数理解析研究所講究録 1460 巻 2005 年 43-52

43

STABILITY AND RIGIDITY OF CERTAIN
SPECIAL LAGRANGIAN CONES
阪市大・理学研究科

義裕 (Yoshihiro Ohnita)
Department of Mathematics,
Osaka City University

大仁田

ABSTRACT. Ill this artcle we explain some results from the theory of special Lagrangian submanifolds with conical singluarities
recently develoved by D. D. Joyce, and we provide new examples
of stable anci rigid special Lagrangian cones in higher dimensions.

INTRODUCTION
Recently Dominic Joyce has provided the profound theory of special
Lagrangian submanifolds with isolated conical conical singularities andthier moduli spaces in a series of his papers. The notion of the stabilityindex of special Lagrangian cones was introduced by Joyce. It is related
to the deformation of special Lagrangian submanifolds with isolated
conical singularities and the regularity of special Lagrangian integral
currents. In this artcle we explain some results from the theory of
special Lagrangian submanifolds with conical singluarities develoved
by Joyce, and we shall provide new examples of stable and rigid special
Lagrangian cones in higher dimensions.
In November 2004, Mark Haskins has visited Kyushu University andTokyo Metropolitan University. The author was able to have nice discussion with him about this subject there. The author would like tothank Dr. Mark Haskins for his valuable suggestion about a problem
on the existence of stable special Lagrangian cones in higher dimension.

1. SPECIAL LAGRANGIAN SUBMANIFOLDS OF CALAEI-YAU
M ANIFOLDSSpecial Lagrangian submanifolds have two aspects of Lagrangiansubmanifolds in symplectic geometry and calibrated submanifolds inRiemannian geometry. Calibrated submanifolds are minimal submanifolds in the sense that the mean curvature vector fields vanish. Morestrongly they are homologically volum minimizing submanifolds, andhence stable minimal submanifolds.
, we recall the notion of special LaIn Euclidean space
grangian submanifolds. The natural group action of $SU(m)\subset U(m)$$\mathrm{e}$$\mathrm{R}^{2m}\cong \mathrm{C}^{m}$44
preserves the standard K\"ahler form (symplectic form) defined as$\omega$$:= \sqrt{-1}\sum_{i=1}^{m}dz^{i}\Lambda d\overline{z}^{i}$and the standard complex volume form defined as$\Omega$$:=dz^{1}\Lambda\cdots$

A $dz^{m}$ .

We decompose 0 into real and imaginary parts as$\Omega={\rm Re}(\Omega)+\sqrt{-1}{\rm Im}(\Omega)$.

.
are parallel real form on
andThen
are characterized by the conThe calibrated submanifoids by
to the submanifold vandition that the restrictions of andare defined as such
. The special Lagrangian submanifoids insubmanifoids, Harvey and Lawson showed that minimal Lagrangian
are special Lagrangian submanifoids.
submanifoids in
In general, suppose that $(M,g)$ is a Riemannian manifold with holonomy group contained in $SU(m)$ , and such Riemannian manifolds
Kahier manifolds of complex dimension $m$ . Thenbecome Catabithe parallel Kahier form and the parallel complex volume form aredefines a calibration on $M$ . The
defined on the whole $M$ , andare characterized by the condition
calibrated submanifoids by
to the submanifold vanish.
that the pull-backs of and$\mathrm{C}^{m}$$n$${\rm Im}(\Omega)$${\rm Re}(\Omega)$${\rm Re}(\Omega)$$\omega$${\rm Im}(\Omega)$$\mathrm{C}^{m}$$\mathrm{i}\mathrm{s}\mathrm{h}$$\mathrm{C}^{m}$$\mathrm{Y}au$$\Omega$$\omega$${\rm Re}(\Omega)$${\rm Re}(\Omega)$${\rm Im}(\Omega)$$\omega$

Definitiop 1.1. A $m$-dimensional submanifold in a Calabi-Yau manifold is called a special Lagrangian submanifold if the pull-backs of both$L$$\alpha\}$and${\rm Im}(\mathrm{f}\mathrm{i})$

to

$L$

vanish.

Proposition 1.1. Let be an oriented Lagrangian submanifold of
Calabimanifold M. Then is a minimal submanifold in $M$ if andonly if is a special Lagrangrian submanifold calibrated by
$L$

$a$

$L$

$\mathrm{Y}au$${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$$L$for some
Proof If$\theta\not\in \mathrm{R}$.

is a minimal submanifold of $M$ , thenhavefor some constantspecial Lagrangian submanifold calibrated by$L$$\Omega(e_{1}, \ldots,e_{m})=e^{-\sqrt{-1}\theta}$$f$$\mathit{0}\in$

is constant. We
R. Thus is a
.
Cl
$L$

${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$

Lemma 1,1. Suppose that is an oriented Lagrangian submanifold
immersed in a Calabi-Yau manifold M. If we define a smooth function
on with values in unit complex numbers by
$L$

$f$

$L$

$f(x)\mathrm{i}=\Omega(e_{1}$, . ..,$e_{m})$

,

(1.1)

45
is an orthonormal basis ofwhere\cdots ,orientation of L, then the formula$e_{m}\}$$\{e_{1},$df$=-f\sqrt{-1}\alpha_{H}$$T_{x}L$compatible with the

,

(1.2)

namelyd $\log(f)=-\sqrt{-1}\alpha_{H}$(1.3)

holds. Here H denotes the mean curvature vector field of L. Note that$\log(f)$ is multi-valued. Locally, we have$d\psi$(1.4)$=-\alpha_{H\prime}$

Proof The second fundamental form a of a Lagrangian submanifold

in a K\"ahler manifold $(M,g, J)$ is a symmetric bilinear form on $TL$
with values in the normal bundle
defined by$L$$T^{[perp]}L$$\alpha(X, \mathrm{Y}):=\nabla_{X}^{M}\mathrm{Y}$ -for each X, Y $\in X(L$ }$\nabla_{X}^{L}\mathrm{Y}$(L5)and is known to satisfy the equation$g(\alpha(X,$(1.6)

Y), JZ) $=g(\alpha(X,$Z), JY), that is,, $JZ)$ is symmetric withfor each $X$ , ,respect to $X$ , , ,be a local orthonormal frame field on andLet
be its dual coframe field. Then$\mathrm{Y}$$Z$$\mathrm{Y}$$\in T_{p}L$$g(\alpha(X, \mathrm{Y})$

$Z$

$L$

$\{e_{1_{\mathrm{t}}}\ldots, e_{m}\}$$\{\theta_{1}, \ldots, \theta_{m}\}$$\Omega=e^{\mathrm{v}\overline{-1}\psi}(\theta^{1}+\sqrt{-1}J\theta^{1})\Lambda\cdots\Lambda(\theta^{m}+\sqrt{-1}J\theta^{m})$.

is a smooth function defined on a neighborhood of .
is well-defined on (up to
The smooth function. Let+1). Note that
denote the Levi-Civita connectionof with respect to the induced metric from M. Let be an arbitrary. For arbitrary vector
point of . We may assume that

Here

$L$

$\psi$$L$$f:=\Omega(e_{1}, \ldots , e_{m})$$f=e^{\sqrt{-1}\psi}$$\nabla^{L}$

$L$

$p$

$L$

$(\nabla^{L}e_{i})_{p}=0$

48field $X$ on , we compute$L$$Xf=X(\Omega(e_{1}$,...,$e_{m}))$$=( \nabla_{X}^{M}\Omega)(e_{1}, \ldots, e_{m})+\sum_{i=1}^{m}\Omega(e_{1}, \ldots , \nabla_{X}^{M}e_{i}, \ldots , e_{m})$$= \sum_{i=1}^{m}\Omega(e_{1}, \ldots, \alpha(X, e_{i}), \ldots, e_{m})$$= \sum_{i=1}^{m}f(p)(\theta^{1}+\sqrt{-1}J\theta^{1})$A...

A $(\theta^{m}+\sqrt{-1}J\theta^{m})(e_{1}, \ldots, \alpha(X, e_{i}), \ldots, e_{m})$$=f(p) \sqrt{-1}\sum_{i=1}^{m}g(Je_{i}, \alpha(X, e_{i}))$$=f(p) \sqrt{-1}\sum_{i=1}^{m}g(JX, \alpha(e_{i}, e_{i}))$$=f(p)\Gamma-\overline{1}g(JX, H)=-f(p)\sqrt{-1}\alpha_{H}\{X)$.
$\square$

2. SPECIAL LAGRANGIAN cones. Let
Let $S^{2m-1}(1)$ denote the unit standard hypersphere of$S^{2m-1}$ (1)be an $(m-1)$ -dimensional smooth submanifold embedded in$S^{2m-1}(1)$ . The cone over I in
is defined as$\Sigma\subset$$\mathrm{C}^{m}$$\mathrm{C}^{m}$.
Let : $S^{2m-1}(1)-\mathrm{C}P^{m-1}$ be the Hopf fibration over the $(m-1)-$
is adimensional complex projective space. Then we know that
$C=C\Sigma$$:=\{t\sigma\in \mathrm{C}^{m}|t|\geq 0, \sigma\in\Sigma\}$$\pi$$C\Sigma$

special Lagrangian submanifold with an isolated singularity at if andonly if ) is a minimal Legendrian submanifold with respect to thestandard contact structure of $S^{2m-1}$ (1), and equivalently $L=\pi(\Sigma)$ is.
a minimal Lagrangian submanifold in$0$$\mathrm{C}P^{m-1}$

Example 2.1. In Harvey-Lawson [5] the first example of a special
Lagrangian cone with an isolated singularity at was given as$0$$C_{HL}^{m}:=\{(z^{1}, \cdot\cdot’, z^{m})\in \mathrm{C}^{m}|(\sqrt{-1})^{m+1}z_{1}\cdot\cdot ‘ z_{m}\in \mathrm{R}, |z^{1}|=\cdot\cdot 4 =|z^{m}f\}$Then$\Sigma_{HL}^{m-1}:=C_{HL}^{m}\cap S^{2m-1}(1)\subset S^{2m-1}(1)$is isometric to an $(m$-1 -dimensional flat toru$)$$T^{m-1}$

.

.

47
be the Laplacians of $(C’, g)$
Set $C’:=C\backslash \{0\}$ . Let A andon functions, respectively. A function on is called aandfor eachhomogeneous function of order on $C’$ if satisfies$t>0$ . Then such a function can be expressed as $u(r\sigma)=r^{a}v(\sigma)$ foris given bysome function on I. The relationship between A andthe formula$\Delta_{\Sigma}$$u$$(\Sigma, g_{\Sigma})$$\mathrm{w}\circ \mathrm{i}=tau$$u$$\alpha$$C^{t}$$\Delta_{\mathrm{E}}$$v$

Au(ra) $=r^{\alpha-2}(\Delta_{\Sigma}v(\sigma)-\alpha(\alpha+m-2)v(\sigma))$ .

Hence we see that is harmonic if and only if is an eigenfunction onwith eigenvalue a $(\alpha+m-2)$ .
Assume that $m$ $>2$ , Set
$v$

$u$

$\Sigma$$D_{\Sigma};=${$\alpha\in \mathrm{R}$|$\alpha(\alpha+m$-2) is an eigenvalue of$\Sigma_{\mathrm{E}}$},, denote
which is a countable and discrete subset of R. For each
, which isbythe multiplicity of eigenvalue $\alpha(\alpha+m-2)$ of
equal to the dimension of vector space of all homogeneous harmonic:
functions of order on $C’$ Then we define a function$\alpha\in D_{\Sigma}$$\Delta_{\Sigma}$$m_{\Sigma}(\alpha)$.

$\alpha$as$N_{\Sigma}$$\mathrm{R}arrow \mathrm{Z}$$N_{\Sigma}( \delta):=-\sum_{\alpha\in \mathcal{D}_{\Sigma}\cap(\delta,0)}m_{\Sigma}(\alpha)$

if$\delta$$<0$ and$N_{\Sigma}( \delta):=\sum_{\alpha\in D_{\mathrm{E}}\cap[0,\delta]}m_{\Sigma}(\alpha)$

if$\delta$$\geqq 0$.

$N_{\Sigma}$

is monotone increasing and upper semicontinuous.

Definition 2.1. The stability-index of the special Lagrangian coneis defined by$C$

s-ind(C ) $:=N_{\Sigma}(2)-b^{0}(\Sigma)-m^{2}-2m+1+\dim G$ .

Here G denotes a compact Lie subgroup of $SU(n)$ defined by$G:=${a $\in SU(n)$ |$a(\Sigma)=\Sigma$}.

, $m\Sigma(1)\geqq 2m$ , $m\Sigma(2)\geqq m^{2}-1-\dim G$ .
Note thatIndeed, the first equality is trivial. The second inequality follows fromthe famous Tsunero Takahashi’s theorem ([15]). The third inequality
is the n-dimensionalcan be shown as follows :Assume thatholomorphic
sectional curvacomplex projective space with constantture 4. Then $M$ is a compact Einstein-K\"ahler manifold with positive
Einstein constant $\kappa=2n$ . Let $V_{1}(M)$ denote the vector space of all$=4n$ . Let
the first eigen-functions of $CPn$ with the first eigenvalue$m_{\Sigma}(0)=b^{0}(\Sigma)$$M=\mathrm{C}P^{n}$$2\kappa$

48
denote the vector space of all eigen-functions of the Laplacianon I with eigenvalue $=2n$ . Then we can define a linear map$V_{\hslash}(\Sigma)$$\kappa$$V_{1}(M)\ni f-f|_{L}0\pi$$- \frac{\int_{L}f|_{L}dx}{\mathrm{V}\mathrm{o}1(L)}\in V_{\hslash}\{\Sigma)$.

Then its kernel is{ $f\in V_{1}(M)|f|_{L}$ is constant}the flow of$\cong\{V\in su(n)\}$$V$preserves },$L$which is isomorphic to the Lie algebra of . Since $\dim V_{1}(M)=n^{2}-1$ ,we obtain $m(2)=\dim V_{\kappa}(\kappa)\geq\dim su(n)-\dim G=n^{2}-1-\dim G$ .
A special Lagrangian cone $G$ is called stable if s-ind((”) $=0$ . A
special Lagrangian cone is called rigid if $m_{\Sigma}(2)=m^{2}-1-\dim G*$
Obviously we have that if is stable, then is rigid.
$G$

$C$

$C$

$C$

Remark 1. Assume that is a stable special Lagrangian cone. If
is a Hamiltonian, thenis an embedded submanifold in. For example, see
stable minimal Lagrangian submanifold in[1] for Hamiltonian stability problem of Lagrangian submanifolds.
$C$$p\mathrm{i}(\Sigma)$$\mathrm{C}P^{n-1}$$pi(\Sigma)$$\mathrm{C}P^{n-1}$

3. SPECIAL LAGRANGIAN SUBMANIFOLD S WITH ISOLATED CONICALSINGULARITIESWe shall mention the results of Joyce on the deformations of ,or the local structure of moduli spaces around , and the regularityof special Lagrangian varieties, which are related to stability-index of
special Lagrangian submanifolds.
Let $A4$ be the moduli space of compact special Lagrangian submanifolds with isolated conical singularities embedded in $M$ . Mclean showedthat if is smooth (i.e. without singularities), then the moduli space
is a smooth manifold of dimension
around .
Joyce showed that if is a special Lagrangian submanifold withisolated conical singularities ,, , then the dimension of the obstruction space of is equal to the sum of stability-indices of special
Lagrangian cones Ci,
, :
$L$

$L$

$L$

$b^{1}(L)$$\mathcal{M}$

$L$

$L$

$C_{1}$$\cdots$$C_{k}$$L$$\cdots$$C_{k}$$\dim \mathcal{O}_{L}=\sum_{i=1}^{k}$s-ind(C i).

Next we shall mention the Joyce’s regularity results of special
grangian integral currents, or special Lagrangian varieties. Geometricmeasure theory implies the compactness of the space of such objects.
Suppose that is a special Lagrangian integral current and has themultiplicity 1 tangent cone at, Joyce show ed that if the$\mathrm{L}\mathrm{a}\sim$$L$$x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{L}$4S
tangent cone of L at x is a rigid special Lagrangian cone; then L has
an isolated conical singularity at x.

4. STABILITY-INDEX OF SPECIAL LAGRANGIAN cones OVER
CERTAIN COMPACT SYMMETRIC SPACES

It is really important to study concretely the stability and rigidityof special Lagrangian cones.
is stable andis unstable if
Joyce, Marshall proved that$m\neq8,9$$m\geqq 4$ , and.
is rigid if and only if
Lagrangian cones
special
We shall discuss stability and rigidity of. Let
constructed by the Lie theoretic method, which include$(U, G)$ be an Hermitian symmetric pair of compact type with the canon. Set $\dim(U/G)=2m$ . Let , denote
ical decomposition
the Ad(U)-invariant inner product of defined by (-1)-times KillingCartan form of . We decompose into the dirrect sum of the semisim. There is an
as follows :ple part
and the center
defines the invariant complex strucsuch thatelementture of $(U, G)$ . Relative to the complex structure the subspace can be. We take the decompoidentified with a complex Euclidean space$(U,G)$
into irreducible Hermitian symmetric pairs of compact
sition oftype :$C_{HL}^{3}$$C_{HL}^{m}$$C_{HL}^{m}$$C_{HL}^{m}$$\mathrm{u}=\mathfrak{g}$$\rangle_{\mathrm{u}}$$\langle$$+\mathfrak{p}$$\mathrm{u}$$\mathrm{u}$$\mathfrak{g}$$=\mathfrak{g}_{ss}\oplus \mathrm{c}(\mathfrak{g})$$\mathrm{c}(\mathfrak{g})$$\mathfrak{g}_{ss}$$\mathfrak{g}$$\mathrm{a}\mathrm{d}Z$$Z\in \mathrm{c}(\mathfrak{g})$$\mathfrak{p}$$\mathrm{C}^{m}$(U,$G)=(U_{1}, G_{1})\oplus\cdots\oplus(U_{s}, G_{s})$ .

Set $\dim(U_{i}/G_{i})=2m_{i}$ far$\mathrm{i}=1$,,$\cdots$

$s$

. Let

(4.1)

be the canonical, . Assume that there$\mathrm{u}_{i}=\mathfrak{g}_{i}+\mathfrak{p}_{i}$, 2,for each.
is an element: satisfying the condition.
, $c_{s}>0$ with
Choose positive numbers $c_{1}>0$ ,$a_{i}=1/\sqrt{2c_{i}m_{i}}$ for each
Ad(G ), . Set,Put$R$, which is an irreducible symmetric -space standard
embedded in a complex Euclidean space .
Ad(G)(yy). Set
Set y7, which is a symmetric $R$-space standard embedded in a complex. Note that we have the inclusions
Euclidean spacedecomposition of$\mathrm{i}=1$$(U_{i}, G_{i})$$\cdots$$s$$(\mathrm{a}\mathrm{d}\eta_{i})^{3}+4(\mathrm{a}\mathrm{d}\eta_{i})=0$$\eta_{i}\in \mathrm{P}$$\sum_{\dot{\mathfrak{g}}=1}^{s}1/c_{i}=1/c$$\cdots$$\mathrm{i}=1$$\cdots$$\hat{L}_{i}=$

$s$

$i$

$(a_{i}\eta_{i})\subset$$S^{2m_{j}-1}(\mathrm{c}_{\dot{\mathfrak{g}}}/4)\subset \mathrm{P}:$$\mathfrak{p}_{i}$$=a_{1}\eta_{1}+\cdots+a_{s}\eta_{s}\in \mathrm{p}$$\hat{L}=$$\subset S^{2m-1}(c/4)\subseteq$$\mathrm{p}$$\cong \mathrm{C}^{n+1}$$\mathfrak{p}$$\hat{L}=L_{1}^{\mathrm{A}}\mathrm{x}$\cdots$\mathrm{x}\hat{L}_{s}\subset S^{2m_{1}-1}(c_{1}/4)\mathrm{x}$

., .

x

’

$S^{2m_{s}-1}(c_{s}/4)\subset S^{2m-1}(c/4)(4.2)$

Note that is a compact $H$-minimal Lagrangian submanifold embed(see [3]).
ded in. Letofc’
We take an orthogonal decomposition
and $G’$ denote the analytic subgroup of generated by, Set. Then Ya is a Legendria$\hat{L}$$\mathrm{C}^{n+1}$$\mathrm{c}(\mathrm{g})=$$\mathfrak{g}’:=\mathfrak{g}_{\mathit{8}S}\oplus \mathrm{c}’$$\mathfrak{g}’$$\ovalbox{\tt\small REJECT}\=\mathrm{A}\mathrm{d}(G’)(\eta)\subseteq S^{2m-1}(c/4)\subset \mathfrak{p}$$\oplus\{Z\}_{\mathrm{R}}$$G$$\mathrm{c}(\mathrm{B})$50submanifold in $S^{2m-1}(c/4)$ . Moreover ) is a minimal submanifold in, .
,if and only if $c_{i}m_{i}=cm$ for eachIn the case where $(U, G)$ is irreducible, i.e. $s=1$ , by the classification, $SU(p)$ , $SU(p)/SO(p)$ ,theory of symmetric -spaces, is one of$SU(2p)/Sp(p)$ ,.
is one of compact irreducible symmetric
Now we assume thatspaces standardly embedded in $S^{2m-1}(1)$ as minimal Legendrian submanifolds in the above way :.
$=SU(p)$ , $SU(p)/SO(p)$ , $SU(2p)/Sp(p)$ $(p\geq 3)$ , or$\mathrm{i}=1,2$$S^{2m-1}(\mathrm{c}/4)$$R$$\cdots$$s$$S^{\mathrm{n}\mathrm{z}-1}$$\Sigma$$E_{6}/F_{4}$$\Sigma$$E_{6}/F_{4}$$\Sigma$

Note that the rank of these symmetric spaces is equal to $p-1$ , and thebe the special Lagrangian cone
is equal to 2. Let
rank ofover X. Then by using results of [14],[1],[2], we can show thein$C\Sigma$$E_{6}/F_{4}$$\mathrm{C}^{m}$following.

are all rigid.
(1), then$\Sigma=SU(3)$$SU(3)/SO(3)$, $SU(6)/Sp(3)(p=3)$ ,,(2) Ifis stable, and hence Legendrian stable.
(3) If $\Sigma=SU(p)$ , $SU(p)/SO(p)$ , $SU(2p)/Sp(p),p\geq 4_{l}C\Sigma$ is notstable, in fact not Legendrian stable.

Theorem 4.1.

$C\Sigma$$E_{6}/F_{4}$$C\Sigma$

5. STABILITY-INDEX OF A SPECIAL LAGRANGIAN CONE IN
OVER A MINIMAL LEGENDRIAN $SU(2)$ -ORBIT
Here we mention about the stability and rigid of an other example$\mathrm{C}^{4}$in$\mathrm{C}^{4}$.

be the complex vector space of all complex homogeneousLetof degree 3. We equip the standard
polynomials with two variables
product
such that
Hermitian inner$V^{3}$$z_{1},z_{2}$$\{v_{k}=\frac{1}{\sqrt{k!(3-k)!}}z_{1}^{3-k}z_{2}^{k}|k=0,1,2,3\}$. We know that is an
is a unitary basis of , andirreducible unitary representation of $SU(2)$ . Now we consider the orbit. Then the orbit $\Sigma=\rho_{3}(SU(2))v\subset$
of $SU(2)$ throughis a 3-dimensional minimal Legen drian submanifold embedded in$V^{3}$$V^{3}$$V^{3}\cong \mathrm{C}^{4}\cong \mathrm{R}^{8}$$v= \frac{1}{\sqrt{2}}(v_{0}+v_{3})$$S^{7}(1)$$S^{7}(1)$.

over the orbit
Theorem 5.1. The special Lagrangian conein$\Sigma=\rho_{3}(SU(2))(v)$ is not Legendrian stable, and hence not stable, but
it is rigid. Its stability-index and Legend rian-indez of are given by$C$$\mathrm{C}^{4}$$C$

s-ind(C) $=3$

andl-ind(C) $=11(\overline{\sim}8+3)$ .

51Remark 2. We can show that $\pi(\Sigma)=\pi(\rho_{3}(SU(2))(v))$ is a 3-dimensional
compact Hamiltonian stable minimal Lagrangian submanifold embed.
ded in
Question 1. Investigate asymptotically conical special Lagrangian subasymptotic to special Lagrangian cones with high symmanifolds inmetry which were discussed in this article ?
Question 2. Can we find compact special Lagrangian submanifolds in
Calabi-Yau manifolds with conical singularities which were discussed$\mathrm{C}P^{3}$$\mathrm{C}^{n}$in this article ?

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$\mathrm{K}\mathrm{U}$

DEPARTMENT OF MATHEMATICS, OSAKA CITY UNIVERSITY, SUGIMOTO, sUMIYOSHI, OSAKA, 65S-8585, JAPAN-mait address: ohnita(Ds ci . osaka-co . .jp$E$$\mathrm{a}\mathrm{c}$