Math. Z. DOI 10.1007/s00209-016-1663-4

Mathematische Zeitschrift

Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans Sachiko Hamano1,2

Received: 16 May 2015 / Accepted: 15 February 2016 © Springer-Verlag Berlin Heidelberg 2016

Abstract Every planar open Riemann surface R carries the Schiffer span s(ζ ) and the harmonic span h(ζ ) with respect to a point ζ ∈ R which are closely related to the conformal 2 ) mappings. They induce the metrics s(ζ )|dζ |2 and ∂∂ζh(ζ |dζ |2 on R, respectively. We show ∂ζ that both metrics in fact coincide on R, have negative curvature everywhere and are complete. For a complex parameter t in a disk B, we study the variation of the metrics on R(t). We prove that if π : R → B is a 2-dimensional Stein manifold with π −1 (t) = R(t), t ∈ B, then 2 ) are plurisubharmonic in R. log s(t, ζ ) and log ∂ ∂ζh(t,ζ ∂ζ Keywords Metric

Pseudoconvexity · Plurisubharmonic function · Principal function · Span ·

Mathematics Subject Classification

32U05 · 30F15 · 32G08

1 Introduction and main results The purpose of this paper is to study the metric deformations induced by the Schiffer span and the harmonic span on planar open Riemann surfaces. To recall the Schiffer span in the potential theory of one complex variable, we begin with some background of the slit mapping theory. Let R be a planar Riemann surface with a finite number of C ω -smooth contours C j  Let a ∈ R and let Ua = {|z − ζ0 | < ra } ( j = 1, . . . , ν) in a larger Riemann surface R.

This work was supported by JSPS Grant-in-Aid for Scientific Research(C)15K04914.

B

Sachiko Hamano [email protected]; [email protected]

1

Department of Mathematics, Faculty of Human Development and Culture, Fukushima University, Kanayagawa, Fukushima 960-1296, Japan

2

Present Address: Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

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S. Hamano

be the local coordinate of a neighborhood Va of a in R. We denote by P (R) the set of all 1 univalent functions P(z) on R with the given singularity z−ζ at ζ0 and normalized so that 0

lim z→ζ0 (P(z) −

1 z−ζ0 )

= 0, namely, P(z) has the following expression ∞

P(z) =

 1 +0+ An (z − ζ0 )n z − ζ0

at ζ0 .

(1)

n=1

For w = P(z) ∈ P (R) let E P be the Euclidean area of Cw \P(R), and set E (R) = sup{E P | P(z) ∈ P (R)}.

Koebe constructed two special functions P0 (z) and P1 (z) in P (R) which are the horizontal slit mapping and the vertical slit mapping for (R, a) with respect to Ua (simply, for (R, ζ0 )), respectively. Grunsky introduced the functions P0 (z) ± P1 (z), and studied their extremal properties. Schiffer discovered that the quantity s := A01 − A11 is always real and nonnegative,  1 i n where Pi (z) = z−ζ + ∞ n=1 An (z − ζ0 ) at ζ0 (i = 0, 1). We call s the Schiffer span for 0 (R, a) with respect to Ua (simply, for (R, ζ0 )). Schiffer [12] proved the following: Proposition 1 Let P0 (z) (resp. P1 (z)) be the horizontal (resp. vertical) slit mapping for (R, ζ0 ), and set M(z) = 21 (P0 (z)+ P1 (z)). Then (i) M(z) ∈ P (R), (ii) Cw \M(R) consists of ν convex domains, and (iii) the complementary area E (R) = E M = π2 s. Here s is the Schiffer span for (R, ζ0 ). Further, the function satisfying (i), (ii), and (iii) is uniquely determined. In this paper we call M(z) the maximizing function of the Schiffer span s for (R, ζ0 ). By the standard approximation argument we define the Schiffer span and the maximizing function of Schiffer span for any planar Riemann surface (to be precise, see [1, §4 Chap.III]). To state our results we shall first change the normalization (1) as follows: We fix a point b ∈ R, and choose any a ∈ R\{b}. Let Ua = {|z − ζ0 | < ra } be the local coordinate of a neighborhood Va of a in R. We denote by Pb (R) the set of all univalent functions P(z) on 1 R such that P(b) = 0 and P(z) has the given singularity z−ζ at ζ0 , namely, 0 ⎧ ⎨ P(b) = 0, ⎩ P(z) =

1 z−ζ0

+ B0 +

∞ 

Bn (z − ζ0 )n

at ζ0 .

n=1

For the slit mapping Pi (z) (i = 0, 1) in P (R) with the normalization (1), we consider the function Pib (z) := Pi (z) − Pi (b) (i = 0, 1) and set Mb (z) =

1 b (P (z) + P1b (z)), 2 0

so that Mb (z) is uniquely determined. Since the Schiffer span s is invariant under the parallel translations, we have: (i) Mb (z) ∈ Pb (R), (ii) Cw \Mb (R) consists of ν convex domains, and (iii) E (R) = E Mb = π2 s. From the uniqueness of the maximizing function of s we have Mb (z) = M(z) − M(b)

on R,

(2)

where M(z) is in Proposition 1 satisfying (1). Now we shall move the singular point ζ0 in the fixed local coordinate Ua , and regard Mb (z, ζ ) as a function of the two complex variables (z, ζ ) in R × Ua . Our first result is as follows:

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Theorem 1 Let the notation be as above. Then the maximizing function Mb (z, ζ ) = 1 b b 2 (P0 (z, ζ ) + P1 (z, ζ )) of the Schiffer span for (R, ζ ) is meromorphic as a function of 1 the two complex variables (z, ζ ) in R × Ua with the pole { z−ζ | ζ ∈ Ua }, i.e., Mb (z, ζ ) is holomorphic as a function of the two complex variables (z, ζ ) in [R ×Ua ]\{z = ζ | ζ ∈ Ua }. Remark 1 We note that P0b (z, ζ ) and P1b (z, ζ ) are not holomorphic for ζ . We give such  := an example in Sect. 4. Theorem 1 means that R := R × Ua is biholomorphic to R {Mb (R, ζ ) | ζ ∈ Ua } by the holomorphic transformation T : (z, ζ ) ∈ R × Ua  → (w, ζ ) = (Mb (z, ζ ), ζ ) ∈ Pw × Ua .  := {∂ Mb (R, ζ ) | ζ ∈ Ua } is Levi flat in Cw × Ua . Thus ∂ R We shall move ζ in the fixed local coordinate Ua , and regard the Schiffer span s(ζ ) for (R, ζ ) as a function on Ua . Proposition 1(iii) implies that the Schiffer span s(ζ ) induces the metric s(ζ )|dζ |2 on R. Next, we shall direct our attention to the case of two logarithmic singularities with opposite residues. We take two distinct points a1 and a2 on R, and let Ua1 = {|z| < r1 } and Ua2 = {|z − ζ0 | < r2 } be the fixed local coordinates of neighborhoods Vak of ak (k = 1, 2) in R, respectively (where Ua1 and Ua2 have no relations). Among all harmonic functions q(z) on R\{a1 , a2 } with two logarithmic poles of − log |z| at 0 and log |z − ζ0 | at ζ0 normalized so that lim z→0 (q(z) + log |z|) = 0, there uniquely exist two functions qi (z) (i = 1, 0) with the L i -boundary conditions (i = 1, 0): for each C j ( j = 1, . . . , ν),  ∂q1 (z) (L 1 ) q1 (z) = c j (constant) on C j and dsz = 0; C j ∂n z (L 0 )

∂q0 (z) = 0 on C j . ∂n z

We call qi (z) the L i -principal function for (R, a1 , a2 ) with respect to the local coordinates Ua1 and Ua2 (simply, for (R, 0, ζ0 )). The constant term βi := lim z→ζ0 (qi (z) − log |z − ζ0 |) is called the L i -constant for (R, 0, ζ0 ), and h := β1 − β0 is called the harmonic span for (R, 0, ζ0 ). Here we shall fix one singular point 0 and move the other singular point ζ0 in the fixed Ua2 . By putting at 0, − log |z| + u 0 (z, ζ ) (3) qi (z, ζ ) := qi (z, 0, ζ ) = log |z − ζ | + βi (ζ ) + u ζ (z, ζ ) at ζ, where u 0 (z, ζ ) and u ζ (z, ζ ) are harmonic near 0 and ζ , respectively, and u 0 (0, ζ ) = u ζ (ζ, ζ ) = 0 on Ua2 , we can regard the L i -principal function qi (z, ζ ) (i = 1, 0) as a function of the two complex variables (z, ζ ) in (R\{a1 , a2 }) × Ua2 . Similarly, the L i -constant βi (ζ ) and the harmonic span h(ζ ) = β1 (ζ ) − β0 (ζ ) for (R, 0, ζ ) can be regarded as functions of ζ in Ua2 . From the definition of harmonic spans and the property of logarithmic functions, h(ζ ) does not depend on the local coordinate Ua2 , which is a function on R\{a1 }: for every holomorphic transition function ζ = φ(ξ ) on the intersection U ,  h(ξ ) := h(φ(ξ )) = h(ζ ). It h(ξ ) ∂ 2 h(ζ ) ∂ 2 1 automatically implies that ∂ζ ∂ζ = ∂ξ ∂ξ |φ  (ξ )|2 on every U in R\{a1 }. By using Lemma 4 in Sect. 3 based on the variational formula of second order of h(ζ ) (see Sect. 2 and [9, § 4]), 2 2 2 ) we can show ∂∂ζh(ζ > 0 on every U in R\{a1 }. Putting ∂ζ∂ ∂hζ¯ (a1 ) := limζ →0 ∂ζ∂ ∂hζ¯ (ζ ), we ∂ ζ¯ can show it is positive (see [9, (3.10)]), so that

∂ 2 h(ζ ) ∂ζ ∂ ζ¯

is a positive function on every U in R.

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Thus, the harmonic span h(ζ ) also induces the metric H (ζ )|dζ |2 := bordered Riemann surface R. How are they related? Our second result is as follows:

∂ 2 h(ζ ) |dζ |2 ∂ζ ∂ ζ¯

on a planar

Theorem 2 Let R be a planar bordered Riemann surface. Let s(ζ )|dζ |2 and H (ζ )|dζ |2 = ∂ 2 h(ζ ) |dζ |2 be the metric on R induced by the Schiffer span s(ζ ) and the harmonic span ∂ζ ∂ ζ¯ h(ζ ), respectively. Then (i) s(ζ )|dζ |2 and H (ζ )|dζ |2 are identical on R. (ii) s(ζ )|dζ |2 (= H (ζ )|dζ |2 ) is of negative curvature at every point ζ ∈ R. (iii) s(ζ )|dζ |2 (= H (ζ )|dζ |2 ) is complete on R. Remark 2 Let D = {|z| < r } and let ζ ∈ D. We computed the Schiffer span s(ζ ) for (D, ζ ) in [7, § 4]; s(ζ ) = r 2 {1−(|ζ2 |/r )2 }2 . By the computation in [9, (3.9)] we got the harmonic span h(ζ ) for (D, 0, ζ ); h(ζ ) = −2 log(1 − (|ζ |/r )2 ). Thus we exactly have H (ζ ) := s(ζ )|dζ |2

H (ζ )|dζ |2

∂ 2 h(ζ ) ∂ζ ∂ζ

= s(ζ ),

and then the metrics and are equal to the Poincaré metric and the Bergman metric up to multiplicative constants. In the case when D is not a simply connected domain, s(ζ )|dζ |2 is different from them in view of the reproducing kernels as we see later in Sect. 3.  → B be a Finally, we shall introduce one complex parameter t to the subject. Let π : R  is a 2-dimensional complex manifold, π is a holomorphic holomorphic family such that R  = π −1 (t), t ∈ B is irreducible and  onto a disk B in Ct , and each fiber R(t) projection from R  . We set R  = ∪t∈B (t, R(t)). non-singular in R Let R = ∪t∈B (t, R(t)) be a subdomain with   R(t)  = ∅ for t ∈ B, R(t)  such that R(t) C ω smooth boundary ∂ R = ∪t∈B (t, ∂ R(t)) in R   consists of a finite number is a planar bordered Riemann surface in R(t), and ∂ R(t) in R(t) ω of C smooth contours C j (t) ( j = 1, . . . , ν). Then each R(t) carries the metric s(t, ζ )|dζ |2 , ) i.e., H (t, ζ )|dζ |2 := ∂ ∂ζh(t,ζ |dζ |2 , induced by the Schiffer span s(t, ζ ) and the harmonic ∂ζ span h(t, ζ ). Here is our result in two complex variables: 2

Theorem 3 Let s(t, ζ )|dζ |2 be the metric induced by the Schiffer span s(t, ζ ) for (R(t), ζ ). . Then log s(t, ζ ), We assume that the total space R = ∪t∈B (t, R(t)) is pseudoconvex in R namely log H (t, ζ ), is plurisubharmonic on R. Remark 3 This is the same phenomenon as the plurisubharmonic variation of the Bergman metric K (t, ζ )|dζ |2 under pseudoconvexity, which was showed by Maitani-Yamaguchi in [11, Corollary 4.1]. Berndtsson [2] extended it to the higher dimension case, which recently led to beautiful and strong results by using the variation of K (t, ζ ) (see Guan-Zhou [4] and Berndtsson-Lempert [3]). Although Theorems 1 and 2 are the results in one complex variable, the approach to these turns out to be by several complex variables. The key tools for the proofs of Theorems in this paper are the new identities of Lemmas 3 and 4, which are based on the variational formulas for the Schiffer and harmonic spans in Sect. 2 (for more details, see [6,8,9])) and pseudoconvexity.

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2 Variational formulas for Schiffer and harmonic spans To prove Lemmas 3 and 4, we recall the variational formulas for Schiffer and harmonic  → B be a holomorphic family of Riemann spans. Let B be a disk in Ct . Let π : R  = π −1 (t), t ∈ B such that R(t)  is irreducible and non-singular in R , and surfaces R(t)



  = t∈B (t, R(t)).  satisfies the following set R If a subdomain R = t∈B (t, R(t)) in R conditions:   R(t)  = ∅ for t ∈ B, and R(t) is a connected planar Riemann surface such that (I) R(t)  consists of a finite number of C ω smooth contours C j (t) ( j = 1, . . . , ν); ∂ R(t) in R(t)

 is C ω smooth and ∂ R is transverse to (II) the boundary ∂ R = t∈B (t, ∂ R(t)) of R in R  each fiber R(t), t ∈ B,  by regarding R as the variation R : t ∈ B → then we say that R is a smooth variation in R R(t). Each C j (t) is oriented by ∂ R(t) = C1 (t) + · · · + Cν (t). For a C 2 defining function ϕ(t, z) of ∂ R, we set  2

2 2  −3 ∂ϕ ∂ ϕ ∂ϕ ∂ ϕ ∂ϕ ∂ϕ ∂ 2 ϕ ∂ϕ 2 − 2Re + k2 (t, z) = ∂t∂ t¯ ∂z ∂ t¯∂z ∂t ∂ z¯ ∂t ∂z∂ z¯ ∂z on ∂ R. Note that k2 (t, z) does not depend on the choice of defining functions ϕ(t, z) of ∂ R (see [10, (1.2)]). We see that R is a smooth variation if and only if there exists a C ω defining function ϕ(t, z) of ∂ R such that ∂ϕ ∂z  = 0. Assume that there exists a section a := {a(t) ∈ R(t) | t ∈ B} of R over B. Let V := B × {|z − ζ | < r } be a π-local coordinate of a neighborhood U of a in R such that a corresponds to B × {ζ }. Let t ∈ B be fixed. Then among all harmonic functions u(t, z) on 1 1 R(t)\{a(t)} with singularity z−ζ at a(t) normalized so that lim z→ζ (u(t, z) − z−ζ ) = 0, there are uniquely determined functions pi (t, z) (i = 1, 0) with the boundary conditions (L i ): for 1 ≤ j ≤ ν,  ∂ p1 (t, z) dsz = 0; (L 1 ) p1 (t, z) = c j (t) (constant) on C j (t) and ∂n z C j (t) ∂ p0 (t, z) = 0 on C j (t). ∂n z    1 i n at ζ (i = 1, 0). p (t, z) and α (t) := + ∞ We have pi (t, z) = z−ζ i i n=1 An (t)(z − ζ ) (L 0 )

{Ai1 (t)} are called the L i -principal function and the L i -constant for (R(t), a(t)) with respect to the local coordinate {|z − ζ | < r }, respectively (simply, for (R(t), ζ (t))). We established in [8, (4.1)] the following variational formula for the Schiffer span s(t) = α0 (t) − α1 (t) for (R(t), ζ (t)): Lemma 1

  ∂ p1 (t, z) 2 ∂ p0 (t, z) 2 + k2 (t, z) ∂z dsz ∂z ∂ R(t)    ∂ 2 p1 (t, z) 2 ∂ 2 p0 (t, z) 2 4 + ∂t∂z + ∂t∂z d xd y. π R(t)

∂ 2 s(t) 1 = ∂t∂t π



Moreover, we assume that R(t) {0, ζ (t)} and z = ζ (t)( = 0) is holomorphic on B. Then R(t), t ∈ B carries the L i -principal function qi (t, z) for (R(t), 0, ζ (t)) (i = 1, 0). In

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S. Hamano

[9, Lemma 4.1] the variational formula for the harmonic span h(t) for (R(t), 0, ζ (t)) was proved. We apply it to the variation of planar Riemann surfaces, and have: Lemma 2

  ∂q1 (t, z) 2 ∂q0 (t, z) 2 dsz k2 (t, z) + ∂z ∂z ∂ R(t)    ∂ 2 q1 (t, z) 2 ∂ 2 q0 (t, z) 2 4 + ∂t∂z + ∂t∂z d xd y. π R(t)

∂ 2 h(t) 1 = ∂t∂t π



3 The relation between principal functions and reproducing kernels Let R be a planar bordered Riemann surface with a finite number of C ω -smooth con Let A(R) be the space of tours C j ( j = 1, . . . , ν) in a larger Riemann surface R. L 2 -analytic differentials on R. For f (z)dz, g(z)dz ∈ A(R), we define the inner product  by ( f dz, gdz) R = R f gd xdy, so that A(R) is a Hilbert space. For a point a ∈ R and the fixed local coordinate Ua = {|z − ζ | < ra }, we consider the following bounded linear functional K on A(R): K : A(R) f (z)dz  → f (ζ ) ∈ C. It follows from the Riesz representation theorem that there uniquely exists a reproducing kernel function K (z, ζ )dz ∈ A(R) such that  f (ζ ) = f (z)K (z, ζ )d xd y for every f (z)dz ∈ A(R) R

(see [1, (44)]). We thus have the Bergman kernel K (ζ, ζ ) and the Bergman metric ds 2 := K (ζ, ζ )|dζ |2 on R. We consider the Green function g(z, ζ ) for (R, ζ ), i.e., g(z, ζ ) is harmonic on R\{ζ } 1 with pole at ζ such that g(z, ζ ) − log |z−ζ | is harmonic near ζ , and g(z, ζ ) continuously vanishes on ∂ R. λ(ζ ) := lim z→ζ (g(z, ζ ) + log |z − ζ |) is called the Robin constant for (R, ζ ). Schiffer [13, Formula (14)] and Suita [14, Theorem 2] showed the following: K (z, ζ ) = −

2 ∂ 2 g(z, ζ ) (by Schiffer), π ∂z∂ζ

K (ζ, ζ ) = −

1 ∂ 2 λ(ζ ) (by Suita). π ∂ζ ∂ζ

We shall consider thesubspace S(R) of A(R) consisting of semi-exact L 2 -analytic differentials on R, namely, γ f dz = 0 for every dividing cycle γ in R. Then S(R) is a closed subspace of A(R). For a ∈ R and Ua = {|z − ζ | < ra }, we consider the following bounded linear functional L on S(R): L : S(R) f (z)dz  → f  (ζ ) ∈ C.  Then  there exists a unique reproducing kernel function L(z, ζ )dz ∈ S(R) such that f (ζ ) = f (z)L(z, ζ )d xd y for every f (z)dz ∈ S(R), so that we have the reproducing kernel R  ∂L ∂L ∂L 2 2 R |L(z, ζ )| d xd y > 0, which induces the metric ∂z (ζ, ζ )|dζ | ∂z (ζ, ζ ) on R: ∂z (ζ, ζ ) = on R. We obtain the useful identities:

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Log-plurisubharmonicity of metric deformations

Lemma 3 (The L-kernel representation by the Schiffer span) 2 ∂ 2 p1 (z, ζ ) 2 ∂ 2 p0 (z, ζ ) = . π ∂z∂ζ π ∂z∂ζ ∂L 1 ∂ 2 s(ζ ) . (ζ, ζ ) = ∂z 2π ∂ζ ∂ζ L(z, ζ ) = −

(4) (5)

Here pi (z, ζ ) (i = 1, 0) is the L i -principal function for (R, ζ ), and s(ζ ) is the Schiffer span for (R, ζ ). In view of the orthogonal relations A(R) = S(R) ⊕ S(R)⊥ , let  L denote the restricted Bergman kernel K on A(R) to S(R):  L : S(R) f (z)dz  → f (ζ ) ∈ C, which induces the metric  L(ζ, ζ )|dζ |2 on R. We show the following: Lemma 4 (The  L-kernel representation by the harmonic span) 2 ∂ 2 q1 (z, ζ ) 2 ∂ 2 q0 (z, ζ )  L(z, ζ ) = =− . π ∂z∂ ζ¯ π ∂z∂ ζ¯ 1 ∂ 2 h(ζ )  . L(ζ, ζ ) = 2π ∂ζ ∂ ζ¯

(6) (7)

Here qi (z, ζ ) (i = 1, 0) is the L i -principal function for (R, 0, ζ ), and h(ζ ) is the harmonic span for (R, 0, ζ ). By the standard use of the immersion theorem for open Riemann surfaces due to Gunning to an unramified domain D  Narasimhan [5], we have a biholomorphic mapping T from R  over Cz . We may assume that for a domain R  R, D := T (R) is the unramified domain  over Cz with a finite number of C ω smooth contours C j (1  ≤ j ≤ ν) in D. It suffices to show j ∼ Lemmas 3 and 4 for D and S(D) = { f (z)dz ∈ A(D) | Cj f (z)dz = 0 (1 ≤ j ≤ ν), C C j in D}. Proof of (4) in Lemma 3 For any fixed ζ ∈ D, the L i -principal function pi (z, ζ ) (i = 1, 0) for (D, ζ ) is determined. Since ∂ D is of class C ω on D, we note pi (z, ζ ) ∈ C ω (D × D\{z = ζ }). Let f dz ∈ S(D) := S(D) ∩ C ω (D). Since f is a holomorphic function on D and ∂ 2 p1 ∈ C ω (D), we see ∂z∂ζ  I := D

  2    ∂ p1 ∂ p1 1 f (z) dz f (z)dz . d xd y = 2i ∂ζ ∂z∂ζ D

Since p1 has a pole only at ζ , it follows from Green’s formula that    ∂ p1 1 dz I = lim f (z)dz 2i →0 D−U (ζ ) ∂ζ ⎛ ⎞   ν  ∂ p ∂ p 1 1 1 = lim ⎝ f (z)dz − f (z)dz ⎠ , 2i →0 C j ∂ζ ∂U (ζ ) ∂ζ j=1

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S. Hamano

where U (ζ ) = {|z − ζ | < }. Since p1 is a constant c j (ζ ) on each C j which is independent of ζ and since f dz ∈ S(D), we see that   ∂c j (ζ ) ∂ p1 f (z)dz = 0. f (z)dz = ∂ζ C j ∂ζ Cj 1 of p1 we have that From the pole z−ζ

∂ p1 ∂ = ∂ζ ∂ζ

  1 1 1 1 + u(z, ζ ) at ζ, + + u(z, ζ ) = 2 z−ζ 2(z − ζ )2 z−ζ

where u is regular near ζ . Letting ε → 0, we have by the Cauchy theorem that 

  1 f (z) π I = lim 0 − + u(z, ζ ) f (z) dz = − f  (ζ ). 2 2i →0 2 |z−ζ |= 2(z − ζ ) Further, since each C j is a closed curve in D which does not depend on ζ and since p1 has flux 0 along C j , we have 

∂ 2 p1

Cj

∂z∂ζ

dz =

∂ ∂ζ

 Cj

   1 ∂ p1 i ∂ ∂ p1 d p1 + i dsz = dsz = 0, 2 ∂n z 2 ∂ζ C j ∂n z

∂ p1 p1 (z,ζ ) dz ∈ S(D). Thus, we see that L(z, ζ ) = − π2 ∂ ∂z∂ζ . so that ∂z∂ζ We remark that the harmonic conjugate p0∗ (z, ζ ) of the L 0 -principal function p0 (z, ζ ) satisfies the L 1 -condition on each C j and has the following pole:   1 1 1 ∗ p0 (z, ζ ) = + + (regular part) at ζ, 2i z − ζ z−ζ 2

so that

2

∂ p0∗ ∂ζ

 D

−

1 2i(z−ζ )2

is regular at ζ . By using

∂ p0∗ ∂z

= i ∂∂zp0 on D, we have

⎛ ⎞  2   ν   ∂ p0∗ ∂ p0∗ ∂ p0 1 f (z) d xd y = − lim ⎝ f dz − f dz ⎠ 2 →0 ∂ζ ∂ζ ∂z∂ζ C |z−ζ |= j j=1  f (z) π 1 dz = f  (ζ ). = 2 |z−ζ |= 2i(z − ζ )2 2

 2p 0 From the L 0 -condition of p0 : ∂∂np0z = 0 on each closed curve C j , we have C j ∂∂z∂ζ dz =  2 ∂ p0 1 ∂ 2 ∂ p0 (z,ζ ) 2 ∂ζ C j ∂n z dsz = 0. Therefore, L(z, ζ ) = π ∂z∂ζ , so that the proof of (4) is completed.   Proof of (5) in Lemma 3 Let pi (z, ζ ) (i = 1, 0) and s(ζ ) be the L i -principal function and the Schiffer span for (D, ζ ), respectively. We consider the following variation of domains: D:

ζ ∈ D → D(ζ ) := {w = z − ζ | z ∈ D} ⊂ Cw ,

and set p˜i (w, ζ ) (i = 1, 0) and s˜ (ζ ) the L i -principal function and the Schiffer span for (D(ζ ), 0), respectively. Then we apply Lemma 1 for s˜ (ζ ) to the Levi flat D, i.e., k2 (w, ζ ) = 0 on ∂ D, so that

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∂ 2 s˜ (ζ ) ∂ζ ∂ζ

=

4 π



 2  ∂ 2 p˜ 0 2 ∂ p˜ 1 2 | +| | dudv. | ∂w∂ζ ∂w∂ζ D(ζ )

Since pi (z, ζ ) = p˜ i (w, ζ ) for w = z − ζ , we have ∂ 2 p˜ i (w, ζ ). ∂w∂ζ

ζ, ζ ) =

∂ 2 pi (z, ζ ) ∂z∂ζ

=

(8)

∂ 2 p˜ i ∂ 2 p˜ i (z − ζ, ζ ) − ∂w∂w (z − ∂w∂ζ

From Proposition 1(iii), (8) is equivalent to the following: ∂ 2 s(ζ ) ∂ζ ∂ζ

Combing with (4), we have

4 π

=

∂ 2 s(ζ ) ∂ζ ∂ζ

   2 ∂ 2 p0 2 ∂ p1 2 | +| | d xd y. | ∂z∂ζ ∂z∂ζ D = 2π

 D

|L(z, ζ )|2 d xd y = 2π ∂∂zL (ζ, ζ ).

 

Proof of (6) in Lemma 4 For any fixed ζ ∈ D\{0}, the L i -principal function qi (z, ζ ) (i = 1, 0) for (D, 0, ζ ) is determined. We note qi (z, ζ ) ∈ C ω (D × D\{z = 0, ζ }). Since q1 has two poles at ζ and 0, we see for f dz ∈ S(D),    2   ∂q1 ∂ q1 1 f (z) dz d xd y = lim f (z)dz 2i →0 D−U (ζ )−U (0) ∂ζ ∂z∂ζ D ⎛ ⎞   ν  ∂q1 ∂q1 1 = lim ⎝ f (z)dz − f (z)dz ⎠ , 2i →0 ∂ζ Cj ∂U (ζ )+∂U (0) ∂ζ 

J :=

j=1

where U (ζ ) = {|z − ζ | < } and U (0) = {|z| < }. By the L 1 -condition of q1 and ∂q1 1 1 f dz ∈ S(D), C j ∂q ∂ζ f dz = 0 holds. From the pole (3) of q1 , ∂ζ + 2(z−ζ ) is regular at ζ and

∂q1 ∂ζ

is regular at 0. Letting ε → 0, we have by the Cauchy theorem that J =−

1 2i

 |z−ζ |=

− f (z) π dz = f (ζ ). 2(z − ζ ) 2

∂ q1 q1 (z,ζ ) Since ∂z∂ζ dz ∈ S(D), we have  L(z, ζ ) = π2 ∂ ∂z∂ζ . ∗ Similarly, the harmonic conjugate q0 (z, ζ ) of q0 (z, ζ ) satisfies the L 1 -condition on C j and has the pole at ζ and 0 such that 2

2

q0∗ (z, ζ ) =



arg(z − ζ ) + b0 (ζ ) + u ∗ζ (z, ζ ) at ζ, at 0, − arg z + u ∗0 (z, ζ )

where u ∗ζ and u ∗0 are harmonic near ζ and 0, respectively, and u ∗ζ (ζ, ζ ) = u ∗0 (0, ζ ) = 0.  We note that q0∗ is not always single-valued on D (mod2π) but C j dq0∗ (z, ζ ) = 0. Thus,  ∂q ∗ ∂q ∗ for the fixed C j (1 ≤ j ≤ ν) which is independent of ζ, C j d ∂ζ0 = 0 holds, i.e., ∂ζ0 is a single-valued function on D. Since

∂q0∗ ∂ζ

=

1 ∂q0 i ∂ζ

on D, it follows that

∂q0∗ ∂{arg(z − ζ )} = + (regular part) ∂ζ ∂ζ −1 = + (regular part) at ζ 2i(z − ζ )

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S. Hamano

and

∂q0∗ ∂ζ

is regular at 0. We analogously have

 D

Since

⎛ ⎞  2   ν   ∂q0∗ ∂q0∗ ∂ q0 1 f (z) f dz − f dz ⎠ d xd y = − lim ⎝ 2 →0 ∂ζ ∂z∂ζ U (ζ )+U (0) ∂ζ j=1 C j  π − f (z) 1 dz = − f (ζ ). = 2 |z−ζ |= 2i(z − ζ ) 2

∂ 2 q0 dz ∂z∂ζ

∈ S(D), we have  L(z, ζ ) = − π2

∂ 2 q0 , ∂z∂ζ

so that (6) is proved.

 

Proof of (7) in Lemma 4 We shall use the definition of the harmonic span instead of the geometrical meaning of the Schiffer span in the proof of (5). Let qi (z, ζ ), βi (ζ ) (i = 1, 0), and h(ζ ) be the L i -principal function, the L i -constant, and the harmonic span for (D, 0, ζ ), respectively. We consider the variation D : ζ ∈ D → D(ζ ) := {z −ζ ∈ Cw | z ∈ D}, and set q˜i (w, ζ ) and β˜i (ζ ) (i = 1, 0) the L i -principal function and the L i -constant for (D(ζ ), −ζ, 0), respectively. To change the normalization we set qˆi (w, ζ ) = −q˜i (w, ζ ) + β˜i (ζ ) (i = 1, 0), so that qˆi (w, ζ ) is the L i -principal function for (D(ζ ), 0, −ζ ) with the L i -constant βˆi (ζ ), ˆ ) is equal to h(ζ ). We note that which is the same as βi (ζ ), and hence the harmonic span h(ζ ∂ 2 qˆi ∂ 2 qi ∂ 2 qi ∂ 2 qi ˆ ) for (D(ζ ), 0, −ζ ) under the Levi = ∂z∂z − ∂z∂ζ = − ∂z∂ζ . Applying Lemma 2 to h(ζ ∂w∂ζ flat variation D, we have    ˆ ) ∂ 2 qˆ1 2 ∂ 2 qˆ0 2 ∂ 2 h(ζ 4 ∂ 2 h(ζ ) = = ∂w∂ζ + ∂w∂ζ dudv π ∂ζ ∂ζ ∂ζ ∂ζ D(ζ )     ∂ 2 q1 2 ∂ 2 q0 2 4 = − ∂z∂ζ + − ∂z∂ζ d xd y. π D By using (6) we have

∂ 2 h(ζ ) ∂ζ ∂ζ

= 2π  L(ζ, ζ ), which is desired.

 

4 Proof and example of Theorem 1 Proof of Theorem 1 We may assume that R is biholomorphic to an unramified domain D over Cz . By (4) we see that   ∂ P1 + P0 ∂ 2 p1 ∂ 2 p0 ∂ ∂ ∂M 0≡ + = = 2 ∂ζ ∂z ∂ζ ∂z ∂ζ ∂z ∂ζ ∂z on (D × D)\{z = ζ }. Thus, 1 (∗) ∂∂zM is meromorphic for (z, ζ ) on D × D with the pole − (z−ζ . )2

Fix an arbitrary point ζ ∈ D\{b}. For z ∈ D\{ζ } we take a piecewise C 1 -curve C(z) in D ) connecting b and z, and set I (z, ζ ) := C(z) ∂ M(ξ,ζ dξ. This is invariant under the homotopy ∂ξ of C(z). Since C(z) is independent of ζ , it follows from (∗) that I (z, ζ ) is holomorphic for ζ , and hence I (z, ζ ) is holomorphic for (z, ζ ) in [D × (D\{b})]\{z = ζ }. On the other hand, we have I (z, ζ ) = M(z, ζ ) − M(b, ζ ) = Mb (z, ζ ) by (2), which implies the assertion.   Here we shall give the example of Theorem 1.

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Log-plurisubharmonicity of metric deformations

Example 1 Let D = {|z| < 1} and b = 0 in Theorem 1. For any fixed ζ ∈ D the vertical 1 slit mapping P1 and horizontal slit mapping P0 for (D, ζ ) are written as P1 = z−ζ − z−ζ , (1−ζ z)(1−|ζ |2 )

P0 =

1 z−ζ

+

z−ζ . (1−ζ z)(1−|ζ |2 )

Then, for ζ  = 0 we have

  z−ζ 1 ζ 1 − , + − = z−ζ ζ 1 − |ζ |2 (1 − ζ z)(1 − |ζ |2 )   1 ζ z−ζ 1 , + + + P00 (z, ζ ) = z−ζ ζ 1 − |ζ |2 (1 − ζ z)(1 − |ζ |2 ) P 0 + P10 1 1 M0 (z, ζ ) = 0 = + . 2 z−ζ ζ P10 (z, ζ )

Thus Pi0 (i = 0, 1) is not holomorphic for ζ on D, but M0 is certainly meromorphic for (z, ζ ) on D × (D\{0}).

5 Proof of Theorem 2 To prove Theorem 2 we prepare the following propositions: Proposition 2 Let the notation be as in Sect. 3. ∂ 3 q1 (z, ζ ) ∂z∂ζ

2

=−

∂ 2 p1 (z, ζ ) ∂z∂ζ

=

∂ 2 p0 (z, ζ ) ∂z∂ζ

=−

∂ 3 q0 (z, ζ ) ∂z∂ζ

2

.

Proof Under the same notation in Lemmas 3 and 4, for every ζ ∈ D, (4) says f  (ζ ) =

2 π



and (6) says 2 f (ζ ) = π

D

 D

  2 ∂ p1 (z, ζ ) d xd y, f (z) − ∂z∂ζ

(9)

  2 ∂ q1 (z, ζ ) f (z) d xd y, ∂z∂ζ

(10)

for every f dz ∈ S(D). Now we differentiate both sides of (10) by ζ , and obtain 2 f (ζ ) = π 





f (z) D

∂ 3 q1 (z, ζ ) ∂z∂ζ

 d xd y.

2

∂ q1 dz ∈ S(D), it follows that Since the contour C j does not depend on ζ and since ∂z∂ζ   ∂ 3 q1 2 3 ∂ q1 ∂ q1 ∂ 2 dz = ∂ζ C j ∂z∂ζ dz = 0, i.e., 2 dz ∈ S(D). From the uniqueness of L(z, ζ ) Cj 2

∂z∂ζ

∂z∂ζ

we see the first equality. Similarly, we can show the third equality. By (4) we have the second equality.   Proposition 3 Let the notation be as in Sect. 3. ∂ 3 p1 (z, ζ ) ∂z∂ζ ∂ζ

=

∂ 3 p0 (z, ζ ) ∂z∂ζ ∂ζ

=

∂ 3 q1 (z, ζ ) ∂z∂ζ ∂ζ

=

∂ 3 q0 (z, ζ ) ∂z∂ζ ∂ζ

≡ 0.

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S. Hamano

Proof We differentiate both sides of (9) by ζ , and obtain  ∂ 3 p1 (z, ζ ) 2 d xd y for ζ ∈ D. f (z) 0=− π ∂z∂ζ ∂ζ D

(11)

∂ p1 ∂ p1 Since C j does not depend on ζ and since ∂z∂ζ dz ∈ S(D), ∂z∂ζ dz ∈ S(D). Substituting ∂ζ 2  3 3 ∂ p1 ∂ p1 for f in (11), we have 0 = D ∂z∂ζ d xd y, so that the assertion for p1 follows. ∂z∂ζ ∂ζ ∂ζ 2

3

Then, it holds for p0 by (4). Similarly, we differentiate both sides of (10) by ζ , and have  ∂ 3 q1 0 = π2 D f ∂z∂ζ d xd y for ζ ∈ D. Using the same idea as above and (6), we have ∂ζ ∂ 3 q1 ∂z∂ζ ∂ζ

=

∂ 3 q0 ∂z∂ζ ∂ζ

≡ 0.

 

Proof of Theorem 2 (i) By Lemma 3 and Proposition 2, we see 2  2   3 ∂ p1 (z, ζ ) 2 8 8 ∂ 2 s(ζ ) ∂ q1 (z, ζ ) = d xd y = d xd y. (12) ∂z∂ζ 2 π π ∂ζ ∂ζ D D ∂z∂ζ  ∂ 2 q1 (z,ζ ) 2 2 ) 8 = By Lemma 4 we have H (ζ ) := ∂∂ζh(ζ D ∂z∂ζ d xd y, and differentiate both π ∂ζ

sides by ζ and ζ , and see by Proposition 3 that    3 ∂ H (ζ ) ∂ q1 ∂ 2 q1 8 ∂ 2 q1 ∂ 3 q1 = + d xd y 2 π ∂ζ ∂z∂ζ ∂z∂ζ ∂ζ D ∂z∂ζ ∂z∂ζ  ∂ 3 q1 ∂ 2 q1 8 d xd y, = 2 π D ∂z∂ζ ∂z∂ζ   3 2 8 ∂ 2 H (ζ ) ∂ q1 = d xd y. 2 π ∂ζ ∂ζ D ∂z∂ζ By (12) we obtain ∂ 2 H (ζ ) ∂ζ ∂ζ

=

∂ 2 s(ζ ) ∂ζ ∂ζ

for ζ ∈ D,

(13)

so that H (ζ ) − s(ζ ) is harmonic on D. Let w = f (z) be an arbitrary univalent function on D such that f (0) = 0, and set  = f (D) and ξ = f (ζ ). Setting the L i -principal function  D pi (w, ξ ) (i = 1, 0) and the  ξ ), and the L i -principal function  Schiffer span  s(ξ ) for ( D, qi (w, ξ ) (i = 1, 0) and the  0, ξ ), we have harmonic span  h(ξ ) for ( D, (ξ ) ∂2 H ∂ξ ∂ξ

:=

∂ 4 h(ξ ) 2

∂ξ ∂ξ 2

=

∂ 2 s(ξ ) ∂ξ ∂ξ

,

 for ξ ∈ D.

(14)

2  Since two metrics satisfy the following relation:  s(ξ ) = s(ζ ) dζ dξ and H (ξ ) = 4 2 2 2 ∂  s(ξ ) ∂ s(ζ ) dζ , where ξ = f (ζ ), we differentiate them by ξ and ξ : = H (ζ ) dζ + ∂ξ ∂ζ ∂ζ dξ ∂ξ dξ        2  2 2 4 2 d 2 ζ ∂ 2 H(ξ ) ) dζ d2ζ ∂ 4 h(ζ ) dζ ∂ 3 h(ζ ) dζ d ζ ; = +2

2 ∂s(ζ +s(ζ ) dξ 2 ∂ξ ∂ξ 2 ∂ζ dξ dξ dξ 2 dξ 2 ∂ζ 2 ∂ζ ∂ζ ∂ζ 2 dξ     2 2 2 d 2 ζ 2 ) d2ζ d2ζ ∂s dζ + s(ζ ) . It follows from (13) and (14) that 2

+ ∂∂ζh(ζ = 2 2 2 ∂ζ dξ dξ dξ dξ ∂ζ

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Log-plurisubharmonicity of metric deformations

2

∂3h ∂ζ 2 ∂ζ

and then



dζ dξ

∂u(ζ ) ∂ζ

2  =

d2ζ dξ 2



∂ 3 h(ζ )

−

+

2 ∂ 2 h(ζ ) d 2 ζ . ∂ζ ∂ζ dξ 2

For simplicity, set u(ζ ) :=

∂ 2 h(ζ ) ∂ζ ∂ζ

− s(ζ ) on D,

∂s(ζ ) ∂ζ ,

so that the above equation can be written as  2 2  2  2  d ζ d ζ dζ ∂u u(ζ ) 2 + 2

=0 for ζ ∈ D. dξ ∂ζ dξ dξ 2 ∂ζ ∂ζ 2

From ξ = f (ζ ) and d 2 ξ dζ 2 dξ −1 − dζ 2 ( dξ ) ( dζ )

=

dζ dξ dξ dζ  − ff 3 .

= 1, we have

d 2 ζ dξ dξ 2 dζ

+

d 2 ξ dζ 2 ( ) dζ 2 dξ

(15)

= 0 on D, so that

Thus (15) is equivalent to

∂u(ζ ) f  (ζ ) u(ζ ) = 2

for every w = f (z). ∂ζ f  (ζ )

d2ζ dξ 2

=

(16)

Therefore u(ζ ) = 0 for ζ ∈ D. In fact, for any fixed 0 ≤ θ0 < 2π and 0 <  << 1 we take  ) eiθ0 +2z f (z) = eiθ0 z + z 2 . Then, f (0) = 0, f is univalent on D, and u(ζ ) = 2 ∂u(ζ . ∂ζ 2   ) iθ0 From lim→0 u(ζ ) = ∂u(ζ and lim→0 u(ζ ) = 0 (since u(ζ ) does not depend on e   ∂ζ ∂u(ζ ) iθ0 ) ), we have ∂ζ e = 0 on D, and then ∂u(ζ ∂ζ = 0 on D. Substituting it with (16), we get u(ζ ) = 0 on D. Consequently,

∂ 2 h(ζ ) ∂ζ ∂ζ

− s(ζ ) = 0, which concludes s(ζ )|dζ |2 = H (ζ )|dζ |2 .  

Proof of Theorem 2 (ii) It is sufficient to prove that for the Schiffer span s(ζ ) for (D, ζ ), log s(ζ ) is strictly subharmonic on D. It is divided into three steps.  2 ) = 2π D |L(z, ζ )|2 d xd y > 0, so that we see that the Schiffer span (i) From (5), ∂∂ζs(ζ ∂ζ s(ζ ) for (D, ζ ) is strictly subharmonic on D. (ii) We shall show that for any non-vanishing holomorphic function ϕ(ζ ) on D,  s(ζ ) := |ϕ(ζ )|s(ζ ) is strictly subharmonic on D. We consider the following variation of domains: D:

ζ ∈ D → D(ζ ) := {w = ϕ(ζ )(z − ζ ) | z ∈ D} ⊂ Cw .

Let us set  pi (w, ζ ) (i = 1, 0) and  s(ζ ) the L i -principal function and the Schiffer span for (D(ζ ), 0), respectively. Applying s(ζ ) to the Levi flat D, we have  2 Lemma 1 for   ∂  p1 2 ∂2  p0 2 ∂ 2 s(ζ ) 4 | dudv, so that  s(ζ ) is strictly subharmonic on B. = | + | | D(ζ ) π ∂ζ ∂ζ ∂w∂ζ ∂w∂ζ dw −2 From Proposition 1(iii) we have  s(ζ ) = dz s(ζ ) = |ϕ(ζ )|−2 s(ζ ) for ζ ∈ D. Thus −2 |ϕ(ζ )| s(ζ ) is strictly subharmonic on B. (iii) We shall prove log s(ζ )is strictly subharmonic on D. We proceed by contradiction:  2π 1 suppose there exist a point ζ0 ∈ D and Δ0 := {|z −ζ0 | < r } ⊂ D such that 2π 0 log s(ζ0 + r eiθ )dθ ≤ log s(ζ0 ). By solving the Dirichlet problem, we construct a harmonic function u(ζ ) on Δ0 with u(ζ ) = log s(ζ ) on ∂Δ0 . Then u(ζ0 ) ≤ log s(ζ0 ). We consider ϕ(ζ ) = ∗ exp−(u(ζ )+iu (ζ )) , so that ϕ is a non-vanishing holomorphic function on Δ0 . If we set s(ζ ) := |ϕ(ζ )|s(ζ ) on Δ0 , then s(ζ0 ) ≥ 1 and s(ζ ) = 1 on ∂Δ0 , so that s(ζ ) is not strictly subharmonic on Δ0 . This is contradiction to the above (ii).   To prove the completeness of s(ζ )|dζ |2 , we prepare the following: Lemma 5 Let D be a simply connected domain in Cz and D  = Cz . For ζ ∈ D, we denote by s(ζ ) the Schiffer span for (D, ζ ) and by d(ζ ) the distance from ζ to ∂ D. Then we have 1 that s(ζ ) ≥ 8d(ζ > 0. )2

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S. Hamano

Proof By the Riemann’s mapping theorem, there exists a holomorphic function w = f (z) on D such that f (ζ ) = 0, f  (ζ ) = 1, and Δ := f (D) = {|w| < r (ζ )} ⊂ Cw . Let  s be the Schiffer span for (Δ, 0). From Proposition 1(iii) we have s(ζ ) = | f  (ζ )|2 s = s. By computation we get  s = r (ζ2)2 (see [8, §6]). Thus s(ζ ) = r (ζ2)2 . It follows from the Koebe’s 1 4 -theorem

for the univalent functions that d(ζ ) ≥

r (ζ ) 4 ,

which implies s(ζ ) ≥

1 8d(ζ )2

> 0.  

Proof of Theorem 2 (iii) Let D = T (R) be the unramified domain bounded by a finite num = T ( R)  in Sect. 3. We denote by C1 ber of simple closed curves C j (1 ≤ j ≤ ν) in D ν ˆ the outer boundary; ∂ D = C1 − j=2 C j , and by D the domain bounded by C1 . For any ζ0 ∈ ∂ D, let {ζn }n be a sequence of points in D such that ζn → ζ0 (n → ∞). Let d(ζn ) be the distance from ζn to ∂ D, and let s D (ζn ) be the Schiffer span for (D, ζn ). From Proposition 1(iii) ˆ we see that s D (ζn ) ≥ s ˆ (ζn ). Since Dˆ is simply connected in D,  Lemma 5 and D ⊂ D, D 1 implies s Dˆ (ζn ) ≥ 8d(ζ )2 → ∞ (n → ∞). Thus we conclude lim D ζ →ζ0 ∈∂ D s D (ζ ) = ∞. n  

6 Proof of Theorem 3 Since D is pseudoconvex, k2 (t, z) ≥ 0 on ∂ D. We fix (t0 , ζ0 ) ∈ D, and make the following four cases. (i) Let B0 := {|t − t0 | < ρ0 } with B0 × {ζ0 } ⊂ D. We consider the parallel transformation  := T (D| B0 ), then D  T : (t, z) ∈ D → (t, w) = (t, z − ζ0 ) ∈ B0 × Cw , and set D  is also pseudoconvex in B0 × Cw . For the Schiffer span  s(t, ξ ) for ( D(t), 0), where  := {z − ζ0 ∈ Cw | z ∈ D(t)}, we have  s(t, 0) = s(t, ζ0 ) on B0 by Proposition D(t) ∂ 2 s(t,0) 1(iii). Using Lemma 1, we have ∂t∂ t¯ ≥ 0 on B0 , which implies that s(t, ζ0 ) is subharmonic on B0 . (ii) We consider the Hartogs transformation T : (t, z) ∈ D → (t, w) = (t, ϕ(t)(z − ζ0 )) ∈ B0 × Cw ,  := T (D| B0 ), where ϕ(t) is any non-vanishing holomorphic function on B0 . We set D   is also pseudoconvex in B0 × Cw . For the Schiffer span  then D s(t, ξ ) for ( D(t), 0),  := {ϕ(t)(z − ζ0 ) ∈ Cw | z ∈ D(t)}, we have  where D(t) s(t, 0) = s(t, ζ0 )|ϕ(t)|−2 on B0 by Proposition 1(iii). Since  s(t, 0) is subharmonic on B0 by (i) above, and since ϕ(t) is any non-vanishing holomorphic on B0 , log s(t, ζ0 ) is subharmonic on B0 from the similar idea to the proof of Theorem 2 (ii). (iii) Let l : ζ = l(t) = ζ0 + a(t − t0 ), where a  = 0, be any complex line passing through (t0 , ζ0 ) in D, and set B1 = {|t − t0 | < ρ1 } in B such that l| B1 ⊂ D. We consider the translation: Tˆ : (t, z) ∈ D → (t, w) = (t, z − ζ0 − a(t − t0 )) ∈ B1 × Cw , and set Dˆ := Tˆ (D| B1 ). Then Dˆ is pseudoconvex in B1 ×Cw and Tˆ (l(B1 )) corresponds to ˆ ˆ 0), where D(t) := {z −ζ0 −a(t − B1 ×{0} ⊂ Dˆ . For the Schiffer span sˆ (t, ξ ) for ( D(t), t0 ) ∈ Cw | z ∈ D(t)}, we have sˆ (t, 0) = s(t, l(t))|a|−2 on B1 by Proposition 1(iii). It follows from (ii) that log sˆ (t, 0) is subharmonic on B1 , and so is log s(t, l(t)). (iv) Let l : t = t0 be the complex line in D passing through (t0 , ζ0 ). Applying Theorem 2 (ii) to the Schiffer span s(t0 , ζ ) for (D(t0 ), ζ0 ), we see log s(t0 , ζ ) is (strictly) subharmonic on D(t0 ).

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Log-plurisubharmonicity of metric deformations

Consequently, log s(t, ζ ) is plurisubharmonic on D.

 

Acknowledgments The author would like to express her sincere gratitude to Professor Hiroshi Yamaguchi for his essential advice and discussion on the variational formulas. The author is deeply grateful to Professor Junjiro Noguchi for his great advice and warm encouragement throughout the years. The author sincerely thanks the referee(s) for the careful reading and the accurate comment in Remark 3.

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