数理解析研究所講究録
第 1694 巻 2010 年 121-130

121

VARIATION FORMULAS FOR PRINCIPALFUNCTIONS AND HARMONIC SPANSSACHIKO HAMANO
ABSTRACT. The purpose of this article is to give a summary of theseminar lecture with title Variation formulas for principal functions and harmonic spans by the author in the conference held at
RIMS, Kyoto, Japan, December 2009. The former half of this article is in Hamano 4 and the latter half is in the manuscript 6
which is a joint work with Maitani and Yamaguchi.
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1. INTRODUCTION
be an unramified (two-dimensional)
Let $B=\{|t|<\rho\}$ and let,. We write
Riemann domain sheeted over, sois a fiber over each $t\in B$ , i.e.,
whereconsists of unramified Riemann surfaces sheeted over .
that each,insuch that, if we put
Consider a subdomain$t\inB$$R(t)$
,
then
over
where
is a fiber of$\tilde{\mathcal{R}}$$\tilde{\mathcal{R}}=\bigcup_{t\in B}(t,\tilde{R}(t))$$B\cross \mathbb{C}_{z}$$\tilde{R}(t)=\{z:(t, z)\in\tilde{\mathcal{R}}\}$$\tilde{R}(t)$$\tilde{R}(t)$$\mathbb{C}_{z}$$\tilde{\mathcal{R}}$$\mathcal{R}=\bigcup_{t\in B}(t, R(t)))$$\mathcal{R}$$\mathcal{R}$$t\in B$ ;(1)
issmooth inof in(2) the boundary;(3) each $R(t),$ $t\in B$ is a connected Riemann surface of genus $g\geq 0$ ,where is independent of $t\in B$ ;$t\in B$ in
consists of a finite number of(4) each, where is independentsmooth contours $C_{j}(t),j=0,1,$$t\inB$such that $\partial R(t)=$. We give the orientation ofof$C_{0}(t)+C_{1}(t)+\cdots+C_{\nu}(t)$ .
$\tilde{R}(t)\Supset R(t)\neq\emptyset,$$\tilde{\mathcal{R}}$$\partial \mathcal{R}=\bigcup_{t\in B}(t, \partial R(t))$$C^{\omega}$$\mathcal{R}$$\tilde{\mathcal{R}}$$g$$C^{\omega}$$\tilde{R}(t)$$\partial R(t),$$\ldots,$

$\nu$

$\nu$

$C_{j}(t)$over
We usually regard two-dimensional Riemann domain$R(t)$
withover
surfaceofRiemannsmooth variation
as a
$t\in
B$
:
with complex parametersmooth boundary$B\cross \mathbb{C}_{z}$$\mathcal{R}$$C^{\omega}$$\mathbb{C}_{z}$$C^{\omega}$$\partial R(t)$$\mathcal{R}:t\in Barrow R(t)$.

Variation formula of the Green function for $(R(t), 0)$ We ascontains $B\cross\{0\}$ , precisely, there exists at least one
sume thatof over $B\cross\{0\}$ . We consider the Green function
constant section$\mathcal{R}$$O$$\mathcal{R}$2000 Mathematics Subject Classification. $30C85;30F15;32F05;32H10$
The author is supported by JSPS Grant-in-Aid for Young Scientists (B).

122
$g(t, z)$ with pole at $z=0$ and the Robin constant$\lambda(t)$that$g(t, z)= \log\frac{1}{|z|}+\lambda(t)+h(t, z)$Here $h(t, z)$ is harmonic for
that

$z$

for $(R(t), 0)$ , so.

in a neighborhood of $z=0$ in $R(t)$ suchfor $t\in B$ .

$h(t, 0)=0$

Let
be a defining function of
consider the following quantities:$\varphi(t, z)$$\partial \mathcal{R}$in$B\cross \mathbb{C}_{z}$$k_{1}(t, z)= \frac{\partial\varphi}{\partial t}/|\frac{\partial\varphi}{\partial z}|$. For$(t, z)\in\partial \mathcal{R}$

, we

,

$k_{2}(t, z)=( \frac{\partial^{2}\varphi}{\partial t\partial\overline{t}}|\frac{\partial\varphi}{\partial z}|^{2}-2{\rm Re}\{\frac{\partial^{2}\varphi}{\partial\overline{t}\partial z}\frac{\partial\varphi}{\partial t}\frac{\partial\varphi}{\partial\overline{z}}\}+|\frac{\partial\varphi}{\partial t}|^{2}\frac{\partial^{2}\varphi}{\partial z\partial\overline{z}}I,$$| \frac{\partial\varphi}{\partial z}|^{3}$

We note that they do not depend on the choice of defining functionsof. We denote by
the arc length element ofat. The function $k_{2}(t, z)$ on
is due to Maitani-Yamaguchi in [8]which is based on [7], which is called the Levi curvature for. Thenthe following variation formulas for the Robin constans are shown in
Hadamard [3], Maitani-Yamaguchi [8].
$\varphi(t, z)$$\partial \mathcal{R}$$ds_{z}$$z$$\partial R(t)$$\partial \mathcal{R}$$\partial \mathcal{R}$Fact. It holds for $t\in B$ that$\frac{\partial\lambda(t)}{\partial t}=-\frac{1}{\pi}\int_{\partial R(t)}k_{1}(t, z)|\frac{\partial g(t,z)}{\partial z}|^{2}ds_{z}$,.

$\frac{\partial^{2}\lambda(t)}{\partial t\partial\overline{t}}=-\frac{1}{\pi}\int_{\partial R(t)}k_{2}(t, z)|\frac{\partial g(t,z)}{\partial z}|^{2}ds_{z}-\frac{4}{\pi}\int\int_{R(t)}|\frac{\partial^{2}g(t,z)}{\partial\overline{t}\partial z}|^{2}dxdy$

2. VARIATION FORMULA OF$L_{1}$-PRINCIPAL FUNCTION FOR$(R(t),$ $0,$ $C_{0}(t))$

This and the next sections are quoted from S. Hamano [4]. Under
the same conditions for the unramified domaininoverand $\partial R(t)=\sum_{j=0}^{\nu}C_{j}(t)$ , we assume that the total spacecontains $B\cross\{0\}$ . For each $t\in B$ , we conventionally write for the
point $O\cap R(t)$ .
Let $t\in B$ be fixed. In the theory of one complex variable, it is known
that there exists a unique real-valued function $u(t, z)$ on
satisfying the following four conditions:(1) $u(t, z)$ is harmonic onand is continuous on;(2) $u(t, z)- \log\frac{1}{|z|}$ is harmonic at $z=0$ ;(3) $u(t, z)=0$ on;$\mathcal{R}=\bigcup_{t\in B}(t, R(t))$$\tilde{\mathcal{R}}$$B\cross \mathbb{C}_{z}$$\mathcal{R}$$0$$R(t)\backslash \{0\}$$R(t)\backslash \{0\}$$C_{0}(t)$$\overline{R(t)}$123(4) for each $i=1,$$\nu$$\ldots,$, we have(i) $u(t, z)=a_{i}(t)$ : constant on$C_{i}(t)$;(ii) $\int_{C.(t)}*du(t, z)=0$ .

We note that $u(t, z)$ extends harmonically acrossas a harmonic$V(t)$
function onsuch that. By (2), we find aneighborhoodof $z=0$ such that$\partial R(t)$$\partial R(t)\Subset V(t)\Subset\tilde{R}(t)$$U_{0}(t)$on$u(t, z)= \log\frac{1}{|z|}+\gamma(t)+h(t, z)$$U_{0}(t)$

,

where
is the constant term and $h(t, z)$ is harmonic forsuch that$\gamma(t)$$h(t, 0)=0$ ,$t\in B$$z$on

$U_{0}(t)$.

The function $u(t, z)$ is called the -principal function on $R(t)$ withlogarithmic pole at with respect to, andis called theconstant on $R(t)$ with logarithmic pole at with respect to(cf:[1] . In this article, we simply call $u(t, z)$ the-principal function for$(R(t), 0, C_{0}(t))$ , andthe -constant for $(R(t), 0, C_{0}(t))$ . We note
$L_{1}$$0$$C_{0}(t)$$\gamma(t)$$L_{1^{-}}$$0$$C_{0}(t)$$L_{1}$$)$$\gamma(t)$$L_{1}$

that $u(t, z)>0$ in
and $a_{i}(t)>0(i=1, \ldots, \nu)$ .
Then we have the following variation formula for thefor $(R(t), 0, C_{0}(t))$ .
$R(t)\backslash \{0\}$$L_{1}$-constant$\gamma(t)$Lemma 1. It holds for $t\in B$ that$\frac{\partial\gamma(t)}{\partial t}=-\frac{1}{\pi}\int_{\partial R(t)}k_{1}(t, z)|\frac{\partial u(t,z)}{\partial z}|^{2}ds_{z}$,$\frac{\partial^{2}\gamma(t)}{\partial t\partial\overline{t}}=-\frac{1}{\pi}\int_{\partial R(t)}k_{2}(t, z)|\frac{\partial u(t,z)}{\partial z}|^{2}ds_{z}-\frac{4}{\pi}\int\int_{R(t)}|\frac{\partial^{2}u(t,z)}{\partial\overline{t}\partial z}|^{2}dxdy$.

This variation formula is formally the same as that for the Robinconstant
in section 1. The essential difference of the proofs forandcomes from the fact that $u(t, z)$ is not a defining functionofcontrary to the case of the Green function $g(t, z)$ .
$\lambda(t)$$\gamma(t)$$\lambda(t)$$\partial \mathcal{R}$

Theorem 2. Under the same conditions in Lemma 1, if
is pseusuperharmonic
doconvex over
then
is a
function on$B$ .
$\mathcal{R}$$B\cross \mathbb{C}_{z\prime}$$\gamma(t)$$C^{\omega}$.
Remark 1. For Lemma 1, we assumed that is unramified over
However, even if each $R(t),$ $t\in B$ has a finite number of branch points$\zeta_{k}(t)(k=1, \ldots, m)$ for $t\in B$ such thatis a holomorphic function
on $B$ with $\zeta_{k}(t)\neq\zeta_{l}(t)(k\neq l),$ $t\in B$ , then Lemma 1 and hence
Theorem 2 hold. For, this case can be reduced to Lemma 1 by thestandard method by use of Y. Nishimura’s theorem [10].
$\mathcal{R}$$\zeta_{k}(t)$$B\cross \mathbb{C}_{z}$124
In the special case when $R(t)$ is a planar Riemann surface, theprincipal function $u(t, z)$ induces a circular slit mapping $f(t. z)$ . Thatis, if we choose a branch $u^{*}(t, z)$ of harmonic conjugate function of$u(t, z)$ on $R(t),$ $t\in B$ such that
$L_{1^{-}}$$f(t, z)=e^{\gamma(t)-(u(t,z)+xu^{r}(t,z))}$is of the form
$w=f(t, z)=z+ \sum_{j=2}^{\infty}b_{j}(t)z^{j}$on$U_{0}(t)$,then $f(t, z)$ conformally maps $R(t)$ onto a circular slit domain $\{|w|<$, where $\ell_{i}\{t)=f(t, C_{i}(t))$ (an arc of the circle $\{|w|=$, thenis logarithmic. If is pseudoconvex over
space
is a
,
the
so
that
total
on
superharmonic.
Hartogs pseudoconvex domain in$e^{\gamma(t)} \}\backslash (\bigcup_{i=1}^{\nu}\ell_{i})$$e^{\gamma(t)-a_{1}(t)})$$B\cross \mathbb{C}_{z}$$\mathcal{R}$$B$$e^{\gamma(t)}$$\bigcup_{t\in B}\{|w|<e^{\gamma(t)}\}$$B\cross \mathbb{C}_{w}$Remark 2. In the theory of one complex variable, the circular slit mapping and the radial slit mapping have good correspondence. But thesame result for the corresponding radius of the radial slit mapping does
not hold. In fact, we have the following counterexamples (i) and (ii)such that the radii of radialinof pseudoconvex domains
slit mappings are not logarithmic superharmonic or not logarithmic
subharmonic on :(i) The radius of radial slit mapping is not logarithmic superharmonic
on $B$ : Let$\mathcal{R}$$B\cross \mathbb{C}_{z}$$B$$\mathcal{R}=\{|t|<\frac{1}{2}\}\cross\{|z|<1\}\backslash \{(t, z):|z-\frac{1}{2}|\leq|t|<\frac{1}{2}\}$$B= \{|t|<\frac{1}{2}\}$,$R(t)= \{|z|<1\}\backslash \{|z-\frac{1}{2}|\leq|t|\}$,,so that $\partial R(t)=C_{0}(t)+C_{1}(t)$ where $C_{0}(t)=\{|z|=1\}$ and $C_{1}(t)=$$\{|z-12|=|t|\}$ .
(ii) The radius of radial slit mapping is not logarithmic subharmonic
on $B$ : Let$\mathcal{R}=\bigcup_{t\in B}\{|z|<r(t)\}\backslash B\cross(C_{1}\cup C_{2})$,$r(t)>1$ and $\log r(t)$ is superharin
wheremonic on $B$ . Thus $\partial R(t)=C_{0}(t)+C_{1}(t)+C_{2}(t)$ where $C_{0}(t)=\{|z|=$$C_{1}=[ \frac{1}{2}, \frac{2}{3}],$$r(t)\},$ $C_{i}(t)=C_{i},$$C_{2}=[ \frac{i}{2}, \frac{2i}{3}]$$\mathbb{C}_{z},$$i=1,2$ .

3. VARIATION FORMULA OF$L_{1}$-PRINCIPAL FUNCTION FOR$(R(t),$ $0,$ $\xi(t))$

Under the same conditions for the unramified domain, we assume that there exist
in
over
and

$\mathcal{R}=\bigcup_{t\in B}(t, R(t))$$\tilde{\mathcal{R}}$$B\cross \mathbb{C}_{z}$$\partial R(t)=\sum_{j=0}^{\nu}C_{j}(t)$125
two holomorphic sections:and$\Xi_{0}:z=0$$\Xi_{1}:z=\xi(t)$of over $B$ such that
. Let $t\in B$ be fixed. In the theory of
one complex variable, there exists a unique real-valued function $p(t, z)$
on $R(t)\backslash \{0, \xi(t)\}$ satisfying the following four conditions:(1) $p(t, z)$ is harmonic on;and continuous on(2) $p(t, z)-\log 1/z|$ is harmonic at $z=0$ and$\Xi_{0}\cap\Xi_{1}=\emptyset$$\mathcal{R}$$R(t)\backslash \{0, \xi(t)\}$$\overline{R(t)}$,$\lim_{zarrow 0}(p(t, z)-\log\frac{1}{|z|})=0:$(3) $p(t, z)-\log|z-\xi(t)|$ is harmonic at $z=\xi(t)$ ;(4) for each $j=0,1,$, we have$\ldots,$$\nu$(i) $p(t, z)=a_{j}(t)$ : constant on$( ii)\int_{C_{j}(t)}*dp(t, z)=0$$C_{j}(t)$;.

We note that $p(t, z)$ extends harmonically acrossas a harmonic$-$ oo $<p(t, z)<+\infty$ ,function on $V(t)$ such that
and-oo $<a_{j}(t)<+\infty$ .
By (2), we find a neighborhoodof $z=0$ such that$\partial R(t)$$\partial R(t)\subset V(t)\Subset\tilde{R}(t),$$U_{0}(t)$on$p(t, z)= \log\frac{1}{|z|}+h_{0}(t, z)$where $h_{0}(t, z)$ is harmonic for

$z$

on

$U_{0}(t)$$h_{0}(t, 0)=0$ ,By (3), we find a neighborhood.

of $z=\xi(t)$ such thaton$p(t, z)=\log|z-\xi(t)|+\alpha(t)+h_{\xi}(t, z)$where$\alpha(t)$is a real constant and$h_{\xi}(t, z)$$h_{\xi}(t, \xi(t))=0$

,

,

and
$t\in B$

$U_{\xi}(t)$$U_{0}(t)$is harmonic for$t\in B$

$U_{\xi}(t)$

$z$

on

,
$U_{\xi}(t)$and.

In this article, we simply call $p(t, z)$ the-principal function for$(R(t), 0, \xi(t))$ , andthe-constant for $(R(t), 0, \xi(t))$ .
$L_{1}$$\alpha(t)$$L_{1}$

Under these situations, we have
Lemma 3. It holds for $t\in B$ that$\frac{\partial\alpha(t)}{\partial t}=\frac{1}{\pi}\int_{\partial R(t)}k_{1}(t, z)|\frac{\partial p(t,z)}{\partial z}|^{2}ds_{z}+2\frac{\partial h_{\xi}}{\partial z}|_{(t,\xi(t))}\cdot\xi’(t)$,$\frac{\partial^{2}\alpha(t)}{\partial t\partial\overline{t}}=\frac{1}{\pi}\int_{\partial R(t)}k_{2}(t, z)|\frac{\partial p(t,z)}{\partial z}|^{2}ds_{z}+\frac{4}{\pi}\int\int_{R(t)}|\frac{\partial^{2}p(t,z)}{\partial\overline{t}\partial z}|^{2}dxdy$

126(Levi condition),
If is pseudoconvex in , then $k_{2}(t, z)\geq 0$ on
and the converse is also true. We then apply Lemma 1 to obtain thefollowing$\tilde{\mathcal{R}}$$\partial \mathcal{R}$$\mathcal{R}$is pseuTheorem 4. Under the same conditions in Lemma 3, if
is a
then
subharmonic function on .
doconvex overas in Remark 1.
This is also true under the same condition for$\mathcal{R}$$B\cross \mathbb{C}_{zr}$$B$$C^{\omega}$$\alpha(t)$$\mathcal{R}$Application of Theorem 4 As an application of Theorem 4, we
be a simply connected domain
show that the following fact. Let
in . Let : $Sarrow B$ be a holomorphic family of compact Riemannsurfaces $S(t)=\pi^{-1}(t)$ over such that each fiber $S(t)$ is of genus
and non-singular in . For a fixed $t\in B$ , we consider the Schottky
of each $S(t)$ (cf: Sec. 101 in p.266 in Ford [2], and $19F$
covering
in p.241 in Ahlfors-Sario [1] . We denote by the total space of the. Then we have:variation: $t\in Barrow\tilde{S}(t)$ , namely,
$B$

$\pi$

$\mathbb{C}_{t}$$B$$\geq 2$$S$$\tilde{S}(t)$$\tilde{S}$$)$$\tilde{S}=\bigcup_{t\in B}(t,\tilde{S}(t))$

consisting of the Schottky covering
Theorem 5. The total spaceof compact Riemann surfaces $S(t)$ with complex parameter $t\in B$.
is holomorphically uniformized to a $univaler\iota t$ domain on$\tilde{S}$$\tilde{S}(t)$’$B\cross \mathbb{P}^{1}$isIn [8], Maitani and Yamaguchi proved that, if$R(t)$,
such that eachan unramified pseudoconvex domain over$t\in B$ is planar and parabolic, thenis holomorphically uniformizable. Since the Schottky coveringof a compact
to a domain in$g\geq2$$S(t)$is planar but not parabolic, their
Riemann surface
of genus
theorem and method cannot be applicable to our case. In [12], Yamaguchi wrote a resum\’e about Theorem 5 with a rough sketch of theproof. However his sketch had a (gap” Then I bridge the gap by es-principal function (Lemma 3),
tablishing the variation formula forand obtain Theorem 5.
$\mathcal{R}=\bigcup_{t\in B}(t, R(t))$$B\cross \mathbb{C}_{z}$$\mathcal{R}$$B\cross \mathbb{P}^{1}$$\tilde{S}(t)$$L_{1}$

4. VARIATION FORMULA OF$L_{0}$-PRINCIPAL FUNCTION FOR$(R(t),$ $0,$ $\xi(t))$

This section is quoted from S. Hamano, F. Maitani and H. Yamguchi[6]. Under the same conditions for the unramified domaininoverand $\partial R(t)=\sum_{j=0}^{\nu}C_{j}(t)$ , we assume thathas twoholomorphic section over $B$ :$\mathcal{R}=\bigcup_{t\in B}(t, R(t))$$\tilde{\mathcal{R}}$$B\cross \mathbb{C}_{z}$$\mathcal{R}$and $\Xi_{1}:z=\xi(t)$. Let $t\in B$ be fixed. Then it is known (cf: Sariosuch that
Nakai [9] that $R(t)$ carries the following real-valued function $q(t, z)$ .
(1) $q(t, z)$ is harmonic on;and is continuous on$/|z|$
$q(t,
z)-$
$z=0$
log
l
is
at
harmonic
and(2)$\lim_{zarrow 0}(q(t, z)-\log 1’|z|)=0$ ;$\Xi_{0}:z=0$$\Xi_{0}\cap\Xi_{1}=\emptyset$$)$$R(t)\backslash \{0, \xi(t)\}$$\overline{R(t)}$127(3) $q(t, z)-\log|z-\xi(t)|$ is harmonic at $z=\xi(t)$ ;(4)on.
We call the function $q(t, z)$ the-principal function for $(R(t), 0, \xi(t))$ .
Note that $q(t, z)$ extends harmonically acrossas a harmonic
function on $V(t)$ such that
.
$q(t,
z)$
By (2) for, we find a neighborhoodof $z=0$ such that$\frac{\partial q(t,z)}{\partial n_{z}}=0$$\partial R(t)$$L_{0}$$\partial R(t)$$\partial R(t)\Subset V(t)\Subset\tilde{R}(t)$$U_{0}(t)$on$q(t, z)= \log\frac{1}{|z|}+\mathfrak{h}_{0}(t, z)$where$\mathfrak{h}_{0}(t, z)$is harmonic for

$z$

on

$U_{0}(t)$$\mathfrak{h}_{0}(t, 0)=0$,$U_{0}(t)$and$t\in B$ .

By (3) for $q(t, z)$ , we find a neighborhood$U_{\xi}(t)$$q(t, z)=\log|z-\xi(t)|+\beta(t)+\mathfrak{h}_{\xi}(t, z)$where$\beta(t)$is a constant and$\mathfrak{h}_{\xi}(t, z)$$\beta(t)$the$L_{0}$

of $z=\xi(t)$ such thaton

is harmonic for$\mathfrak{h}_{\xi}(t, \xi(t))=0$

We call,,$t\in B$$z$

$U_{\xi}(t)$on,$U_{\xi}(t)$and.

-constant for $(R(t), 0, \xi(t))$ .

We shall give the variational formulas for -constant. In orderto proves the formula for, we have to add a new idea to the prooffor. In fact, the formulas for
do not concern to the genus of$R(t)$ but the variation formula of the second order fordoes concern
to the genus of $R(t)$ . It seems to be curious that of the first order does
not concern to the genus as below. In case when $R(t)$ is of positive
genus $g\geq 1$ , we takebe usual $A,$ $B$ cycles on $R(t)$with intersection number condition: for $l=1,$,$L_{0}$$\beta(t)$$\beta(t)$$\alpha(t)$$\alpha(t)$$\beta(t)$$\{A_{l}(t), B_{l}(t)\}_{1\leq l\leq g}$$k,$$A_{k}(t)\cross B_{l}(t)=\delta_{k,l}$

Here

,

$A_{k}(t)\cross A_{l}(t)=0$$\ldots,$ $g$,$B_{k}(t)\cross B_{l}(t)=0$.

is Kronecker’s delta; $A_{k}(t)\cross B_{l}(t)$ means that $A_{k}(t)$ crosses
from the left-side to the right-side of the direction; andeach $A_{k}(t)(B_{k}(t)),$ $k=1,$
varies continuously with parameter$t\in B$ such that $A_{k}(t),$ $B_{k}(t)$ do not pass through $\{0, \xi(t)\}$ . On each$R(t),$ $t\in B$ we denote by $*dq(t, z)$ the conjugate differential of $dq(t, z)$ .
Then we have$\delta_{k,l}$$B_{l}(t)$$B_{l}(t)$$\ldots,$ $g$

Lemma 6. It holds for $t\in B$ that,$\frac{\partial\beta(t)}{\partial t}=-\frac{1}{\pi}\int_{\partial R(t)}k_{1}(t, z)|\frac{\partial q(t,z)}{\partial z}|^{2}ds_{z}+2\frac{\partial \mathfrak{h}_{\xi}}{\partial z}|_{(t_{2}\xi(t))}\cdot\xi’(t)$$\frac{\partial^{2}\beta(t)}{\partial t\partial\overline{t}}=-\frac{1}{\pi}\int_{\partial R(t)}k_{2}(t, z)|\frac{\partial q(t,z)}{\partial z}|^{2}ds_{z}-\frac{4}{\pi}\int\int_{R(t)}|\frac{\partial^{2}q(t,z)}{\partial\overline{t}\partial z}|^{2}dxdy$$- \frac{2}{\pi}\Im\sum_{k=1}^{g}(\frac{\partial}{\partial t}\int_{A_{k}(t)}*dq(t, z))\cdot(\frac{\partial}{\partial\overline{t}}\int_{B_{k}(t)}*dq(t, z))$.

128
is pseudoconvex in , then -constantLemma 3 implied that if
subharmonic function on $B$ . On the otherfor $(R(t), 0, \xi(t))$ is ahand, Lemma 6 implied the following:$\tilde{\mathcal{R}}$$\mathcal{R}$$L_{1}$$C^{\omega}$$\alpha(t)$Theorem 7. Under the same conditions in Lemma 6, if
is pseudoand each $R(t),$ $t\in B$ is planar, then theconvex in-constant$\mathcal{R}$$\tilde{\mathcal{R}}$$\beta(t)$$L_{0}$

for $(R(t), 0, \xi(t))$ is a$C^{\omega}$superharmonic function on$B$.

The contrast between the subaharmincity ofand the superharmonicity of
are unified with the notion of the harmonic span $s(t)$$(R(t),
0,\xi(t))$
fordue to M. Schiffer [11].
$\alpha(t)$$\beta(t)$Variations formula for the harmonic spans.
We recall somenotions studied in the theory of one complex variable. Let
be adomain in$j=$bounded by a finite number of closed curves$\xi\neq
0$
$0\in
R$
$0,1,$. For simplicity we assume. For a point, weconsider the -and -function $p(z)$ and $q(z)$ for $(R, 0, \xi)$ and theand -constantand for $(R, 0, \xi)$ . In the function theory of one
complex variable, it is known (cf: [1] and [9])$R$

$C_{j},$$\mathbb{C}_{z}$$\ldots,$$\nu$$L_{0}$$L_{1}$$L_{0}$$\alpha$$L_{1^{-}}$$\beta$$s(R)= \frac{\pi}{2}(\alpha-\beta)$

as the harmonic span $s(R)$ for $(R, 0, \xi)$ .
We return to the variation of Riemann surfaces. Let: $t\in Barrow$$R(t)$ satisfy the conditions in the beginning of Sections 3 and 4. For a
fixed $t\in B$ , we denote by $p(t, z)(q(t, z))$ theprincipal function, by
theconstant and by $s(t)$ the harmonic span$(R(t),
0,\xi(t))$. Then combining Lemmas 3 and 6, we immediatelyfor$\mathcal{R}$$L_{1^{-}}(L_{0^{-}})$$\alpha(t)(\beta(t))$$L_{1^{-}}(L_{0^{-}})$

have the following:

Lemma 8. Assume that $R(t),$ $t\in B$ is planar. Then it holds that$\frac{\partial^{2}s(t)}{\partial t\partial\overline{t}}=\frac{1}{2}\int_{\partial R(t)}k_{2}(t, z)(|\frac{\partial p(t,z)}{\partial z}|2 +| \frac{\partial q(t,z)}{\partial z}|^{2})ds_{z}$$+2 \int\int_{R(t)}(|\frac{\partial^{2}p(t,z)}{\partial\tilde{t}\partial z}|2 +| \frac{\partial^{2}q(t_{2}z)}{\partial\overline{t}\partial z}|^{2})dxdy$

Lemma 8 implied the following:
Theorem 9. Under the same conditions in Lemma 6, if is pseudoconvex inand each $R(t),$ $t\in B$ is planar, then the harmonic span$s(t)$
subharmonic function on $B$ .
for $(R(t), 0, \xi(t))$ is a$\mathcal{R}$$\tilde{\mathcal{R}}$$C^{\omega}$

5. EXAMPLESWe begin with a simple example of our general result shown in thisarticle. Let $B=\{|t|<\rho\}$ be a disk in . For each $t\in B$ , let $R(t)$ be, where $\log r(t)$ is a superharmonic function
a disk $\{|z|<r(t)\}$ in$\mathbb{C}_{t}$$\mathbb{C}_{z}$129, then. If we set the Hartogs domain of disks
existsa
therethat. Assume
is a pseudoconvex domain in
holomorphic section : $t\in Barrow\xi(t)(\neq 0)\in R(t)$ .
We consider the following function:[Example of Theorem 4.]on$\mathcal{R}=\bigcup_{t\in B}(t, R(t))$$B$$B\cross \mathbb{C}_{z}$$\mathcal{R}$$\xi$

on $R(t)$ .

$f(t, z)=- \frac{1}{\xi(t)}\cdot\frac{r(t)^{2}(z-\xi(t))}{z(r(t)^{2}-\overline{\xi}(t)z)}$

Then is a circular slit mapping on $R(t)$ with zero at $z=\xi(t)$ andon $B$ is written intopole at $z=0$ . The -constant$f$$\alpha(t)$$L_{1}$$\alpha(t)=\log|\frac{\partial f}{\partial z}(t, \xi(t))|$$=$$\log|-\frac{1}{\xi(t)^{2}}\cdot\frac{r(t)^{2}}{r(t)^{2}-|\xi(t)|^{2}}|$.

is holomorphic on $B$ and since $\log r(t)$ is superharmonic on$B,$
is subharmonic on $B$ , so is the second term in the right-handis a subharmonic function on $B$ .
side. Hence,Since$\xi(t)$$\log\frac{|\xi(t)|}{r(t)}$$\alpha(t)$[Example of Theorem 7.]

We put $\theta(t)=\arg\xi(t)$ . Then

$F(t, z)= \frac{1}{2}(\frac{z}{r(t)e^{i\theta(t)}}+\frac{r(t)e^{i\theta(t)}}{z})-\frac{1}{2}(\frac{|\xi(t)|}{r(t)}+\frac{r(t)}{|\xi(t)|})$

is the radial slit mapping on $R(t)$ with zero at $z=\xi(t)$ and pole at
is written into
-constant$z=0$ . The$L_{0}$$\beta(t)$$\beta(t)=\log|\frac{\partial Q}{\partial z}(t, \xi(t))|=-2\log|\xi(t)|+\log[1-(\frac{|\xi(t)|}{r(t)}I^{2}]$,which certainly is superharmonic on $B$ .
We also see that the harmonic span
[Example of Theorem 9.]$s(t)= \frac{\pi}{2}(\alpha(t)-\beta(t))$ for $(R(t), 0, \xi(t))$ is$s(t)= \log\frac{1}{1-(\frac{|\xi(t)|}{r(t)})^{2}}$,which is subharmonic on $B$ .
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