A FAMILY OF MAXIMAL SURFACES IN LORENTZ-MINKOWSKI THREE-SPACE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number, November 26, Pages S (6)8543- Article electronically published on May 8, 26 A FAMILY OF MAXIMAL SURFACES IN LORENTZ-MINKOWSKI THREE-SPACE YOUNG WOOK KIM AND SEONG-DEOG YANG (Communicated by Richard A. Wentworth) Abstract. We prove the existence of an infinite family of complete spacelike maximal surfaces with singularities in Lorentz-Minkowski three-space and study their properties. These surfaces are distinguished by their number of handles and have two elliptic catenoidal ends.. Introduction Spacelike maximal surfaces in the Lorentz-Minkowski three-space L 3 arise as solutions of the variational problem of locally maximizing the area among spacelike surfaces. By definition, they have everywhere vanishing mean curvature. There are many interesting local properties of maximal surfaces similar to those of minimal surfaces. In particular, maximal surfaces possess a Weierstrass-type representation formula [2]. The most significant difference between minimal and maximal surfaces is the existence of singularities for maximal spacelike surfaces. It has been known that spacelike planes are the only spacelike maximal surfaces which are complete [2, 3], which is probably the main reason why people have not paid much attention to maximal surfaces, and there are not so many known examples of interesting maximal surfaces. If we allow some sort of singularities for maximal surfaces, however, the situation changes. Recently many interesting examples with isolated singularities have been found and studied by F. J. López, R. López, Fernández, and Souam [5, 6, 7, 4]. Umehara and Yamada showed that if admissible singularities are included, then there is an interesting class of objects which they call maxfaces [7]. In this article, we show that there exists an infinite family of maximal surfaces with singularities (Figure ) and investigate their properties. Our main results are Theorem (Main Theorem). For every natural number k, there exists a maximal surface with singularities, which we call M k, with the following properties: (i) M k is parameterized on a compact Riemann surface of genus k with two points removed. Received by the editors April 8, 25 and, in revised form, May 23, Mathematics Subject Classification. Primary 53A, 53C5. Key words and phrases. Lorentz-Minkowski space, spacelike maximal surface, elliptic catenoidal ends c 26 American Mathematical Society Reverts to public domain 28 years from publication

2 338 YOUNG WOOK KIM AND SEONG-DEOG YANG Figure. Genus and 2 examples, their upper halves and vertical slices (ii) M k is a complete generalized maximal surface [4]. Inparticular,M is a complete maxface of finite type [7]. (iii) M k has two ends, both of which are elliptic catenoidal. (iv) M k is not embedded. (v) M k intersects the xy-plane in k + quadruple line segments (of finite length), which meet at equal angles at the origin. Removal of the line segments disconnects the maximal surface into a union of two topological annuli. (vi) There are 4(k +) swallowtail singularities. All the other singularities except the ones in the xy-plane are cuspidal edges. The singularities in the xy-plane are cuspidal crosscaps. (vii) M k has a symmetry group of order 8(k +), which are generated by () K =, L = k+ π), M = R( where R(θ) is the matrix of rotation by θ in the xy-plane. k+2 R( k+ π) According to Schoen [6], catenoids in E 3 are the only complete minimal surfaces of finite total curvature with 2 embedded ends. Existence of M shows that Schoen s nonexistence result does not hold for complete maximal surfaces with singularities of finite type, with the definition of being complete and of finite type as in Definition 3. It should be remarked that the integral of the Gauss curvature on M k outside the singular set is infinite. Rossman and Sato previously showed that there exists a genus immersed catenoid in H 3 ( ) [5].,

3 A FAMILY OF MAXIMAL SURFACES IN L The method of construction is to use the Weierstrass representation theorem. The Weierstrass data as well as the main ideas for the existence proof have been motivated by the work of Hoffman and Meeks III [9], and the idea is explained in [8]. In view of the ways in which Costa-Hoffman-Meeks family of minimal surfaces are deformed, it is expected that these maximal surfaces can be deformed in various ways. We thank Fujimori, Rossman, Umehara and Yamada for many helpful conversations as well as pointing out an error in an earlier version of this article, and Jae Ung Yu for drawing the pictures in this article. 2. Preliminaries In this article, the coordinates system of the Lorentz-Minkowski three-space L 3 is set as follows: L 3 = {(x, y, t) R 3 : ds 2 = dx 2 + dy 2 dt 2 }. An immersion X : M 2 L 3 of a 2-manifold is called spacelike if the induced metric is positive definite. A spacelike immersion is called maximal if the mean curvature is everywhere. A maximal map is a smooth map from a 2-manifold into L 3 if there is an open dense subset W of M 2 such that f W is a maximal immersion. A maxface is defined to be a maximal map all of whose singular points are admissible. See [7] for details. It is in this category that we state the following Weierstrass representation formula for maximal spacelike surfaces. Theorem 2 ([2, 7]). Let g and η be a meromorphic function and a holomorphic -form on a Riemann surface M 2 such that ( + g 2 ) 2 η 2 is a Riemannian metric on M 2 and ( g 2 ) 2 η 2 does not vanish identically. Then the following defines a maxface from M 2 to L 3 : p (2) X : M 2 L 3 (, X(p) =Re ( + g 2 )η, i( g 2 )η, 2gη ) p if all the periods vanish. Furthermore, (i) the Gauss map ν is given by ν = ( 2Re g g 2, 2Img g 2, + g 2 g 2 (ii) the induced metric is ds 2 =( g 2 ) 2 η 2, (iii) the Gauss curvature of the surface is K = 4 ( g 2 ) 4 dg 2 η 2. The converse also holds. The simplest examples of maximal surfaces with singularities are the rotationally invariant maximal surfaces. There are three kinds of such surfaces depending upon the causal character of the axis of rotation [] (Figure 2). Definition 3 ([7]). AmaxfaceX : M 2 L 3 is complete (resp. of finite type) if there exist a compact set C M 2 and a symmetric 2-tensor T on M 2 such that T vanishes on M 2 \ C and ds 2 + T is a complete metric (resp. a metric of finite total Gaussian curvature) on M 2,whereds 2 is the pull-back of the Lorentz-Minkowski metric by X. ),

4 3382 YOUNG WOOK KIM AND SEONG-DEOG YANG Figure 2. Elliptic, parabolic, and hyperbolic catenoids We use x, y, t : M 2 R to denote the coordinate functions. Then, it is immediate from (2) that (3) (x + iy)(p) = p p η + p p g 2 η, t(p) = p p gη + p p gη. 3. The Weierstrass data Let k {, 2, 3, } be an arbitrary natural number. Consider the closed Riemann surface of genus k, M k = {(α, β) (C { }) 2 : β k+ = α k (α +)(α )}. The maximal surfaces we construct have the following Weierstrass data: (4) η = dα β, g = σ β α, M k = M k \{(, ), (, )}, (α,β )=(, ), where σ is a positive real number to be determined later. (α,β )=(, ) is the reference point for the integration of the associated one-forms in the Weierstrass representation theorem, hence the image of (α,β )=(, ) is the origin (,, ) L Symmetries of the Weierstrass data Definition 4. Let c := e π k+ i,andletκ, λ, µ :(C { }) 2 (C { }) 2 be κ(α, β) :=(ᾱ, β), λ(α, β) :=( α, c k β), µ(α, β) :=(α,cα 2 β). They preserve M k. In the rest of this article, we think of them as defined on M k.notethatλ does not preserve the base point of integration (, ) M k, unlike κ and µ. It is straightforward to see that the order of κ is 2, that the order of λ and µ is 2(k +), and that κ, τ, µ form a group of order 8(k + ). We will see later that these conformal diffeomorphisms induce the symmetries of the resulting maximal surfaces. As a first step to observe it, we prove Lemma 5. κ, λ, µ preserve M k.furthermore, g κ =ḡ, g λ = c k g, g µ = cg, κ η = η, λ η = c k η, µ η = c η.

5 A FAMILY OF MAXIMAL SURFACES IN L Figure 3. FD and its image by X The proof is straightforward. An immediate consequence is the following. Corollary 6. Let φ =(φ,φ 2,φ ) T = ( ( + g 2 )η, i( g 2 )η, 2gη ) T. Then κ φ = K φ, λ φ = Lφ, µ φ = Mφ, where K, L, M are the matrices in (). 5. Resolution of the period problem For convenience, we sometimes regard M k as obtained by gluing k +number of α-planes with the branch cuts between and andbetweenandalong the real line. We fix once and for all the α-plane with β(α) > forα>, and a set FD in this α-plane (Figure 3): FD := {α C : arg α π, α }. 2 8(k +)copiesoffd are needed to cover M k through the action of κ, τ, andµ. Let γ(t) = 2 + eit, t 2π. Then, a homology basis of M k is generated by applying κ and µ to γ. Therefore, to show that the immersion X is well defined on M k, we only need to check whether the periods on γ and on loops around (, ), (, ) vanish. Lemma 7. The periods around (, ) and (, ) vanish. Proof. Argue as in the proof of [9, Proposition 3.4] with κ, µ and K, M. To deal with the period around γ, we first observe from (3) and (4) that (5) t(α, β) =2σ ln α, which immediately implies that the t-component of the period around γ vanishes. Now we show the following: Lemma 8. The xy-components of the period around γ vanish if and only if A σ = 2 B, where A = dt k+ t k ( t 2 ), B = k+ t k ( t 2 )dt t 2.

6 3384 YOUNG WOOK KIM AND SEONG-DEOG YANG Proof. From (3), it follows that the xy-components of the period around γ vanish if and only if γ η + γ g2 η =. We will compute each of the integrals as γ collapses to the double line segment connecting and. We immediately see that γ η = (c c )A. The situation for γ g2 η = σ 2 β γ α dα is a bit different since β 2 c ɛ α dα 2 does not converge as ɛ, where c ɛ is a circle of radius ɛ centered at. We observe, however, that β (6) α 2 dα = (k +)d( β α )+ 2β α 2 dα. Since 2β γ α 2 dα =2(c c)b, we get the conclusion. Therefore the map X : M k L 3 is well defined. 6. Geometry of the surfaces 6.. Symmetries of X(M k ) generated by K, L, M. Let K, L, M be the matrices in (). Recall that λ does not preserve the base point of integration, hence L may not be a symmetry of X(M k ). However, we have Lemma 9. X(, ) = (,, ). Proof. Let γ(θ) =e iθ for θ [,π]. Then (x + iy)(, ) = γ η + γ g2 η. We evaluate these integrals by collapsing γ to the line segment from (, ) to (, ). We immediately see that γ η =( c)aand that, using (6), γ g2 η =(c )2σ 2 B.Here A, B are the constants in the statement of Lemma 8. Therefore, (x+iy)(, ) =. It is easy to see that t(, ) =. Then, the following is now immediate. Lemma. (i) X(M k ) is not embedded. (ii) L isalsoasymmetryofx(m k ), hence the group of order 8(k+) generated by K, L, M is contained in the symmetry group of X(M k ). ( ) k π (iii) MK is the reflectional symmetry with respect to the line y =tan k+ 2 x, t =. ( ) (iv) KL is the reflectional symmetry with respect to the plane y =tan x. k+ (v) K is the reflectional symmetry with respect to the xt-plane. (vi) L k KMKL is the reflectional symmetry with respect to the xy-plane. (Note, however, that the xy-plane does not intersect X(M k ) orthogonally.) 6.2. Image of the boundary of the fundamental domain. Now let us look at X( FD). Let us define γ (t) :=t and γ 3 (t) :=it for <t<, γ 2 (θ) :=e iθ for θ π 2. We do not distinguish γ i from its image in FD. Then, FD = γ γ 2 γ 3.First, let us take a look at X(γ 2 ). Lemma. t(γ 2 (θ)) =, and(x + iy)(γ 2 (θ)) = F (θ)e k π k+ 2 i for some nonnegative real valued function F with F () = F ( π 2 ) =, which is strictly increasing on [,θ ] and strictly decreasing on [θ, π 2 ],whereθ =arcsin. In particular, 2σ k+ X(i) =X(γ 2 ( π 2 )) = (,, ). π 2

7 A FAMILY OF MAXIMAL SURFACES IN L Proof. Given θ [,π], let γ(t) =e it,t [,θ]. Then, where C(θ) := θ γ where D(θ) = θ γ η = θ ie it dt k+ e kit (e 2it ) = i C(θ), k+ 2i k+ dt. On the other hand, sin t k+ e i(k+)θ (e iθ e iθ ) g 2 η = σ 2 θ e 2iθ ie iθ dθ = i k+ 2iσ 2 D(θ), k+ sin tdt. So,iff(θ) :=C(θ) k+ 4σ 2 D(θ), then (x + iy)(e iθ )= i f(θ) = k π k+ k+ 2i 2 e k+ 2 i f(θ). We immediately see that f is real-valued and f() =. From Lemma 9, we know that f(π) =. Since 2f( π 2 )=f(π) wealsohavef( π 2 ) =. We also see that f (θ) > if<θ<θ,andf (θ) < ifθ <θ< π 2. By letting F (θ) =f(θ)/ k+ 2, we have the conclusion. Hence, X(γ 2 ) is a double line segment in the xy-plane. Next we show that X(γ ) and X(γ 3 ) are curves in vertical planes. Lemma 2. (i) X(γ ) lies in the quarter plane with x>, y =, t >, and is the graph of a strictly increasing function( t = t(x) ) with t() =. π (ii) X(γ 3 ) lies in the quarter plane x>, y =tan k+ 2 x, t >, andis the graph of a strictly increasing function t = t( x 2 + y 2 ) with t() =. Proof. For X(γ ), we observe that (x + iy)(u) = = u u η + u g 2 η dt u k+ t k (t 2 ) + σ2 k+ t k (t 2 ) dt := g(u). Since g(u) R + for u>, we have x(u) >, y(u) =. Since t(u) =2σ ln u>, we see that X(γ ) lies is the quarter plane stated in the lemma. Furthermore, since g() = and g (u) > foru>, we see that x is a strictly increasing function of u, hence we may regard u as a strictly increasing function of x. The conclusion for X(γ ) follows. For X(γ 3 ), we observe that (x + iy)(iv) = iv i η + iv ( v = e π k+ 2 i i g 2 η dt v k+ t k (t 2 +) + σ2 t 2 ) k+ t k (t 2 +) dt. Here we used the fact that i can also be used as the reference point of integration since the image of α = i is also the origin, as was proved in Lemma. The conclusion follows in similar ways as above. t 2

8 3386 YOUNG WOOK KIM AND SEONG-DEOG YANG The above lemma along with the following shows that the image by X of a neighborhood of γ and of γ 3 is indeed perpendicular to vertical planes y =and ( y =tan( π σ α+α k+ 2 ) x, respectively. Note that ηdg = dα ) 2 k+ α α α is real on γ and γ 3, and purely imaginary on γ 2. Lemma 3 (Gauss map on X( FD)). The unit timelike normal vectors of X(M k ) along γ lie in the vertical plane y =. The unit timelike normal vectors of X(M k ) along γ 2 and on γ 3 lie in the vertical plane y =tan( π k+ 2 ) x. Proof. We have g γ (t) =g(t) =σ k+ t t for t, g γ 2 (θ) =g(e iθ )=σ k+ 2sinθe π k+ 2 i for θ [,π/2], g γ 3 (t) =g(it) =σ k+ t + t e π k+ 2 i for t, which imply the claims Asymptotic behavior of the ends. Define z by α = z (k+). Then, z in a neighborhood of is a local coordinate for the end (α, β) =(, ). Using this, we see that ( ( + g 2 ),i( g 2 ), 2g ) η ( ) = σ2 (k +) z 2 + O(z),i σ2 (k +) 2σ(k +) z 2 + O(z), + O(z) dz. z Therefore, z =,orequivalently(α, β) =(, ), is a simple end of type I in the language of [], which converges to an elliptic catenoid Singularities of the surfaces. Note that, for any M k,themetricds 2 vanishes at the points where g = σ β α =. Excluding those points, the metric does not vanish on M, but it vanishes at (α, β) =(±, ) on M k with k 2. We now want to locate the points with g = on FD. σ β α = implies α k+ = σ k+ β k+ = σ k+ α k α 2, hence α α = σ (k+). By letting α = re iθ,thisequals (7) r 2 + r 2 = σ 2(k+) +2cos2θ := t(θ). Since r 2 + r 2 2 for any r>, we conclude that t 2. Now t(θ) 2 θ θ, where θ =arcsin 2σ k+. Note that this θ has already appeared in Lemma. The above shows that the singularities are in FD { arg α θ } (Figure 4). For each θ [,θ ], there is t(θ)+ t(θ) a unique value of r whichsolves(7):r(θ) = Fujimori, Saji, Umehara and Yamada showed that the generic singularities of maximal surfaces in L 3 consist of cuspidal edges, swallowtails, and cuspidal crosscaps, whose typical examples are given by (u 2,u 3,v), (3u 4 + u 2 v, 4u 3 +2uv, v), and (u, v 2,uv 3 ), respectively [8, 7]. They cover all the singularities of X(M k ).

9 A FAMILY OF MAXIMAL SURFACES IN L FD Singularities θ Figure 4. Singularities for X(M ),X(M 2 ) Lemma 4. Let r :[,θ ] R be the function defined above. (i) The image by X of a neighborhood of α = r(θ)e iθ for any θ (,θ ) is locally diffeomorphic to a cuspidal edge. The image by X of a neighborhood of α = r() is locally diffeomorphic to a swallowtail. The image by X of a neighborhood of α = e iθ is locally diffeomorphic to a cuspidal crosscap. (ii) The singularities form two closed curves in M k. In each curve, there are 2(k+) swallowtail singularities. Half of them have constant t =2lnr() >, and the other half have constant t = 2lnr() <. be the differentiation by α. By direct computa- Proof. Let ˆη = η tions, we see that dα g g 2ˆη = σ(k +) =/β, and α 2 + α 2, g g =(k +)αα2 α 2 +, ( ) g g g g 2ˆη = 4 α 2 σ α 4. Hence, ( ) g α 4 ( ) g Re g 2ˆη = σ(k +) α 2 2, Im i α 2 α 2 g 2ˆη = σ(k +) α 2 2, { ( ) } g g Re g g 2 = 2 ( ) α 2 ˆη σ α 4 + α2 α 4, { ( ) } g g Im g g 2ˆη = 2i ( ) α 2 σ α 4 α2 α 4. Now we apply the criteria in [7, Theorem 3.] and [8, Theorem 2.3] to conclude (i). (ii) is obvious from Lemma Horizontal cross sections of X(M k ). The horizontal cross sections of the maximal surfaces are obtained as the image of the circles α = constant because of (5) (Figure 5). (α, β) =(, ) is the bottom end, and (α, β) =(, ) isthetop end since lim α t(α, β) = and lim α t(α, β) = About the orientation. We see that g(, ) = g(, ) =, whichmeans that both of the normals at the top and at the bottom catenoidal ends are past pointing (or future pointing depending upon our choice). Note that the normals at the two ends of the elliptic catenoids are in opposite directions. This is in complete analogy with the relation between Costa-Hoffman-Meeks surfaces and catenoids in Euclidean three-space.

10 3388 YOUNG WOOK KIM AND SEONG-DEOG YANG Figure 5. Horizontal cross sections of X(M ),X(M 2 )ast The symmetry group of X(M k ). Having understood the geometry of X(M k ), we can now determine the full symmetry group of X(M k ). Lemma 5. The symmetry group G of X(M k ) is of order 8(k+), and is generated by K, L, M. Proof. Note that g(, ) = g(, ) =. Since a symmetry sends ends to ends, it preserves vertical normals. On X(M k ), we see that X(, ) and X(, ) are the only points where the Gauss map is vertical. Since both of them are mapped to L 3 by X, any element of G must fix and leave the t-axis invariant. Let G be the subgroup of G of time orientation-preserving isometries. Then the index [G : G ]ofg in G is 2. Let G 2 be the subgroup of G of orientationpreserving isometries. Then [G : G 2 ]=2andG 2 is a cyclic rotation group around π k+. the t-axis. The order of G 2 is at least 2(k +) since it contains the rotation by On the other hand, since any element in G 2 must send a swallowtail singularity to another swallowtail singularity of the same time value, and since there are no or 2(k + ) swallowtail singularities on horizontal planes, the order of G 2 is 2(k +). Combining them all, we see that G is of order 8(k + ). Since G contains the group generated by K, L, M which is of order 8(k + ), we get the conclusion. Appendix A. Singly periodic maximal and minimal surfaces Integrating the Weierstrass data (4) with k =andwithα = i as the reference point of integration yields the null holomorphic curve (Ψ, Ψ 2, Ψ )inc 3 after α is substituted by w+i w i, while the null holomorphic curve (Φ, Φ 2, Φ 3 )inc 3 is obtained by chance: Ψ = 2 ln w +2σ2 w2 w 2 +, Φ = 2 ln w 2σ2 w2 w 2 +, Ψ 2 = i 2 ln w 2iσ2 w2 w 2 +, Φ 2 = i 2 ln w +2iσ2 w2 w 2 +, ( Ψ =2σln i w + i ) (, Φ 3 =2σln i w + i ). w i w i So, for any nonzero real number σ, the real parts of (Ψ, Ψ 2, Ψ )and(φ, Φ 2, Φ 3 ) give maximal and minimal surfaces, respectively, well defined on the universal cover

11 A FAMILY OF MAXIMAL SURFACES IN L Figure 6. Singly periodic maximal and minimal surfaces of Ĉ \{,,i, i}., are planar ends, and i, i are catenoidal ends. They are singly periodic (Figure 6). The Weierstrass data (4) for k produces doubly periodic minimal surfaces in E 3. References [] Luis J. Alías, Rosa M. B. Chaves, and Pablo Mira, Björling problem for maximal surfaces in Lorentz-Minkowski space, Math. Proc. Camb. Phil. Soc., 34 (23), MR9724 (24d:5376) [2] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math., 5 (97), MR2642 (4:886) [3] S.-Y. Cheng and S.-T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 4 (976), MR436 (55:463) [4] F. J. M. Estudillo and A. Romero, Generalized maximal surfaces in Lorentz-Minkowski space L 3, Math. Proc. Camb. Phil. Soc., (992), MR5327 (93b:53) [5] I. Fernández and F. López, Periodic maximal surfaces in the Lorentz-Minkowski space L 3, arxiv:math.dg/4246 v3. [6] I. Fernández, F. López and R. Souam, The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space L 3, arxiv:math.dg/333 v2. [7] I. Fernández, F. López and R. Souam, The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space L 3, arxiv:math.dg/429 v. [8] S. Fujimori, K. Saji, M. Umehara, K. Yamada, Cuspidal crosscaps and singularities of maximal surfaces, Preprint. [9] D. Hoffman and W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math., 3 (99), 34. MR38356 (9i:53) [] T. Imaizumi, Maximal surfaces with conelike singularities of finite type, KobeJ.Math.,8 (2), 5 6. MR (23c:534) [] T. Imaizumi, Maximal surfaces with simple ends, Kyushu J. Math., 58 (24), MR25379 (25a:53) [2] O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space L 3,TokyoJ.Math., 6 (983), MR73285 (85d:533) [3] O. Kobayashi, Maximal surfaces with conelike singularities, J. Math. Soc. Japan, 36 (984), MR75947 (86d:538) [4] F. J. López, R. López, and R. Souam, Maximal surfaces of Riemann type in Lorentz- Minkowski space L 3, Michigan J. of Math., 47 (2), MR8354 (22c:539) [5] W. Rossman and K. Sato, Constant mean curvature surfaces with two ends in hyperbolic space, Experimental Math., 7, no. 2 (998), 9. MR6773 (2b:534) [6] R. Schoen, Uniqueness, symmetry, and embeddeness of minimal surfaces, J. Diff. Geom., 8 (983), MR73928 (85f:53)

12 339 YOUNG WOOK KIM AND SEONG-DEOG YANG [7] M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space, arxiv:math.dg/3739 v6; to appear in Hokkaido Mathematical Journal. [8] S.-D. Yang, Elliptic catenoids in L 3 with an arbitrary number of handles, Proceedings of the International Workshop on Integral Systems, Geometry, and Visualization, Nov. 24, Kyushu University, Fukuoka, Japan. Department of Mathematics, Korea University, Seoul 36-73, Korea address: ywkim@korea.ac.kr Department of Mathematics, Korea University, Seoul 36-73, Korea address: sdyang@korea.ac.kr