + d1 d3 + d1 d5 = f (t),
d1 d2 + d1 d4 = g(t).
Thus by (6.4) and (6.5), we obtain Γμ;μ = nd0 + td1 d2 + d1 d3 + d1 d5 + d1 d4 s z, v = (n − 1)d0 + f (t) + tg(t) + d1 d4 (s − t) z, v = ρ(t, s)z, v, where we denote ρ(t, s) = (n − 1)d0 + f (t) + tg(t) + (s − t)d1 d4 . Thus we have
(6.5)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
19
Ric(z, v) = −δα¯ (Γμ;μ )v α = −(∂α¯ − Γβ;α ∂˙β¯ )(Γμ;μ )v α
! ∂ρ α β ∂ρ α z z, v + ρv − Γ;α sβ¯ z, v v α =− ∂t ∂s ! ∂ρ β ∂ρ 2 2 |z, v| + ρv − 2 G sβ¯ z, v . =− ∂t ∂s
(6.6)
By (6.4) and (6.5), we get 2Gβ = Γβ;γ v γ = f (t)z, vv β + g(t)(z, v)2 z β . Note that since sβ¯ v β = 0 and sβ¯ z β =
z,v r (t
(6.7)
− s), it follows that
2Gβ sβ¯ = g(t)s(t − s)z, v.
(6.8)
Substituting (6.8) into (6.6) yields (6.3). This completes the proof. 2 Theorem 6.3. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary ˘ of the complex Berwald connection associated invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric with F is r d (n + 1)f (t) + 2tg(t) ˘ Ric(z, v) = − s + (n + 1)f (t) + 2tg(t) . 2 dt
(6.9)
Proof. Differentiating (6.7) with respect to v μ , we obtain 2Gγμ = f (t)z μ v γ + f (t)z, vδγμ + 2g(t)z, vz μ z γ , from which we get 2Gμμ = (n + 1)f (t) + 2tg(t) z, v. Note that 2Gμμ is holomorphic with respect to v, we get 1 1 ˘ Ric(z, v) = − Xα¯ (2Gμμ )v α = − ∂α¯ (2Gμμ )v α 2 2 n d (n + 1)f (t) + 2tg(t) α 1 z z, v + (n + 1)f (t) + 2tg(t) v α v α =− 2 α=1 dt 1 d (n + 1)f (t) + 2tg(t) 2 2 |z, v| + (n + 1)f (t) + 2tg(t) v =− 2 dt r d (n + 1)f (t) + 2tg(t) 2 s + (n + 1)f (t) + 2tg(t) . =− 2 dt Corollary 6.4. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary ˘ invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric(z, v) = 0 if and only if f (t) = −
2 tg(t). n+1
20
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
Acknowledgement This work is supported by NCET-13-0510; NNSFC (Grant No. 11271304, 11171277); the Fujian Province Natural Science Funds for Distinguished Young Scholar (Grant No. 2013J06001); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry; the Fundamental Research Funds for the Central Universities (Grant No. 2012121006). The authors would like to thank the referee for his/her careful reading and helpful comments. References [1] M. Abate, G. Patrizio, Finsler metrics of constant curvature and the characterization of tube domains, in: David Bao, Shiing-Shen Chern, Zhongmin Shen (Eds.), Finsler Geometry: Joint Summer Research Conference on Finsler Geometry, Seattle, Washington, July 16–20, 1995, in: Contemporary Mathematics, vol. 196, 1996, pp. 101–107. [2] M. Abate, G. Patrizio, Finsler Metrics—A Global Approach with Applications to Geometric Function Theory, Lecture Notes in Mathematics, vol. 1591, Springer-Verlag, Berlin/Aeidelberg, 1994. [3] T. Aikou, On complex Finsler manifolds, Rep. Fac. Sci., Kagoshima Univ. (Math., Phys. Chem.) 24 (1991) 9–25. [4] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. Fr. 109 (1981) 427–474. [5] G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Academic Publishers, 2004. [6] L. Sun, C. Zhong, Characterization of complex Finsler connections and weakly complex Berwald metrics, Differ. Geom. Appl. 31 (2013) 648–671. [7] C. Zhong, On real and complex Berwald connections associated to strongly convex weakly Kähler–Finsler metric, Differ. Geom. Appl. 29 (2011) 388–408. [8] C. Zhong, On unitary invariant strongly pseudoconvex complex Finsler metrics, Differ. Geom. Appl. 40 (2015) 159–186.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature Hongchuan Xia, Chunping Zhong ∗ School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 March 2015 Received in revised form 20 July 2015 Available online xxxx Communicated by Z. Shen
In this paper, we give a characterization of strongly pseudoconvex complex Finsler metric F which is unitary invariant. A necessary and sufficient condition for F to be a weakly complex Berwald metric and a necessary and sufficient condition for F to be a weakly Kähler Finsler metric are given, respectively. We also give a classification of unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures. © 2015 Elsevier B.V. All rights reserved.
MSC: 53C60 53C40 Keywords: Unitary invariant Weakly complex Berwald metric Weakly Kähler Finsler metric Constant holomorphic curvature
1. Introduction and main results Let ·, · be the canonical complex Euclidean inner product in the complex n-dimensional linear space C , · be the norm induced by ·, ·, that is, n
z, v =
n
zi vi ,
z =
z, z,
∀z = (z 1 , · · · , z n )T , v = (v 1 , · · · , v n )T ∈ Cn ,
i=1
where and in the following bars and overlines denote conjugations of complex numbers, and (· · · )T denotes the transpose of vectors in Cn . Denote U (n) the set of n-by-n unitary matrices over the complex number field C. A domain D ⊂ Cn is called unitary invariant if Az ∈ D whenever z ∈ D and A ∈ U (n). * Corresponding author. E-mail addresses: [email protected] (H. Xia), [email protected] (C. Zhong). http://dx.doi.org/10.1016/j.difgeo.2015.08.001 0926-2245/© 2015 Elsevier B.V. All rights reserved.
2
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
Definition 1.1. Let F be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . F is called unitary invariant if F (Az, Av) = F (z, v), ∀A ∈ U (n), z ∈ D, v ∈ Tz1,0 D.
(1.1)
It is well known that the canonical complex Euclidean inner product ·, · in Cn is unitary invariant. For strongly pseudoconvex complex Finsler metrics in the sense of Abate and Patrizio [2], we have the following characterization. Theorem 1.2. Let F be a strongly pseudoconvex complex Finsler metric on a unitary invariant domain D ⊂ Cn . Then F is unitary invariant if and only if there exists a smooth function φ(t, s): [0, +∞) × [0, +∞) → (0, +∞) such that F (z, v) =
rφ(t, s),
r := v2 ,
t := z2 ,
s :=
|z, v|2 r
(1.2)
for every z ∈ D and every nonzero holomorphic tangent vector v ∈ Tz1,0 D. In [3], a strongly pseudoconvex complex Finsler metric is called a complex Berwald metric if the horiγ zontal Chern–Finsler connection coefficients Γν;μ (z, v) associated with F depend only on the base manifold 1 n coordinates z = (z , · · · , z ). In [7], a strongly pseudoconvex complex Finsler metric is called a weakly complex Berwald metric if the complex Berwald connection coefficients Gγνμ (z, v) associated with F depend only on the base manifold coordinates z = (z 1 , · · · , z n ). It was also shown in [7] that the complex Wrona metric is a weakly complex Berwald metric, but not a complex Berwald metric. In [8], unitary invariant strongly pseudoconvex complex Finsler metrics of the form (1.2) were systematically studied. It was proved that Theorem 1.3. (See [8].) A strongly pseudoconvex complex Finsler metric F = D ⊂ Cn is a weakly complex Berwald metric if and only if
rφ(t, s) defined on a domain
φ(φst + φss ) − φs (φt + φs ) = g(t) (φ − sφs ) φ + (t − s)φs + s(t − s)φφss
(1.3)
for a real-valued smooth function g(t). In this paper we are able to simplify the second-order nonlinear PDE (1.3) to a first-order linear PDE and solve it completely. We obtain Theorem 1.4. Let F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly complex Berwald metric if and only if φt + 1 − tg(t)s + g(t)s2 φs = f (t) + g(t)s φ
(1.4)
holds for some real-valued smooth functions f (t) and g(t). The solution of (1.4) satisfying φ(0, s) = ϕ(s)
(1.5)
for a given smooth positive function ϕ(s) of s ∈ R is necessary of the form
t φ(t, s) = exp 0
s−t h(t, s), f (τ )dτ ϕ h(t, s)
(1.6)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
3
where h(t, s) is
t h(t, s) = exp
τ g(τ )dτ
σ
t + (s − t)
0
g(σ) exp 0
τ g(τ )dτ dσ.
(1.7)
0
In Section 3, we also obtain a sufficient condition for the solution (1.6) of (1.4) to be a weakly complex Berwald metric on a unitary invariant domain D ⊂ Cn . A fundamental result of Lempert [4] states that the Kobayashi and Carathéodory metrics on strongly convex bounded domains with smooth boundaries in Cn are weakly Kähler Finsler metrics, whether they are Kähler Finsler metrics is still open [1]. These metrics, however, don’t have explicit formulae. So far there is no explicit weakly Kähler Finsler metric found in literature. It was prove in [8] that among unitary invariant strongly pseudoconvex complex Finsler metrics, there is neither non-Hermitian complex Berwald metric nor Kähler Finsler metric. For weakly Kähler Finsler metrics, a necessary and sufficient condition for F = rφ(t, s) to be a weakly Kähelr Finsler metric was obtained, which is characterized by a second-order nonlinear PDE (cf. Theorem 4.3 in [8]). Among weakly complex Berwald metrics, however, the author in [8] also obtained a negative result under the assumption that φ satisfying φ(φst + φss ) − φs (φs + φt ) = 0.
(1.8)
In this paper, we are able to get rid of the assumption (1.8) and prove that Theorem 1.5. Let F = rφ(t, s) be a weakly complex Berwald metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric if and only if it satisfies the following system of PDEs
φss = 0, φs − φt + sφts = 0.
(1.9)
The solution of (1.9) is given by φ(t, s) = a0 (t) + a0 (t)s,
(1.10)
where a0 (t) is a positive smooth function satisfying a0 (t) + ta 0 (t) > 0. Remark 1.6. Theorem 1.5 shows that among unitary invariant weakly complex Berwald metrics there is also no non-Hermitian weakly Kähler Finsler metric. Note that Theorem 1.4 implies that there are lots of non-Hermitian weakly complex Berwald metrics which are unitary invariant. One may wonder among them whether there are metrics of constant holomorphic curvatures. In this paper, we obtain the following classification theorem. Theorem 1.7. Let F = rφ(t, s) be a weakly complex Berwald metric defined on a unitary invariant domain D ⊂ Cn with constant holomorphic curvature KF along a nonzero vector v ∈ Tz1,0 D . Then (i) KF = constant = 0 if and only if φ(t, s) is given by φ(t, s) = ϕ(s − t), where ϕ(w) is a positive smooth function satisfying Proposition 2.2. (ii) KF = constant = 0 if and only if φ is given by
(1.11)
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
φ(t, s) =
B 2 A(s − t) + B , B − At
(1.12)
where A, B are constants with A = 0, B > 0, B − At > 0. In this case KF = −4AB −2 .
(1.13)
2. Characterization of unitary invariant strongly pseudoconvex complex Finsler metric In this section we shall give a characterization of unitary invariant strongly pseduconvex complex Finsler metrics. We have Theorem 2.1. Let F be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is unitary invariant if and only if there exists a smooth function φ(t, s) : [0, +∞) × [0, +∞) → (0, +∞) such that F (z, v) =
rφ(t, s),
r := v2 ,
t := z2 ,
s :=
|z, v|2 r
(2.1)
for every z ∈ D and every nonzero holomorphic tangent vector v ∈ Tz1,0 D. Proof. Since the complex Euclidean inner product ·, · in Cn is unitary invariant, the sufficiency is clear. Next we prove the necessity. Suppose that F is a unitary invariant strongly pseudoconvex complex Finsler metric, i.e., F satisfies (1.1) and is smooth outside of the zero section of T 1,0 D. Denote the unit orthogonal basis of Cn by e1 , · · · , en , where (j th)
ej = (0, · · · , 0, 1 , 0, · · · , 0),
j = 1, · · · , n.
Let z be an arbitrary fixed point in D and v ∈ Tz1,0 D with v = 0. Case 1: If z, v = 0 and v = λz for any λ ∈ C. Then ε1 =
z, vv , |z, v|v
ε2 =
z − z, ε1 ε1 z − z, ε1 ε1
(2.2)
are two unit orthogonal vectors in Cn since v = λz implies that z − z, ε1 ε1 2 = z2 − |z, ε1 |2 = z2 −
|z, v|2
= 0. v2
(2.3)
Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
Aε2 = e2 .
It follows from (2.2)–(2.4) that Az = A(z, ε1 ε1 + z − z, ε1 ε1 ε2 ) |z, v|2 Aε2 = z, ε1 Aε1 + z2 − v2 |z, v| |z, v|2 e1 + z2 − = e2 . v v2
(2.4)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
5
Using (1.1) and the homogeneity of F , i.e., F (z, λv) = |λ|F (z, v) for any λ ∈ C, we get
z, vv = vF Az, A |z, v|v
|z, v|2 |z, v| 2 e1 + z − e 2 , e1 = vF (Az, Aε1 ) = vF v v2 √ √ √ = rF ( se1 + t − se2 , e1 ).
z, vv F (z, v) = vF z, |z, v|v
Thus one need only taking √ √ φ(t, s) = F 2 ( se1 + t − se2 , e1 ).
(2.5)
Case 2: If z, v = 0 and z = 0, then ε1 =
v , v
ε2 =
z z
(2.6)
are two unit orthogonal vectors in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
Aε2 = e2 .
(2.7)
Using (1.1) and the homogeneity of F , we get √ √ v F (z, v) = vF z, = vF (Az, Aε1 ) = vF (ze2 , e1 ) = rF ( te2 , e1 ). v Thus one need only taking √ φ(t, s) = F 2 ( te2 , e1 ).
(2.8)
Case 3: If z, v = 0 and z = 0, then ε1 =
v v
(2.9)
is a unit vector in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
(2.10)
so that √ v = vF (A0, Aε1 ) = vF (0, e1 ) = rF (0, e1 ). F (z, v) = vF 0, v Thus one need only taking φ(t, s) = F 2 (0, e1 ) = constant > 0. Case 4: If z = λv for some λ ∈ C. If λ = 0, then it reduces to case 3. If λ = 0, then ε1 =
|λ|v λv
(2.11)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
6
is a unit vector in Cn . Thus there exists a unitary matrix A ∈ U (n) such that Aε1 = e1 ,
(2.12)
from which we get Av =
λv e1 . |λ|
(2.13)
So that Az = λAv = |λ|ve1 = ze1 .
(2.14)
Using (1.1) and the homogeneity of F , we get F (z, v) = vF (z, ε1 ) = vF (Az, Aε1 ) = vF (ze1 , e1 ) =
√
√ rF ( te1 , e1 ).
Thus one need only taking √ φ(t, s) = F 2 ( te1 , e1 ). Since F is smooth outside of the zero section of T 1,0 D, it follows from the discussion of case 1 to 4 that φ(t, s) is a positive smooth function of t and s. This completes the proof. 2 In [8], the author gave the following characterization for a complex Finsler metric F to be a strongly pseudoconvex complex Finsler metric in the sense of Abate and Patrizio [2]. Proposition 2.2. (See [8].) A complex Finsler metric F = pseudoconvex if and only if φ satisfies φ − sφs > 0,
rφ(t, s) defined on a domain D ⊂ Cn is strongly
(φ − sφs ) φ + (t − s)φs + s(t − s)φφss > 0
(2.15)
when n ≥ 3 or (φ − sφs ) φ + (t − s)φs + s(t − s)φφss > 0 when n = 2. Remark 2.3. In fact, the condition φ(0, 0) > 0 and suitable smoothness of φ(t, s) is enough to ensure that F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric on Bn (ε) ⊂ Cn for some ε > 0. 3. Characterization of unitary invariant weakly complex Berwald metric In this section, we shall give a characterization of unitary invariant weakly complex Berwald metric, which is characterized by a first-order linear PDE with its solution explicitly given. Theorem 3.1. Let F = rφ(t, s) be a strongly pseudoconvex complex Finsler metric defined on a unitary invariant domain D ⊂ Cn . Then F is a weakly complex Berwald metric if and only if φt + 1 − tg(t)s + g(t)s2 φs = f (t) + g(t)s φ
(3.1)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
7
holds for some real-valued smooth functions f (t) and g(t). The solution of (3.1) satisfying φ(0, s) = ϕ(s) for a given smooth positive function ϕ(s) of s ∈ R is necessary of the form
t φ(t, s) = exp
s−t h(t, s), f (τ )dτ ϕ h(t, s)
(3.2)
0
where h(t, s) is
t h(t, s) = exp
τ g(τ )dτ
0
σ
t + (s − t)
g(σ) exp 0
τ g(τ )dτ dσ.
(3.3)
0
Proof. We shall simplify the second-order nonlinear PDE (1.3) to a first-order linear PDE. Dividing by φ2 on both sides of (1.3), we have φ2 + (t − 2s)φφs − s(t − s)φ2s + s(t − s)φφss φ(φt + φs )s − φs (φt + φs ) = g(t) φ2 φ2 φφss − φ2s φs = g(t) 1 + (t − 2s) + s(t − s) φ φ2 φs , = g(t) 1 + (st − s2 ) φ s
(3.4)
where and in the following subscript s denotes the derivative with respect to this variable. Since the left hand side of (3.4) can be rewritten as φ(φt + φs )s − φs (φt + φs ) = φ2
φt + φs φ
. s
Thus integrating (3.4) with respect to s, we get φt + φs φs = g(t)s + g(t)(st − s2 ) + f (t) φ φ
(3.5)
for some real-valued smooth function f (t). Rearranging (3.5), we get (3.1), which is a first-order linear PDE. Next we shall solve the equation (3.1) with suitable initial-value. We consider the following Cauchy problem
φt + [1 − s(t − s)g(t)]φs = [f (t) + g(t)s]φ, φ(X(a), Y (a)) = Z(a), a ∈ I,
(3.6)
where I is an closed interval such that 0 is an interior point in I, and Γ : t = X(a), s = Y (a), b = φ(X(a), Y (a)) =: Z(a) is a given noncharacteristic curve. Note that since s ≤ t and t = 0 implies that s = 0. It is natural to choose Γ passing through the point (X(0), Y (0), Z(0)) = (0, 0, 1) on the surface Σ : b = φ(t, s). In the following we choose (X(a), Y (a), Z(a)) = (0, a, ϕ(a)) such that ϕ(a) is a positive function satisfying ϕ(0) = 1. First, let solve the following initial problem:
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
⎧ dt ⎪ t(0, a) = 0 ⎪ ⎨ dl = 1, ds dl = 1 − s(t − s)g(t), s(0, a) = a ⎪ ⎪ ⎩ db = [f (t) + g(t)s]b, b(0, a) = ϕ(a). dl
(3.7)
The above initial-value problem can be solved by using the method of characteristics. Denote the solution to the above equation by t = t(l, a), s = s(l, a), b = b(l, a). Note that by (3.7), we have ∂(t, s) ∂t ∂l (0, a) = ∂s ∂(l, a) ∂l (0, a)
∂t ∂a (0, a) ∂s ∂a (0, a)
1 = 1 + a2 g(0)
0
= 0 1
at least at a = 0. Thus by implicit function theorem we can obtain l = l(t, s), a = a(t, s) so that φ(t, s) = b(l(t, s), a(t, s)) is the unique solution in a neighborhood of the initial curve Γ :
t = 0, s = a, b = ϕ(a).
By (3.7), it follows that t = l is a solution to dt dl = 0 satisfies the initial condition t(0, a) = 0. Substituting 2 t = l into ds = 1 − tg(t)s + g(t)s and then setting u = s − l, we get dl du − lg(l)u = g(l)u2 , dl
(3.8)
which is a Bernoulli equation. 2 If a = 0, then it is clear that s = l is a solution to the equation ds dl = 1 − tg(t)s + g(t)s satisfies the db initial condition s(0, a) = a. In this case, substituting t = l, s = l into dl = [f (t) + g(t)s]b and integrating from 0 to t, we get
t φ(t, s) = φ(t, t) = exp
[f (τ ) + τ g(τ )]dτ .
(3.9)
0
If a = 0, then by (3.7) and (3.8), it follows that s = l. In this case it is necessary that a < 0 since 1 a = s(0, a) ≤ t(0, a) = 0. Setting y = u1 = s−l , we are able to obtain a first-order linear ODE with respect to y as follows: dy + lg(l)y = −g(l). dl
(3.10)
Solving equation (3.10) with the initial-value y|l=0 = a1 , we get
l
y = exp −
τ g(τ )dτ
1 a
−
0
l
σ g(σ) exp
0
τ g(τ )dτ dσ ,
0
from which we obtain exp s=l+ where we denote
l 0
1 a
τ g(τ )dτ
− c(l)
,
(3.11)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
l c(l) =
σ
0
τ g(τ )dτ
g(σ) exp
9
dσ.
0
Substituting t = l and (3.11) into the third equation in (3.7), and solving it with the initial-value b(0, a) = φ(t(0, a), s(0, a)) = φ(0, a) = ϕ(a), we get
l b(l, a) = ϕ(a) exp
exp
f (τ ) + g(τ ) τ +
τ 0
σg(σ)dσ
dτ − c(τ )
l τ g(τ ) exp 0 σg(σ)dσ f (τ ) + τ g(τ ) + = ϕ(a) exp dτ 1 a − c(τ ) 0
l l d 1 − c(τ ) a = ϕ(a) exp f (τ ) + τ g(τ ) dτ − 1 a − c(τ ) 1 a
0
0
0
l = ϕ(a) exp
f (τ ) + τ g(τ ) dτ
0
1 . |1 − ac(l)|
(3.12)
By (3.11), we get s−l s−l 1 = , = l l 1 − ac(l) a exp 0 τ g(τ )dτ a exp 0 τ g(τ )dτ
(3.13)
or equivalently, a= exp
l 0
s−l . τ g(τ )dτ + (s − l)c(l)
(3.14)
Since l = t and s < t, it follows by (3.12) and (3.14) that
t φ(t, s) = exp
s−t f (τ )dτ ϕ h(t, s), h(t, s)
(3.15)
0
where we denote t h(t, s) =: exp
τ g(τ )dτ + (s − t)c(t).
(3.16)
0
By (3.14) it is necessary that h(t, s) > 0 since a < 0 and s < t. We also note that the solution (3.9) which is corresponding the case a = 0 can be obtained by allowing s = t in (3.15). This completes the proof. 2 In the following, let’s denote φ(t, s) = L(t)ϕ(w)h(t, s), where
(3.17)
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t L(t) = exp
f (τ )dτ ,
w=
s−t , h(t, s)
(3.18)
0
h(t, s) = M (t) + (s − t)N (t),
t M (t) = exp
(3.19)
τ g(τ )dτ ,
(3.20)
0
σ
t g(σ) exp
N (t) = 0
t τ g(τ )dτ dσ = g(σ)M (σ)dσ.
0
(3.21)
0
The following Proposition is a sufficient condition for the solution (3.2) of (3.1) to be a unitary invariant strongly pseudoconvex complex Finsler metric. Proposition 3.2. Let φ(t, s) be defined by (3.17) with L(t) and h(t, s) defined by (3.18) and (3.19) on a unitary invariant domain D ⊂ Cn , respectively. If g(t) and ϕ(w) are smooth functions such that
g(t) ≤ 0
Then F =
and
⎧ ⎪ ⎨ ϕ (w) ≤ 0, ϕ(w) − wϕ (w) > 0, ⎪ ⎩ ϕ (w) ≥ 0.
(3.22)
rφ(t, s) is a weakly complex Berwald metric on D ⊂ Cn .
Proof. By a direct calculation, we have φ + (t − s)φs = L(t)M (t)(ϕ(w) − wϕ (w)),
sϕ (w)M (t) φ − sφs = L(t) ϕ(w) M (t) − tN (t) − , h(t, s) 1 (t − s)φss = φ + (t − s)φs = − 2 L(t)M 2 (t)wϕ (w). h (t, s) s Using our assumption (3.22), it is easy to check that φ − sφs > 0 Thus by Proposition 2.2, F =
and
(φ − sφs )[φ + (t − s)φs ] + s(t − s)φφss > 0.
rφ(t, s) is a weakly complex Berwald metric on D ⊂ Cn . 2
4. Condition for weakly complex Berwald metric to be weakly Kähler Finsler metric Since there are lots of weakly complex Berwald metrics which are unitary invariant, one may wonder among them whether there are weakly Kähler Finsler metrics? In [8], it was proved that Theorem 4.1. (See [8].) A weakly complex Berwald metric F = domain D ⊂ Cn is a weakly Kähler Finsler metric if and only if
rφ(t, s) defined on a unitary invariant
φs − φt + s(φst + φss ) − s2 (t − s)φss g(t) = 0 for a real-valued smooth function g(t).
(4.1)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
11
Under the assumption that φ(φst + φss ) − φs (φt + φs ) = 0,
(4.2)
the author in [8] proved that among weakly complex Berwald metrics which are unitary invariant, there is no non-Hermitian weakly Kähler Finsler metrics. In this paper, we are able to get rid of the assumption (4.2), thus in an equivalent way, solving the equation (4.1) completely. More precisely, we prove that Theorem 4.2. Let F = rφ(t, s) be a unitary invariant weakly complex Berwald metric defined on a domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric if and only if it satisfies the following system of PDEs
φss = 0, φs − φt + sφst = 0,
(4.3)
The solution of (4.3) is given by φ(t, s) = a0 (t) + a0 (t)s,
(4.4)
where a0 (t) is a positive smooth function satisfying a0 (t) + ta 0 (t) > 0. Proof. Assume that F = rφ(t, s) is a weakly complex Berwald metric, then by Theorem 3.1, φ is necessary of the form (3.2) for some smooth functions f (t) and g(t). By the proof of Theorem 3.10 in [8], a weakly complex Berwald metric F = rφ(t, s) is a weakly Kähler Finsler metric if and only if the following equation holds: 2φs + sφ
∂k2 − (φ − sφs )k2 = 0, ∂s
(4.5)
where k2 =
1 (φt + φs ) − s φ + (t − s)φs g(t) . φ
(4.6)
On the other hand, by the proof Theorem 3.4 in [8], a strongly pseudoconvex complex Finsler metric F = rφ(t, s) is a weakly complex Berwald metric if and only if ∂k2 = 0. ∂s
(4.7)
2φs − (φ − sφs )k2 = 0.
(4.8)
2φφs − (φ − sφs ) (φt + φs ) − s φ + (t − s)φs g(t) = 0.
(4.9)
Thus (4.5) reduces to
Substituting (4.6) into (4.8), we get
Again by Theorem 3.1, φ satisfies the following PDE: φt + φs = s φ + (t − s)φs g(t) + f (t)φ. Substituting (4.10) into (4.9), we get
(4.10)
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
2φφs − (φ − sφs )f (t)φ = 0,
(4.11)
which implies that f (t) =
2φs φ − sφs
(4.12)
since φ > 0, φ − sφs > 0. Differentiating (4.12) with respect to s yields 0=
2φφss , (φ − sφs )2
(4.13)
φss = 0.
(4.14)
which is equivalent to
(4.14) together with (4.1) gives (4.3). Solving (4.3) and notice that φ have to satisfy (2.15), we get (4.4). This completes the proof. 2 Remark 4.3. By the proof of Theorem 3.4 in [7], a unitary invariant complex Finsler metric F is a weakly complex Berwald metric if and only if k3 = g(t) for some smooth function g(t). It was also shown in the proof of Theorem 3.4 in [7] that the condition k3 = g(t) implies that k2 is independent of the variable s, 2 i.e., ∂k ∂s = 0. Here k2 and k3 are respectively defined by (3.12) and (3.13) in [7]. Note that (4.6) and (4.10) implies that for a weakly complex Berwald metric, it is necessary that k2 = f (t).
(4.15)
Conversely, if k2 = f (t) and k3 = g(t) for some functions f (t) and g(t), then by (3.14) in [7], we have 2Gγ = k2 z, vv γ + k3 (z, v)2 z γ , which implies that the complex geodesic coefficients Gγ is quadratic with respect to the fiber coordinates v = (v 1 , · · · , v n ). Thus a unitary invariant complex Finsler metric F is a weakly complex Berwald metric if and only if k2 = f (t),
k3 = g(t).
(4.16)
5. Holomorphic curvature of unitary invariant weakly complex Berwald metric In this section, we shall investigate the unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures. Proposition 5.1. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F has vanishing holomorphic curvature, that is, KF = 0 if and only if f (t) = g(t) = 0,
(5.1)
φ(t, s) = ϕ(s − t)
(5.2)
and in this case
for a smooth positive function ϕ satisfying Proposition 2.2.
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
13
Proof. By Proposition 3.6 in [8], the holomorphic curvature KF of a weakly complex Berwald metric F = rφ(t, s) is given by dg(t) dg(t) dk2 2 + k2 + sφ s + 2g(t) + s(t − s)φs s + 2g(t) , KF (v) = − 2 φ s φ dt dt dt
(5.3)
where k2 is given by (4.15). Substituting (4.15) into (5.3) yields 2 KF (v) = − 2 φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) . φ
(5.4)
It is clear that f (t) = g(t) = 0 implies KF = 0. Conversely, if KF = 0, then by (5.4) φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) = 0.
(5.5)
Taking s = 0 in (5.5) we get f (t) = 0.
(5.6)
φ + (t − s)φs sg (t) + 2g(t) = 0.
(5.7)
Substituting (5.6) into (5.5) gives
Since F is a strongly pseudoconvex complex Finsler metric, taking s = 0 in (2.15) shows φ + tφs > 0.
(5.8)
g(t) = 0.
(5.9)
Thus again taking s = 0 in (5.7) yields
Finally, substituting (5.6) and (5.9) into (3.3), we get (5.2). This completes the proof. 2 By Theorem 4.2 and Proposition 5.1, we have the following corollary. Corollary 5.2. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F is a weakly Kähler Finsler metric with vanishing holomorphic curvature if and only if φ(t, s) = constant > 0.
(5.10)
Proof. The sufficiency is obvious. We need only prove the necessity. Assume that the weakly complex Berwald metric F is a weakly Kähler Finsler metric satisfying KF = 0. By (5.2), we have φt + φs = 0,
(5.11)
φt = φs = 0.
(5.12)
which together with (4.3) implies
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H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
Thus we get (5.10), which completes the proof. 2 Definition 5.3. A complex Finsler metric F is said to be of isotropic holomorphic curvature if its holomorphic curvature KF is independent of the fiber coordinates v, that is KF (z, v) = KF (z). Now assume that F is of isotropic holomorphic curvature KF = k(t). Since the case k(t) = 0 has already been considered in Theorem 5.1, we need only investigate the case KF = k(t) = 0, and we shall discuss it in two cases: g(t) = 0 or g(t) = 0. Theorem 5.4. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn satisfying g(t) = 0. Then F has nonzero isotropic holomorphic curvature if and only if φ is either given by φ(t, s) = Be C0 t ;
(5.13)
or φ(t, s) =
B B − At
C1 A
A(s − t) + B
(5.14)
for real constants A, B, C0 , C1 satisfying C0 = 0, C1 = 0, B > 0, B − At > 0. The holomorphic curvatures of F are respectively given by KF = −
2C0 Be C0 t
(5.15)
and KF = −2C1 B −
C1 A
(B − At)
C1 A
−2
.
(5.16)
2 sf (t) + f (t) , φ
(5.17)
Proof. Substituting g(t) = 0 into (5.4), we get KF = k(t) = − or equivalently φ=−
2 sf (t) + f (t) . k(t)
(5.18)
On the other hand, (3.3) together with g(t) = 0 implies h(t, s) = 1.
(5.19)
Substituting (5.19) into (3.2), we have
t φ = exp
f (τ )dτ ϕ(s − t).
0
Note that by (5.18), φ is linear with respect to s, hence by (5.20) we have
(5.20)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
t φ = exp
f (τ )dτ
A(s − t) + B
15
(5.21)
0
for some real constant A and B satisfying B > 0,
B − At > 0,
(5.22)
since φ have to satisfy the strongly pseudoconvex condition (2.15) for F . It follows from (5.21) and (5.18) that ⎧ t 2 ⎨exp f (τ )dτ A = − k(t) f (t), 0 (5.23) t 2 ⎩exp f (τ )dτ (B − At) = − k(t) f (t). 0 If A = 0, then the first equality in (5.23) implies that f (t) = C0 ,
(5.24)
where C0 is nonzero constant. Substituting (5.24) into (5.21) and (5.17), we get (5.13) and (5.15). If A = 0, dividing the first equality of (5.23) by the second one yields A f (t) = . f (t) B − At
(5.25)
Solving equation (5.25), we obtain C1 , f (t) = B − At
t exp
f (τ )dτ
=
0
B B − At
CA1 ,
(5.26)
where C1 is nonzero constant. Substituting (5.26) into (5.21) and (5.23) yields (5.14) and (5.16). This completes the proof. 2 Theorem 5.5. There is no weakly complex Berwald metric of the form F = rφ(t, s) with φ given by (3.2) on a unitary invariant domain D ⊂ Cn and satisfying g(t) = 0 such that F has nonzero isotropic holomorphic curvature KF = k(t) = 0. Proof. Suppose that F = rφ(t, s) is a weakly complex Berwald metric. Denote by ψ(t) = φ(t, 0), here φ(t, s) is given by (3.2). In the following we shall use this ψ(t) as the initial value of the PDE of F = rφ(t, s) which is of isotropic holomorphic curvature. By (4.16), we have k2 = f (t). If KF (z, v) = k(t), then by (5.4), we have 2 (5.27) k(t) = − 2 φ sf (t) + f (t) + s φ + (t − s)φs sg (t) + 2g(t) . φ Taking s = 0 in (5.27) yields k(t) = −
2 f (t). φ(t, 0)
(5.28)
Therefore there exists some t0 = 0 such that k(t0 ) = 0 and f (t0 ) = 0. We still denote t0 by t if it causes no confusion. For this fixed t, we can choose a positive number ε small enough such that s(t −s)(sg (t) +2g(t)) = 0 for each s ∈ (0, ε]. Thus (5.27) can be rewritten as
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φs − p(t, s)φ = −q(t, s)φ2 ,
(5.29)
where p(t, s) =
s2 g (t) + 2sg(t) + sf (t) + f (t) , s(s − t)(sg (t) + 2g(t))
q(t, s) = −
k(t) . 2s(s − t)(sg (t) + 2g(t))
(5.30) (5.31)
Now we consider the following ODE:
φs − p(t, s)φ = −q(t, s)φ2 ,
s ∈ (0, ε],
φ(t, 0) = ψ(t).
(5.32)
By our assumption, the function φ given by (3.2) is a solution to equation (5.32). Furthermore, by the local existence and uniqueness theorem (Picard Theorem) for ODE, the solution to (5.32) is unique and is also given by (3.2). On the other hand, the equation (5.32) is a Bernoulli equation. By a routine exercise, it is easy to check that s exp 0 p(t, τ )dτ φ(t, s) = s τ (ψ(t))−1 + 0 q(t, τ ) exp 0 p(t, σ)dσ dτ
(5.33)
is a solution to (5.32). In the following we shall analyze the numerator of the right side of (5.33) to get a contradiction. In fact, by (5.30) we can view p(t, s) as a rational function with respect to s, then it can be decomposed as p(t, s) =
E(t)s + J(t) D(t) + s (s − t)(sg (t) + 2g(t))
(5.34)
for some functions D(t), E(t) and J(t) to be determined. It easy to see that D(t) is given by D(t) = −
f (t) . 2tg(t)
We conclude that D(t) = 0,
(5.35)
otherwise by (5.27), f (t) = 0 implies that k(t) = − φ22 s[φ + (t − s)φs ][sg (t) + 2g(t)], it contradicts the hypothesis that F has isotropic holomorphic curvature. s By (5.34) and (5.35), the integration 0 p(t, τ )dτ diverges, therefore (5.33) is not the solution to equation (5.32). It is a contradiction. 2 Theorem 5.5 shows that there is also no weakly complex Berwald metric F = rφ(t, s) satisfying g(t) = 0 such that F is of nonzero constant holomorphic curvature. Thus by Theorem 5.4 and Theorem 5.5, taking C1 A = 2 in Theorem 5.4 we get Corollary 5.6. Let F = rφ(t, s) be a weakly complex Berwald metric given by (3.2) on a unitary invariant domain D ⊂ Cn . Then F has nonzero constant holomorphic curvature if and only if φ is given by
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
φ(t, s) =
B 2 A(s − t) + B B − At
17
(5.36)
for real constants A, B satisfying A = 0, B > 0, B − At > 0. In this case the holomorphic curvature of F is given by KF = −4AB −2 .
(5.37)
Remark 5.7. The canonical complex Euclidean metric in D = Cn is obtained by taking A = 0, B = 1 and φ(t, s) = 1 with KF = 0. The Bergman metric on the unit ball D = Bn is obtained by taking A = B = 1 and φ(t, s) =
s 1 + , 1 − t (1 − t)2
KF = −4.
(5.38)
The Fubini–Study metric on a local coordinate on CPn is obtained by taking A = −1, B = 1 and φ(t, s) =
s 1 − , 1 + t (1 + t)2
KF = 4.
(5.39)
Theorem 5.8. Let F = rφ(t, s) be a weakly complex Berwald metric on a unitary invariant domain D ⊂ Cn with holomorphic curvature KF . Then (i) KF = constant = 0 if and only if φ(t, s) is given by φ(t, s) = ϕ(s − t),
(5.40)
where ϕ(w) is a positive smooth function satisfying Proposition 2.2. (ii) KF = constant = 0 if and only if φ is given by φ(t, s) =
B 2 A(s − t) + B , B − At
(5.41)
where A, B are real constants satisfying A = 0, B > 0, B − At > 0. In this case KF = −4AB −2 . Proof. It follows from Proposition 5.1 and Corollary 5.6.
(5.42)
2
6. Ricci scalar curvature of weakly complex Berwald metrics In this section, we shall investigate Ricci scalar curvature of unitary invariant weakly complex Berwald metrics. There are two notions of Ricci scalar curvature in literature, we refer to [5–7] for more details. Definition 6.1. (See [6].) Let F be a strongly pseudoconvex complex Finsler metric defined on a domain D ⊂ Cn . Then the Ricci scalar curvature Ric of the Chern–Finsler connection associated with F is defined by Ric(z, v) = −δα (Γμ;μ )v α ,
(6.1)
˘ of the complex Berwald connection associated with F is defined by and the Ricci scalar curvature Ric ˘ Ric(z, v) = −Xα (Gμμ )v α ,
(6.2)
18
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
α where Γα ;μ and Gμ are the complex nonlinear connection coefficients of the Chern–Finsler connection and the complex Berwald connection, respectively, δα = ∂α − Γβ;α ∂˙β , Xα = ∂α − Gβα ∂˙β and the index μ is summed from 1 to n.
Theorem 6.2. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric of the Chern–Finsler connection associated with F is ∂ρ ∂ρ s− g(t)s2 (t − s) , Ric(z, v) = −r ρ + ∂t ∂s
(6.3)
where ρ = ρ(t, s) = (n − 1)d0 + f (t) + tg(t) + (s − t)d1 d4 , d0 =
φs , φ − sφs
d 1 d4 =
φ2 φss . (φ − sφs ) (φ − sφs )[φ + (t − s)φs ] + s(t − s)φss
Proof. By Proposition 3.1 in [8], we know that the complex nonlinear connection coefficients of a strongly pseudoconvex complex Finsler metric is given by 1 1 Γγ;μ = d0 z, vδμ¯γ + d1 d2 z, vz μ z γ + d3 z μ v γ + d4 (z, v)2 v μ z γ + d5 z, vv μ v γ , r r
(6.4)
where d0 =
φs , φ − sφs
d1 =
1 , (φ − sφs ){(φ − sφs )[φ + (t − s)φs ] + s(t − s)φss }
d2 = φ2 φst − sφφs (φst + φss ) − φs (φ − sφs )(φt + φs ), d3 = (φ − sφs ) φ + (t − s)φs (φt − sφst ) + s(t − s)φt φss + s2 φφs φss , d4 = φ2 φss , d5 = −sφφss [φ + (t − s)φs ]. Since F is a weakly complex Berwald metric, by (4.15) in this paper and Proposition 3.2 in [8], we get d0 + d1 d3 + d1 d5 = f (t),
d1 d2 + d1 d4 = g(t).
Thus by (6.4) and (6.5), we obtain Γμ;μ = nd0 + td1 d2 + d1 d3 + d1 d5 + d1 d4 s z, v = (n − 1)d0 + f (t) + tg(t) + d1 d4 (s − t) z, v = ρ(t, s)z, v, where we denote ρ(t, s) = (n − 1)d0 + f (t) + tg(t) + (s − t)d1 d4 . Thus we have
(6.5)
H. Xia, C. Zhong / Differential Geometry and its Applications 43 (2015) 1–20
19
Ric(z, v) = −δα¯ (Γμ;μ )v α = −(∂α¯ − Γβ;α ∂˙β¯ )(Γμ;μ )v α
! ∂ρ α β ∂ρ α z z, v + ρv − Γ;α sβ¯ z, v v α =− ∂t ∂s ! ∂ρ β ∂ρ 2 2 |z, v| + ρv − 2 G sβ¯ z, v . =− ∂t ∂s
(6.6)
By (6.4) and (6.5), we get 2Gβ = Γβ;γ v γ = f (t)z, vv β + g(t)(z, v)2 z β . Note that since sβ¯ v β = 0 and sβ¯ z β =
z,v r (t
(6.7)
− s), it follows that
2Gβ sβ¯ = g(t)s(t − s)z, v.
(6.8)
Substituting (6.8) into (6.6) yields (6.3). This completes the proof. 2 Theorem 6.3. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary ˘ of the complex Berwald connection associated invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric with F is r d (n + 1)f (t) + 2tg(t) ˘ Ric(z, v) = − s + (n + 1)f (t) + 2tg(t) . 2 dt
(6.9)
Proof. Differentiating (6.7) with respect to v μ , we obtain 2Gγμ = f (t)z μ v γ + f (t)z, vδγμ + 2g(t)z, vz μ z γ , from which we get 2Gμμ = (n + 1)f (t) + 2tg(t) z, v. Note that 2Gμμ is holomorphic with respect to v, we get 1 1 ˘ Ric(z, v) = − Xα¯ (2Gμμ )v α = − ∂α¯ (2Gμμ )v α 2 2 n d (n + 1)f (t) + 2tg(t) α 1 z z, v + (n + 1)f (t) + 2tg(t) v α v α =− 2 α=1 dt 1 d (n + 1)f (t) + 2tg(t) 2 2 |z, v| + (n + 1)f (t) + 2tg(t) v =− 2 dt r d (n + 1)f (t) + 2tg(t) 2 s + (n + 1)f (t) + 2tg(t) . =− 2 dt Corollary 6.4. Let F = rφ(t, s) be a weakly complex Berwald metric with φ given by (3.2) on a unitary ˘ invariant domain D ⊂ Cn . Then the Ricci scalar curvature Ric(z, v) = 0 if and only if f (t) = −
2 tg(t). n+1
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Acknowledgement This work is supported by NCET-13-0510; NNSFC (Grant No. 11271304, 11171277); the Fujian Province Natural Science Funds for Distinguished Young Scholar (Grant No. 2013J06001); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry; the Fundamental Research Funds for the Central Universities (Grant No. 2012121006). The authors would like to thank the referee for his/her careful reading and helpful comments. References [1] M. Abate, G. Patrizio, Finsler metrics of constant curvature and the characterization of tube domains, in: David Bao, Shiing-Shen Chern, Zhongmin Shen (Eds.), Finsler Geometry: Joint Summer Research Conference on Finsler Geometry, Seattle, Washington, July 16–20, 1995, in: Contemporary Mathematics, vol. 196, 1996, pp. 101–107. [2] M. Abate, G. Patrizio, Finsler Metrics—A Global Approach with Applications to Geometric Function Theory, Lecture Notes in Mathematics, vol. 1591, Springer-Verlag, Berlin/Aeidelberg, 1994. [3] T. Aikou, On complex Finsler manifolds, Rep. Fac. Sci., Kagoshima Univ. (Math., Phys. Chem.) 24 (1991) 9–25. [4] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. Fr. 109 (1981) 427–474. [5] G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Academic Publishers, 2004. [6] L. Sun, C. Zhong, Characterization of complex Finsler connections and weakly complex Berwald metrics, Differ. Geom. Appl. 31 (2013) 648–671. [7] C. Zhong, On real and complex Berwald connections associated to strongly convex weakly Kähler–Finsler metric, Differ. Geom. Appl. 29 (2011) 388–408. [8] C. Zhong, On unitary invariant strongly pseudoconvex complex Finsler metrics, Differ. Geom. Appl. 40 (2015) 159–186.