INDEX, NULLITY AND FLUX OF n-noids

Transcription

1 Kato, S. and Tatemichi, K. Osaka J. Math. 53 (06), 0 39 INDEX, NULLITY AND FLUX OF n-noids Dedicated to Professor Atsushi Kasue on his sixtieth birthday SHIN KATO and KOSUKE TATEMICHI (Received June 3, 04, revised December, 04) Abstract In this paper, we give a criterion for 4-noids to have nullity greater than 3 and its applications. We also compute the indices and the nullities of some families of Z N - invariant n-noids, and analyze the correspondence between nullity and a flux map.. Introduction Let M be a Riemann surface, and XÏ M R 3 a complete conformal minimal immersion. The index of X is the supremum of the numbers of negative eigenvalues of the Jacobi operator ½ dg on relatively compact domains of M, where ½ is the Laplacian with respect to the metric ds X ds 3 on M induced by X, and R 3 GÏ M S is the Gauss map of X. Fischer Corbrie [5] and Gulliver-Lawson [6, 7] proved that X has a finite index if and only if it has finite total curvature, and Osserman [] proved that if X has finite total curvature, then M is conformally equivalent with a compact Riemann surface ÆM punctured by a finite number of points, and its Weierstrass data (g, ) extends meromorphically on ÆM. If X has finite total curvature, then its index depends only on the extended Gauss map G Æ gï ÆM S R 3, where we denote the stereographic projection from the north pole by. Indeed, the index coincides with the number of negative eigenvalues of the operator ½, where ½ is the Laplacian with respect to the metric G ds S on ÆM induced by G. Hence we denote the index of X by both Ind(X) and Ind(g). On the other hand, the nullity of X is defined as the dimension of the space of bounded Jacobi functions, elements of the kernel of the Jacobi operator. It also depends only on G since it coincides with the number of zero eigenvalues of ½. Hence we denote the nullity of X by both Nul(X) and Nul(g) in the same way as index. Since there exists a 3-dimensional isometry group of parallel translations in R 3, Nul(X)3 holds for any X. The following fact is very significant since it characterizes nullity completely in a sense. 00 Mathematics Subject Classification. Primary 53C4; Secondary 58E.

2 0 S. KATO AND K. TATEMICHI Theorem. (Ejiri Kotani [4], Montiel Ros []). If X has finite total curvature, then Nul(X) 3 holds if and only if its Gauss map is realized also as the Gauss map of some flat-ended non-branched or branched minimal surface. Other than this result, it is also known that if X has finite total curvature, and if all of its ends are embedded ends and parallel with each other, then Nul(X) 3 holds (cf. [6]). On the other hand, we see, by combining Nayatani s example in [0, 4] and basic facts, that there exists a family of X s such that Nul(X) 3, each of which has N catenoidal ends arranged on the positions of the vertices of a regular N-gonal pyramid (see Example 3.3). By Theorem., each of these X s also has the same Gauss map as that of some flat-ended minimal surface. However, the reasons for nontrivial nullity seem to be different between flat or parallel ones and pyramidal ones, since the former have natural deformations which induce nontrivial bounded Jacobi functions, that is homotheties or rotations (or deformations to their associated family or López Ros deformations if the genus of ÆM is zero), but such deformations for the latter are not so trivial. What happens in the latter case? Which kind of X has the same Gauss map as a flat-ended surface in general? In particular, is some symmetry necessary? Since the eigenvalues depend continuously on any parameter of deformations of X, index is lower semicontinuous, and nullity is upper semicontinuous with respect to the parameter. Therefore, determining the index and the nullity of some sampling point makes a significant role. For instance, Nayatani [8] showed that Ind(X) (n ) n 3 and Nul(X) 3 hold for Jorge-Meeks surface with n ends (n 3). Since the moduli space of maps which are realized as the Gauss maps of some flat-ended minimal surface has codimension greater than as a subset of the space of meromorphic maps of the common degree, Ejiri Kotani [4] showed that if the genus of ÆM is zero, that is ÆM S Ç C Ï C {½}, then Ind(X) d and Nul(X) 3 hold for a generic X such that deg g d. In particular, since there is no flat-ended minimal surface with deg g, Ind(X) 3 and Nul(X) 3 hold for any X such that deg g. On the other hand, since there are many flat-ended minimal surfaces with d deg g 3, Nul(X) 3 (and Ind(X) d also) holds for some X such that deg g 3. In this paper, we study index and nullity of n-noids, complete conformal minimal immersions with n embedded ends. In 3 we summarize basic facts on n-noids and flat-ended minimal surfaces respectively, and in 4 5, we give a criterion for 4-noids to have nullity greater than 3, and its applications. In 6 7, we compute the indices and the nullities of some families of Z N -invariant n-noids. In 8, we discuss the correspondence between nullity and a flux map. Both the authors would like to thank Professor Toshihiro Shoda for fruitful discussions and useful comments. They also thank Professors Norio Ejiri and Shin Nayatani for helpful advices.

3 INDEX, NULLITY AND FLUX OF n-noids 03. Basic facts on n-noids and their flux Let XÏ M ÆM Ò {q,, q n } R 3 be a complete conformal minimal immersion with finite total curvature. We use the Enneper Weierstrass representation formula of the following type: X(z) Re z t ( g, Ô (g ), g). The flux vector of the end q j of X is defined by the integral ³ j Ï j n ds, where j is a loop surrounding q j from the left, n is a unit conormal vector field along j such that ( ¼ j, n) is positively oriented, and ds is the line element of X(M). ³ j is independent È of the choice of j. By divergence formula, or residue theorem, it always n holds that j ³ j 0. We call this equality the flux formula. It is known that, if the end q j is an embedded end, then it is asymptotic to a catenoid or a plane. We call such an end a catenoidal end or planar end respectively. It is also known that the flux vector of any embedded end is parallel to its limit normal. Hence we can define the weight of the embedded end q j by Û(q j ) Ï ³ j (4G(q j )), where G is the Gauss map of X as before. In another word, the weight is the ratio of the size of the asymptotic catenoid of the end to the standard catenoid. Û(q j ) 0 holds if and only if the end q j is a planar end. We call X an n-noid if all the ends q,, q n are embedded ends. For an n-noid X, we can rewrite the flux formula by using the weights as follows: n j Û(q j )G(q j ) 0. We È call a suit of unit vectors Ú,, Ú n and real numbers a,, a n satisfying n j a j Ú j 0 a flux data. We say an n-noid or a flux data is of TYPE III (resp. TYPE I, TYPE II) if the flux vectors span a 3- (resp. -, -) dimensional vector space. Umehara, Yamada and the first author [,, 3] proved that, for generic flux data of TYPE III (or TYPE II with n 8), there exists an n-noid X of genus zero satisfying G(q j ) Ú j, Û(q j ) a j ( j,, n). In general, if the genus of ÆM is zero, that is ÆM C, Ç then the Weierstrass data (g, ) of an n-noid XÏ M C Ç Ò {q,, q n } R 3 with q j ½, g(q j ) p j ½, Û(q j ) a j ( j,, n) is of the following form: (.) g(z) P(z) Q(z), Q(z) dz

4 04 S. KATO AND K. TATEMICHI with (.) P(z) n j p j b j z q j, Q(z) n j b j z q j and (.3) n kák j n kák j b j b k p k p j q k q j a j ¾ R, b j b k p j p k q k q j 0, ( j,, n). Hence, to find an n-noid with the prescribed flux data, we have only to solve (.3) as an algebraic equation. More precisely, For any given p j, a j ( j,, n) satisfying the balancing condition n a j Ú j n j j t Re p j a j pj, Im p j pj, p j p j t (0, 0, 0), if q j, b j ( j,, n) satisfy the equation (.3), and if P(z) and Q(z) have no common zero, then the Weierstrass data (g, ) given by (.) with (.) realizes an n-noid such that g(q j ) p j, Û(q j ) a j, ( j,, n). We note here that it is useful to rewrite the second equalities in (.3) as Ab 0 with A Ï ¼ p p p 0 q q p n q n q p 0 q q p n p p.. q n q p n p q q n p n p q q n 0 ½, b Ï ¼ ½ b b... b n If q,, q n and b,, b n realizes some n-noid, then it must hold that det A 0 and 0 b ¾ Ker A. In particular, in the case n 4, rank A 3 (resp. ) holds if the data is of TYPE III (resp. TYPE II) (cf. [, Proposition 3.]). We also note here that we can define the relative weights of end-pairs (q j, q k ) ( j, k,, n; j k) by Û jk Ï b j b k p k p j q k q j,

5 INDEX, NULLITY AND FLUX OF n-noids 05 which is conformal invariants satisfying Û kj Û jk and È n ká k j Û jk Û(q j ) (cf. [0, 9]). In general, we may assume that q j ½, p j ½ ( j,, n) without loss of generality. However, in some cases, it is more useful to assume that some p j s and q j s are ½. In such case, we need to modify the equation (.3) and (.) as follows: () The case that q p ½ and q j ½, p j ½ ( j,, n): (.4) n k b j b n b b k a, n ká k j b b k ( p k ) 0, k b j b p j n kák j b j b k p k p j q k q j a j, b j b k p j p k q k q j 0, ( j,, n), and (.5) P(z) b n j p j b j z q j, Q(z) n j b j z q j. () The case that q p p ½ and q j ½ ( j,,n), p j ½ ( j 3,,n): (.6) n n k3 k3 b b k a, c b k q k q a, p b j b k p j b j c b j b k q k3ák k q j a j, j b c ( ) n b b k ( p k ) 0, c b n k3 q q j n k3 b j b p j b j c p k c b k q k q 0, p j q q j n k3ák j b j b k p j p k q k q j 0, ( j 3,, n),

6 06 S. KATO AND K. TATEMICHI and (.7) P(z) b c z q n j3 p j b j z q j, Q(z) n j3 b j z q j. 3. Basic facts on flat-ended minimal surfaces In this section, we summarize basic facts on flat-ended minimal surfaces in the style suitable for our situation. Let XÏ M ÆM Ò {q,, q n } R 3 be a complete conformal minimal branched (or non-branched) immersion with finite total curvature. We call the end q j is a flat end if it is asymptotic to a plane. A flat end is not necessarily an embedded end. We say X is flat-ended if all of the ends q,, q n are flat ends. Since minimal surfaces with embedded flat ends, that is planar ends, have a correspondence with Willmore surfaces in R 3, they were studied in early years. In particular, Bryant [] proved many significant results on their moduli spaces. However, to determine the indices and the nullities of minimal surfaces by applying Theorem., we have to consider minimal surfaces with non-embedded flat ends also. Here we call the end q j of X is of order k if at least one of and g has a pole at q j and the maximum of the orders at q j is k. For the well-definedness of X, k must be greater than. The end is an embedded end if and only if k. On the other hand, the end q j of order k is a flat end if and only if q j is a zero of g ¼ of order at least k (see [4, Proposition 3.5]). Now, let X be of genus zero, that is ÆM C. Ç We may assume that q j ½ ( j,, n) without loss of generality as before. If q j is an end of order k j, then k j must hold for the well-definedness of X around q j ( j,, n). On the other hand, since ½ is not an end of X, both and g do not have a pole at ½, that is, both dz and g dz have a zero of order at least at ½. Hence, if deg g d 0, then it must hold that d deg g max deg n j k j. n j (z q j ) k j µ, deg dz n j (z q j ) k g j dz µµ Now we see that n j k j max{n, d }.

7 INDEX, NULLITY AND FLUX OF n-noids 07 È n Moreover, if j k j d È n, then X has j k j d branch points if counting their multiplicities. On the other hand, if X is flat-ended, then it holds that n j (k j ) #{z ¾ ÆM g ¼ (z) 0} d. Combining these facts, we have the following: Lemma 3.. Let X be a flat-ended conformal minimal branched (or nonbranched) immersion of genus zero. Suppose that each end q j of X is of order k j ( j,, n), and that deg g d 0. Then it holds that max{n, d } n k j n d. In particular, it must hold that d 3 and 4 n d. j For instance, in the case d 3, we have max{n, 8} n j k j n 4 and n 4. Hence the orders of the ends must satisfy the following: 8 4 j k j 8, {k j } {,,, }. In 4, we give a classification and a characterization of the surfaces in this class. On the other hand, in the case d 4, we have max{n, 0} n j k j n 8 and n 4, 5 or 6. In this case, there are the following five possibilities: n 4, 0 4 k j 0, {k j }{,,, 4} or {,, 3, 3}, n 5, n 6, j 0 5 k j, j 6 k j, j {k j }{,,,, } or {,,,, 3} ( branch point), {k j }{,,,,, } ( branch points),

8 08 S. KATO AND K. TATEMICHI where we counted the multiplicity of branch points as before. It is known that {k j } {,,,, } is not the case (cf. []; see Remark 4. for a short proof of this fact). However, to give some estimate for nullity, we must consider the remaining cases. The following result has also to be recalled here. For later use, we describe the statement by means of a GL(, C)-action, in place of the SO(3, C)-action Bryant considered. Lemma 3.. Let (g, ) be the Weierstrass data of a flat-ended minimal surface of genus zero. Then «g g Æ, ( g Æ) is also the Weierstrass data of some flat-ended minimal surface for any «,,, Æ ¾ C such that «Æ 0. Ô Proof. By the assumption, all the residues of t (( g ), ( g ), g) vanish. Hence those of t ({( g Æ) («g Ô ) }, {( g Æ) («g ) }, («g )( g Æ)) also, because ¼ {( g Æ) («g ) } Ô {( g Æ) («g ) } («g )( g Æ) ¼ ½ «Ô Æ («Æ ) ( «Ô Æ) ( «Æ ) «Ô Æ («Æ) ( «Ô Æ) ( «Æ) («Æ ) ¼ ( g ) Ô (g ) g ½. Since this transformation is linear, the property that all the ends are flat is preserved (cf. [4, Proposition 3.]). By this lemma, we see that two rational functions g and g have the same index and nullity with each other if there exist Möbius transformations ³ and F of Ç C satisfying g Æ ³ F Æ g. In this paper, we say that these two functions g and g are equivalent with each other. EXAMPLE 3.3 ((N )-noids with pyramidal flux). ½ Nayatani [0] showed that Ind(g N & M )d (N M) and Nul(g N & M )5 hold for the map g N & M (z)ï z N z M (N, M ¾ N, N M 3). Let N be an integer such that N 3, and set N Ï e Ô N. For the data

9 INDEX, NULLITY AND FLUX OF n-noids 09 j,, N N p j p N j ½ a j a a ¼ with p ¾ RÒ{0, }, a ¾ RÒ{0} and a ¼ Na( p )( p ), by solving the equation (.4), we get the following Weierstrass data (cf. [8, Example 3.3]): g pyr (z) (N )(p )z N p N {(N )p (N )} pyr, N p z N a N p (N )p (p z N dz. ) z N p N This data realizes an (N )-noid whose flux vectors are arranged on the positions of the vertices of a regular N-gonal pyramid. Since holds for N p (N )(p ) gpyr(z) z N z (N )p (N ) N (N )(p ) p, g pyr is equivalent with z N z, that is a special case of g N & M, and hence Ind(g pyr ) d N (N ) 4 and Nul(g pyr ) 5 hold. 4. A criterion in the case deg g 3 In this section, we give a criterion for the rational functions of degree 3 to be the Gauss map of some flat-ended minimal surface. As we have already seen in 3, for any flat-ended minimal surface such that deg g 3, each of its ends must be an embedded flat end, namely the surface is a flat-ended 4-noid. The structure of the space of flat-ended n-noids was already studied by Bryant [] (see also Kusner Schmidt [5]), and we can compute the index and the nullity of any flat-ended 4-noid by applying Nayatani s estimate for g N & M with (N, M) (, ). First, we summarlize these facts in the style suitable for our consideration. In the case of flat-ended n-noids, that is the case a j 0 ( j,,n), the algebraic equation (.3) is equivalent with the following equation: (4.) n kák j n kák j b k q k q j 0, p k b k q k q j 0, ( j,, n).

10 0 S. KATO AND K. TATEMICHI Hence, to classify all of the flat-ended 4-noids, we have only to solve (4.) with n 4 completely as an algebraic equation with respect to q j and b j ( j,, 4). Note here that the equation (4.) is rewritten as A 0 b A 0 c 0 with A 0 Ï ¼ 0 q q q n q 0 q q.. q n q q q n q q n 0 ½, b Ï One of the most typical examples is given by the data j 3 4 ¼ ½ b b.., b n p j p p p Ô p Ô c Ï ¼ p b p b. p n b n ½. with p Ï ( Ô 6 Ô ). By solving (4.), we get a family of solutions j 3 4 q j p p p Ô p Ô b j p Ô t p Ô t p Ô t p Ô t where t ¾ CÒ{0} is a parameter of homothety. The Weierstrass data of the flat-ended 4-noids given by these solutions are as follows: g tet (z) Ï Ô 3z Ô z(z 3), Ï Ô Ô z(z 3) t dz (z p )(z Ô z(z 3) 8t p ) z Ô dz. 4 3z In 6, we will analyze a family of functions which includes g tet as a special case. By Lemma 3., we see that the Weierstrass data (4.) «gtet g tet Æ, ( g tet Æ) «Ptet Q tet, ( P P tet tet ÆQ tet ÆQ tet ) dz also realizes a flat-ended 4-noid for any «,,, Æ ¾ C such that «Æ 0, where we set P tet (z) Ï Ô Ô ( 3z ) (z p )(z p ), Q tet(z) Ï Ô Ô z(z 3) (z p )(z p ).

11 INDEX, NULLITY AND FLUX OF n-noids The following fact seems to be well known among the researchers of this field. Bryant [, 5] pointed out it to classify Willmore immersions from S into S 3 with Willmore energy. It follows directly by (4.). Lemma 4.. If X is a flat-ended 4-noid, then its ends q, q, q 3, q 4 satisfy the condition that the cross ratio q 34 Ï (q q )(q 3 q 4 )(q q 3 )(q q 4 ) coincides with 6 e Ô 3 or 6 e Ô 3, that is, the ends can be arranged on the positions of the vertices of a regular tetrahedron. Proof. By the first equalities of (4.), we have 0b¾Ker A 0. Now, since n 4, it holds that This implies our assertion. 0 det A 0 q 34 q 34. q 34 (q q 4 )(q q 3 ) REMARK 4.. By the second equalities of (4.), we also have 0 c ¾ Ker A 0. Since deg g n, b and c are linearly independent. Hence rank A 0 must be smaller than or equal to n. Therefore, also in the case n 5, the cross ratio of each four of {q,, q 5 } must be 6 or 6. However there are no arrangement of the ends which satisfies such a condition. Indeed, if q 34 6 and q 35 6, then q 543 q 34 q , 6. Hence there are no flat-ended 5-noids. This is an essence of the proof of the nonexistence result for n 5 given by Bryant []. The following fact asserts that the inverse of the assertion of Lemma 4. is also true. It is a restatement of the classification by Bryant [, 5] we have already mentioned before. Lemma 4.3. Let g be a rational function of deg g 3. If the cross ratio of the zeroes of g ¼ coincides with 6 or 6, then g is equivalent with the Gauss map of one of the flat-ended 4-noids given by (4.). Proof. By the assumption, CÒ{z Ç g ¼ (z) 0} is conformally equivalent with CÒ Ç {z g ¼ tet (z) 0} CÒ{ p Ç Ô, p }, where p Ô ( 6 Ô ). Hence there exists a Möbius transformation ³ such that {z (g Æ ³) ¼ (z) 0} {z g ¼ tet (z) 0}. Set Then it holds that g Æ ³(z) Ï È 3 j0 «j z j È 3 j0 j z j. (g Æ ³) ¼ (z) («3 «3 )(z Ô 4 3z ) È 3 j0 j z j, «3 «3 0.

12 S. KATO AND K. TATEMICHI Hence we have («3 «3 ) 0, 3(«3 0 «0 3 )(««) Ô 3(«3 «3 ), («0 «0 ) 0, «0 «0 («3 «3 ), from which it follows that «Ô 3«3, «Ô 3«0, Ô 3 3 and Ô 3 0, where we use the assumption deg g 3. Now, we see that where and g Æ ³(z) «3z 3 Ô 3«0 z Ô 3«3 z «0 3 z 3 Ô 3 0 z Ô 3 3 z 0 «0( Ô 3z )«3 z(z Ô 3) 0 ( Ô 3z ) 3 z(z Ô 3) «0g tet (z)«3 0 g tet (z) 3 F Æ g tet (z), F(Û) «0Û «3 0 Û 3 «0 3 «3 0 Ô «3 3 «3 Ô 3 Ô («3 «3 ) 3 0. By combining Lemmas 4. and 4.3, we see that the Weierstrass data of any flatended 4-noid is given by (4.) up to conformal coordinate transformations. In particular, all the elements have the common index and nullity. The function z z, that is one of Nayatani s examples g N & M (z) with (N, M) (, ), is also in this case. Indeed, if we choose Möbius transformations ³(z) Ï 6 z p 3, F(Û) pz Ï pû 6 Û p with p ( Ô 6 Ô ), then we have F Æ g tet (z) g N & M Æ ³(z). Hence, for any flat-ended 4-noid, its index and nullity must be 4 and 5 respectively. Now, we get the following: Lemma 4.4. Let X be a conformal minimal immersion of genus zero such that degg 3. If the cross ratio of the zeroes of g ¼ coincides with 6 or 6, then Ind(X)4 and Nul(X) 5 hold. Otherwise, Ind(X) 5 and Nul(X) 3 hold. Let us give a criterion for the assumption in Lemma 4.4, which we will use in 5. First we prepare a criterion for polynomials.

13 INDEX, NULLITY AND FLUX OF n-noids 3 Lemma 4.5. Set f (z)ï È 4 j0 a j z j (a 4 0). Then Ç CÒ{z f (z)0} is conformally equivalent with Ç CÒ{z g tet ¼ (z) 0} if and only if D tet0 Ï a 3a 3 a a 0 a 4 0 holds. Proof. Let {z,z,z 3,z 4 } be the set of solutions of f (z)0. Then Ç CÒ{z,z,z 3,z 4 } is conformally equivalent with Ç CÒ{z g tet ¼ (z)0} if and only if its cross ratio z 34 Ï (z z )(z 3 z 4 )(z z 3 )(z z 4 ) coincides with either 6 or 6, that is, z 34 z This equality is equivalent with 0 (z z ) (z 3 z 4 ) (z z )(z 3 z 4 )(z z 3 )(z z 4 )(z z 3 ) (z z 4 ) i j z i z j i j,ik, jk z i z j z k 6z z z 3 z 4 Ï D. Denote the elementary symmetric expression of degree j by j, and set È, Ï i j z i z j and,, Ï È i já ki, j z i z j z k. Then, since D,,, 6 4,,,, 6 4 and 3,, 4 4, we have D,,, 6 4 3,, 3( ) a a 4 3 a3 a 4 a a 4 a 0 a 4 a 4 (a 3a 3 a a 0 a 4 ) D tet0 a 4. As a corollary to this lemma, we have a criterion for rational functions. Lemma 4.6. Let g(z)«(z) (z) be a rational function of deg g 3. Set «(z)ï È 3 j0 «j z j and (z) Ï È 3 j0 j z j. Then Ç CÒ{z g ¼ (z) 0} is conformally equivalent with Ç CÒ{z g tet ¼ (z) 0} if and only if D tet Ï 3«3 0 ««3«0 3 0 holds. Proof. By applying Lemma 4.5 to f (z) «¼ (z) (z) «(z) ¼ (z), we have D tet0 (3«3 0 ««3«0 3 ) D tet. By combining Lemmas 4.4 and 4.6, we get the following: Theorem 4.7. Let g(z) È 3 j0 «j z j È 3 j0 j z j be a rational function of degg 3. If D tet 3«3 0 ««3«0 3 0, then Ind(g) 4 and Nul(g) 5 hold. Otherwise, Ind(g) 5 and Nul(g) 3 hold. 5. Index and nullity of 4-noids In this section, we observe which kind of 4-noid has the same Gauss map as that of a flat-ended 4-noid by applying Theorem 4.7.

14 4 S. KATO AND K. TATEMICHI As we have already mentioned in introduction, any n-noid of TYPE I has nontrivial bounded Jacobi functions, and hence Ind(X) 4 and Nul(X) 5 hold for any 4-noid X of TYPE I. On the other hand, these equalities also hold for any 4-noid X of TYPE III whose flux vectors are arranged on the positions of the vertices of a regular trigonal pyramid. It should be remarked here that each 4-noid in these two families is located at a special position in the space of 4-noids from the viewpoint of the equation det A 0. Indeed, for any flux data of TYPE I, det A 0 is automatically satisfied and suitable conformal classes cannot be decided only by det A 0. On the other hand, for any flux data of TYPE III, the number of suitable conformal classes is at most 4, since det A 0 is equivalent with a quartic equation on the cross ratio of the ends (cf. [, 3]). However, for any data of pyramidal type as above, the number is, that is, pyramidal examples are given by double solutions of the equation det A 0. Hence it seems that there is some correspondence between the equation det A 0 and nullity, and the similar phenomenon is also expected in the case of TYPE II. However the condition that the cross ratio of the ends of X is given by a double solution of det A 0 is not a sufficient condition for Nul(X) 3. Indeed, for any flux data of TYPE II, each 4-noid is given by a double solution of det A 0 by the reason we describe below. But, for instance, the nullity of Jorge-Meeks 4-noid is 3. Here we present a result similar to above in the case of quadruple solutions. Theorem 5.. If a 4-noid X is of TYPE II, and if its conformal class is given by a unique quadruple solution of the equation det A 0 on the cross ratio of the ends for some given flux data, then Ind(X) 4 and Nul(X) 5 hold. Proof. Since deg g 4 3 holds for any 4-noid XÏ M CÒ{q Ç,q,q 3,q 4 } R 3, the limit normals p, p, p 3, p 4 must take at least two distinct values. First, we consider the case that at least one of p j s is different from the others. In this case, we may assume that p is different from the others, and in particular p ½ without loss of generality. Since we can also choose three q j s freely, we assume here that q ½, q 0 and q 3. For the data and the assumption j 3 4 p j ½ p p 3 p 4 q j ½ 0 q

15 INDEX, NULLITY AND FLUX OF n-noids 5 with p, p 3, p 4 ¾ R and q ¾ Ç CÒ{½, 0, }, set A Ï ¼ 0 p p 3 p 4 p p 0 p p 3 p 4 q p p 3 (p 3 p 3 p 4 ) 0 q p 4 p 4 p q p 4 p 3 q If the equation (.4) has a solution, then it holds that det A 0 and 0 b t (b, b, b 3, b 4 ) ¾ Ker A. In particular, since A is an alternative matrix, the pfaffian Pf A of A, that is a homogeneous polynomial of components of A satisfying det A (Pf A), is also defined, and given by Set Pf A q(q ) {p (p 3 p 4 )q p 3 (p p 4 )(q ) p 4 (p p 3 )q(q )}. pfa(q) Ï q(q ) Pf A p 4 (p p 3 )q ( p p 3 p 4 p p 3 p 4 )q p 3 (p p 4 ). Then its derivative pfa ¼ (q) and discriminant D pfa of pfa(q) as a polynomial of q are given respectively by 0 ½. pfa ¼ (q) p 4 (p p 3 )q ( p p 3 p 4 p p 3 p 4 ), D pfa Ï pfa ¼ (q) 4p 4 (p p 3 ) pfa(q) 3p p 3 p 4 p p 3 p 4 p p 3 p 4 p p 3 p 4 p p 3 p 4 p p 3 p p 4 p 3 p 4. Now, for any b ¾ Ker A Ò {0}, the corresponding Weierstrass data (g, ) is given by (.) with (.5). Set «(z) Ï z(z )(z q)p(z) and (z) Ï z(z )(z q)q(z). Then we have «(z) b z 3 {(q )b p b p 3 b 3 p 4 b 4 }z { qb (q )p b qp 3 b 3 p 4 b 4 }z qp b, (z) (b b 3 b 4 )z { (q )b qb 3 b 4 }z qb.

16 6 S. KATO AND K. TATEMICHI Since Ker A is spanned by t (p p 3, p 3, p, 0) and t ( (p p 4 ), p 4 q, 0, p q), b is given by b ¼ ½ b b b 3 b 4 ¼ s(p p 3 ) t(p p 4 ) sp 3 tp 4 q sp tp q for some (s, t) ¾ C Ò {(0, 0)}. By direct computation, we have ½ and D tet (q) (b b b b 3 )q { b b (p p 4 )b b 4 (p 3 p 4 )b 3 b 4 }q {b b b b 4 (p p 3 )b b 3 (p 3 p 4 )b 3 b 4 } { (p p 3 )(p p 3 )q p 3 (p p 3 )q p 3 (p )}s { p 4 (p p 3 )q 3 (p p 3 p 4 p p 3 p 4 )q (p p 3 p 4 p p 3 p 4 )q p 3 (p p 4 )}st {p 4 (p )q 3 p 4 (p p 4 )q (p p 4 )(p p 4 )q}t, p 4 (p p 3 ) D tet (q) D (q, s, t) pfa ¼ (q)c (s, t) pfa(q) pfa ¼ (q)c (s, t) pfa(q), where we set D (q, s, t) Ï (p p 3 ) {(p p 3 )q p 3 }s 4(p p 3 ){(p p 3 )q p 3 (p p 4 )}st [{ (p p 3 )(p ) p p 4 (p p 3 )(p p 4 )}q p 3 (p )(p p 4 )]t, C (s, t) Ï (p p 3 )st (p )t, C (s, t) Ï (p p 3 )(p p 3 ) s (p p 3 ){ 3(p p 3 ) p 4 (p p 3 )}st (p ){(p p 3 ) p 4 (p p 3 )}t. Now, if the data (g, ) realizes a well-defined 4-noid X, then pfa(q)0 and hence p 4 (p p 3 ) D tet (q) D (q,s,t)pfa ¼ (q) holds. Moreover, if X is given by a quadruple solution, then D pfa 0 and hence pfa ¼ (q) 0 holds. If p 4 0 or p p 3 0, then pfa ¼ (q) p p 3 0 must hold. However, in the case p 4 0, it is already known that this is not the case (cf. [, Theorem 4.5]), and in the case p p 3 0, this contradicts the assumption that p, p 3 ¾ R. Hence p 4 0 and p p 3 0, and we get D tet (q) 0.

17 INDEX, NULLITY AND FLUX OF n-noids 7 Secondly, we consider the case that p takes the same value with p only. In this case, we may assume that p p ½ without loss of generality. Since we can also choose three q j s freely, we assume here that q ½, q 0 and q 3. For the data and the assumption j 3 4 p j ½ ½ p 3 p 4 q j ½ 0 q with p 3, p 4 ¾ R and q ¾ Ç CÒ{½, 0, }, set ¼ 0 p 3 p 4 ½ ÄA Ï p 4 0 p 3 q p 3 p 3 0 p 4 p 4 q p 4 p 3 q p 3 p 4 q If the equation (.6) has a solution, then it holds that det Ä A 0 and 0 Ä b Ï t (b, c, b 3, b 4 ) ¾ Ker Ä A. In particular, since Ä A is also an alternative matrix, the pfaffian Pf Ä A of Ä A is also defined, and given by Set 0. Pf Ä A q(q ) {(p 3 p 4 )q p 3 p 4 (q ) p 3 p 4 q(q )}. pfa(q) Ï q(q ) Pf Ä A p 3 p 4 q ( p 3 p 4 )q p 3 p 4. Then its derivative pfa ¼ (q) and discriminant D pfa of pfa(q) as a polynomial of q are given respectively by pfa ¼ (q) p 3 p 4 q ( p 3 p 4 ), D pfa Ï pfa ¼ (q) 4p 3 p 4 pfa(q) 3p 3 p 4 p 3 p 4 (3p 3 p 4 )(p 3 p 4 ). Now, for any Ä b ¾ Ker Ä A Ò {0}, the corresponding Weierstrass data (g, ) is given by (.) with (.7). Set «(z) and (z) as in the first case. Then we have «(z) b z 3 {(q )b c p 3 b 3 p 4 b 4 }z { qb (q )c p 3 qb 3 p 4 b 4 }z qc, (z) (b 3 b 4 )z ( qb 3 b 4 )z.

18 8 S. KATO AND K. TATEMICHI Since Ker Ä A is spanned by t (p 3, p 3,, 0) and t ( p 4, p 4 q, 0, q), Ä b is given by Äb ¼ b c b 3 b 4 ½ ¼ sp 3 tp 4 sp 3 tp 4 q s tq for some (s, t) ¾ C Ò {(0, 0)}. By direct computation, we have ½ and D tet (q) (q )( p 3 s p 4 qt ), p 3 p 4 D tet (q) D (q, s, t) pfa ¼ (q)c (t) pfa(q) pfa ¼ (q)c (s, t) pfa(q), where we set D (q, s, t) Ï p 3 q( p 3 s p 4 qt ), C (t) Ï t, C (s, t) Ï p 3 s (p 3 p 4 )t. Now, if the data (g, ) realizes a well-defined 4-noid X, then pfa(q)0 and hence p 3 p 4 D tet (q) D (q, s, t) pfa ¼ (q) holds. Moreover, if X is given by a quadruple solution, then D pfa 0 and hence pfa ¼ (q) 0 holds. If p 3 0 or p 4 0, then pfa ¼ (q) must hold. This contradicts pfa ¼ (q) 0. Hence p 3 0 and p 4 0, and we get D tet (q) 0. Now, in both cases, by applying Theorem 4.7, we get our assertion. It is clear from the proof of Theorem 5. that the sufficient condition pfa ¼ (q) 0 is valid independent of the choice of the parameters s and t. On the other hand, we can see also by the proof of Theorem 5. that D (q, s, t) 0 also implies D tet (q) 0. Since this condition depends on the choice of s and t, it comes from another type of deformation. Before concluding this section, we present a description of the condition D tet 0 by means of relative weights. Theorem 5.. Let X be a 4-noid of genus zero. Then Ind(X) 4 and Nul(X) 5 hold if and only if its relative weights and cross ratios satisfy the following condition: (5.) (Û Û 34 )(Û 3 Û 4 )q 34 (Û 4 Û 3 )q 43 0.

19 INDEX, NULLITY AND FLUX OF n-noids 9 This condition holds if the relative weights satisfy the following condition: (5.) Û () () Û (3) (4) Û () (3) Û () (4) ( ¾ S 4 ), (Û Û 34 )(Û 3 Û 4 Û 4 Û 3 ) (Û 3 Û 4 )(Û 4 Û 3 Û Û 34 ) (Û 4 Û 3 )(Û Û 34 Û 3 Û 4 ) 0. Proof. We may assume that q j ½ ( j,, 3, 4) without loss of generality. Under this assumption, the Weierstrass data of X is given by (.) and (.). Set «(z)ï P(z) É 4 j (z q j) and (z)ï Q(z) É 4 j (z q j). Then, by direct computation, we have D tet (Û Û 34 )(q q ) (q 3 q 4 ) (Û 3 Û 4 )(q q 3 ) (q q 4 ) (Û 4 Û 3 )(q q 4 ) (q q 3 ), from which the condition (5.) follows. Now, if the inequalities in (5.) hold, then the cross ratios of the ends are given by the following: q () () (3) (4) (q () q () )(q (3) q (4) ) (q () q (3) )(q () q (4) ) Û () (4)Û () (3) Û () (3) Û () (4) Û () (4) Û () (3) Û () () Û (3) (4) ( ¾ S 4 ). By applying these equalities to (5.), we get the equality in (5.). As for the second assertion of Theorem 5., the condition given by the inequalities in (5.) is a generic condition. Indeed, it means that the cross ratios of the ends and the limit normals are different from each other, and these cross ratios coincide with each other only if the limit normals p, p, p 3, p 4 satisfies det A (q,q,q 3,q 4 )(p, p, p 3, p 4 ) det p j p k p k p j j,k,,3,4 0. In the case that this equality holds, a 4-noid realizing the corresponding relative weights is not unique, and we cannot determine the index and the nullity of such a 4-noid only by its relative weights. 6. Flat-ended minimal surfaces of deg g 4 In this section, we want to determine the nullities and the indices of a family of rational functions of degree greater than or equal to 3, which includes g tet and g N & M as special cases.

20 0 S. KATO AND K. TATEMICHI Let N, L ¾ N, L N, and let s, s, s, s ¾ CÒ{0}, s s s s 0. Set g(z) Ï s z N s z L (s z N s ). In the case that some of s, s, s, s are equal to 0, g is equivalent with z NL, z N L, z NL z L or z N L z L. The first (resp. second) one is the Gauss map of Jorge-Meeks (N L )- (resp. (N L )-) noid, and Nayatani [8] proved that Ind(g) d (N L ) 3 (resp. (N L ) 3) and Nul(g) 3 hold. Also for the last one, Nayatani [0] proved that Ind(g) d (N ) 4 and Nul(g) 5 hold under the assumption L or L N (see Example 3.3). Here we assume that each of s, s, s, s is not equal to 0. In this case, g is equivalent with (6.) g s (z) Ï sz N z L (z N s) for some s ¾ CÒ{0} satisfying s 0. Indeed, it holds that g( Éz) s s NL g s (Éz) with Ï ( s s s s ) (N) and s Ï ( s s s s ). In the case (N, L)(, ), g Ô 3 coincides with the function g tet which we considered in 4, and it is also equivalent with z z. However g s is not equivalent with g N & M (z) z N z M for a general pair (N, L), since the orders of zeroes of dg s and dg N & M do not coincide with each other in general, where G s Ï Æ g s and G N & M Ï Æ g N & M. Indeed, the derivative of g s is given by g s ¼ (z) z L {N z N (s ) L(z N s)(sz N )} z L (z N s) [L(szN s){(n L)s (N L)}z N ] z L (z N s) Ls(z N t)(z N t ) z L (z N s), where t ¾ C Ò {0} is a solution of the quadratic equation Ls(t ) {(N L)s (N L)}t 0. Since s 0, it holds that t s 0 and st 0. Now, assume that (N L) s (N L) 0 additionally. Then the equation above does not have a double solution, that is, t 0. Hence q, j Ï t N j N and q, j Ï t N j N ( j,, N) are the solutions of the algebraic equation Ls(z N ){(N L)s (N L)}z N 0, and zeroes of dg s of order. Moreover, 0 and ½ are zeroes of dg s of order L. On the other hand, the derivative of g N & M is given by g ¼ N & M (z)

21 INDEX, NULLITY AND FLUX OF n-noids ( j,, N M) are zeroes of dg N & M of order, and 0 (resp. ½) is a zero of dg N & M of order M (resp. N ). Kusner [4] gave an example of flat-ended minimal surface whose Gauss map is given by g s in the case that N, L N and s Ô N. In general, if g s is the Gauss map of some flat-ended minimal surface, then each (N z NM M)z M, and hence (MN) (NM) j NM of the ends of the surface must be a zero of dg s. Moreover, if (g s,) is the Weierstrass data of the surface, then each of q, j, q, j ( j,, N) (resp. 0, ½) is not a pole or a pole of, g s, g s whose order is (resp. at most L if L ). Hence must be of the following form: z L (z N s) h(z) (z N t) (z N t ) dz, where h is a polynomial of degree at most N. Here g s and g s is given by Set g s (z N s)(sz N )h(z) (z N t) (z N t ) dz, g s (sz N ) h(z) z L (z N t) (z N t ) dz. h(z) Ï N l0 h l z l. It is clear that both and g s do not have a pole on z 0. On the other hand, g s has the following Laurent expansion near z 0. g s L dz l0 h l z L l holomorphic part. If z 0 is a regular point or a well-defined flat end, then the residue of this form must be zero, that is, h L 0. By the coordinate transformation Éz Ï z, the Weierstrass data (g s, ) is rewritten as follows: g s (Éz) Éz L (Éz N s) séz N, (séz N ) É h(éz) Éz L (Éz N t ) (Éz N t) déz, where É h is a polynomial defined by Éh(Éz) Ï Éz N h(éz ) N l0 h N l Éz l.

22 S. KATO AND K. TATEMICHI Here g s and g s is given by g s (séz N )(Éz N s)é h(éz) (Éz N t ) (Éz N t) déz, g s Éz L (Éz N s) É h(éz) (Éz N t ) (Éz N t) déz. It is clear that both g s and g s do not have a pole on Éz 0. On the other hand, has the following Laurent expansion near Éz 0. L déz l0 h N l Éz L l holomorphic part. If Éz 0, that is z ½, is a regular point or a well-defined flat end, then the residue of this form must be zero, that is, h N L 0. Now, let us calculate the residues of, g s and g s at z q, j ( j,, N). By direct computation, we have the following expansions: z L t q, j N L q, j z L t N L z N s (t s) L (z q, j q, j ) O((z q, j ) ) L (z q, j q, j ) O((z q, j ) ), Nt (z q, j (t s)q, j ) O((z q, j ) ) sz N (st ) z N t z q, j q, j z N t Nt from which it follows that z L (z N s) t t Nst (st )q, j (z q, j ) O((z q, j ) ) Nt (z N t) (z N t ) t(t s) q, j (z N s)(sz N ) (t t )q, j (z q, j ) O((z q, j ) ) N q, j (z q, j ) O((z q, j ) ) N (t ) NL (z N t) (z (t s)(st )q, j N t ) N (t ) (sz N ) z L (z N t) (z N t ) (st ) q, j N t(t ) N L,,,,0,, O(), (z q, j q, j ) z q, j (z q, j q, j, ) z q, j O(), (z q, j q, j, ) z q, j O(),

23 INDEX, NULLITY AND FLUX OF n-noids 3 where we set By the definition of t, we have,0 Ï (N L ), Ï (N L ), Ï,0,. Nt(st ) (t )(t s), Nt(t s) (t )(st ), Nt(st,0, (N L )(N L ) ) Nt(t s) (t )(t s) (t )(st ) ) {Nt(s L(t s)(st )} (t s)(st 0, ) namely,0,,. Denote this value by. Then we get the following residues: Since NL Res zq, j t(t s) q, j (h N (t ¼ (q ), j )q, j h(q, j )), Res zq, j g s (t s)(st )q, j N (t ) (h ¼ (q, j )q, j h(q, j )), Res zq, j g s (st ) q, j N L N t(t ) (h ¼ (q, j )q, j h(q, j )). h ¼ (z)z h(z) z ( h(z)zh ¼ (z)) z N l0á ll,n L ( l)h l z l, we get the following conditions for the end q, j to be a well-defined flat end: (6.) 0 N l0áll,n L N l0áll,n L ( l)h l q, j l ( l)h l t ln N l( j ).

24 4 S. KATO AND K. TATEMICHI For any integer m such that 0 m N, it holds that (6.3) 0 N j N m( j ) N N l0áll,n L N l)t l0áll,n L ( ln h l j ( l)h l t ln N l( j ) N (l m)( j ) Nt mn {( m)h m ( m N)th mn } (m ¾ Z N,L ), Nt mn ( m)h m (m N L, N ), Nt mn ( m N)th mn (m L ), where we set Z N,L Ï {m ¾ Z 0 m N, m L, N L }, and we use the equality N (l m)( j ) N 0 j N (l m mod N), (l m mod N). The condition (6.3) is equivalent with the original condition (6.) ( j,, N). By replacing t by t, we can also show that the ends q, j ( j,, N) are well-defined flat ends if and only if (6.4) 0 Nt mn m N {( m)h m ( m N)t h mn } (m ¾ Z N,L ), Nt mn m N ( m)h m (m N L, N ), Nt mn m N ( )( m N)t h mn (m L ) holds for any integer m such that 0 m N, where we set Ï (N L ) (N L ) Nt(t s) (t )(st ) Nt(st ) (t )(t s) (N ). Combining the conditions (6.3) and (6.4) for the ends q, j and q, j ( j,, n), we get the following: () For m N L or N (resp. L ), we can choose h m 0 (resp. h mn 0) if and only if m m 0 (resp. m N m N 0). () For m ¾ Z N,L, we can choose (h m, h mn ) (0, 0) if and only if 0 m ( m N)t m ( m N)t ( m)( m N)t ( m N)t( m) t {( m)( m N)( m N)( m)t }.

25 INDEX, NULLITY AND FLUX OF n-noids 5 Since (N ), () is the case if and only if m N. In this case, by N and the definition of t, we have (st ) t L(t s) Nt s st, from which it follows that (st ) (s t). Hence we have t (s )(s ) or (s)(s ). Now, by using the definition of t again, we get s (N L)(N L). For consider the situation (), set D N & L Ï ( m)( m N)( m N)( m)t. t Then, by direct computation, we have D N & L m ( N)m N( t ) t m (N )m (t ) N( t ). t Moreover, by using the equalities ( N L )( N L ) (N L )( ) (N L ) Nt(st ) (t Nt(t s) )(t s) (t )(st (N L )(N ) (N L ) ) 4N t (t (N L )(N L ), ) t ( N L )( N L )t (N L )(t ) Nt{(st ) (t s) t } (t )(t s)(st ) (N L )(t ), we have Set D N & L m (N )m (N L )(N L ) N(N L ) N t{t(t s)(st )(st ) (t s) t } (t ) (t s)(st ) m N (N )m (N L )(L ) t (t s)(st ) m (N )m (N L )(L ) N L s. S N,L (m) Ï N L m (N )m (N L )(L ).

26 6 S. KATO AND K. TATEMICHI Then, D N & L 0 holds if and only if s S N,L (m) holds for some m ¾ Z N,L. Note here that S N,L (N ) (N L)(N L), and that S N,L (m) N L {m (N )} (N L) 4 {m (N L )}{m ( L )} {m (N L )}{m (L )}. The latter implies that S N,L (m ) S N,L (m ) holds if and only if m m or m m N. It is clear that the matrix m ( m N)t m ( m N)t cannot be the zero matrix. Now, we get the following fact on Nul(g s ): Theorem 6.. Let g s be the rational function given by (6.). Then the following assertions hold for its nullity: () If N and s S N,L (N ) (N L)(N L) 0, then Nul(g s ) 5 holds. () If N 4 and s ¾ {S N,L (m)m ¾ Z, (N )m N, m L, N L }, then Nul(g s ) 7 holds. (3) If N is even, N 4, L N, and s S N,L ((N )) (N L) (N L) 0, then Nul(g s ) 5 holds. (4) Nul(g s ) 3 holds for any other s such that s {, (N L) (N L) }. In particular, if Nul(g s ) 3, then s ¾ R Ô R. If m min{l, N L } or max{l, N L } m, then s ¾ R, and if min{l, N L } m max{l, N L }, then s ¾ Ô R. Since the set of s such that Nul(g s ) 3 is connected and includes 0, and since g 0 (z) z L N, it holds that Ind(g s ) Ind(g 0 ) d (N L) for such s. In the case that m N and s (N L)(N L), each of the flat-ended minimal surfaces above has the same symmetry as that of Costa s or Hoffman-Meeks surfaces. Hence we can compute their indices by applying the method in Nayatani [0, 9] (see also [7]). Set (s) Ï (N L)s (N L). Ls Then, for any s, t is given by t (s) Ô (s). In particular, for any s ¾ R, it also holds that t ¾ R. Here we choose t (s) Ô (s). If s 0, then (s) 0 and t 0. Moreover, since (s (s)) ((s) ) N L (s ) 0,

27 we have s t. Set I (z) Ï z, I (z) Ï N z and INDEX, NULLITY AND FLUX OF n-noids 7 Then it holds that m Ï z Lm (z N s) (z N t) (z N t ) dz. g s (I (z)) g s (z), I m m, g s (I (z)) N L g s (z), I m N Lm m. Let X Neu (resp. X Dir ) be the flat-ended N-noid given by the Weierstrass data (g, ) (g Ô (NL)(N L), h N N ) with h N ¾ RÒ{0} (resp. h N ¾ Ô RÒ{0}). Recall here that, for any conformal minimal immersion X(z) t (X (z), X (z), X 3 (z)) whose Weierstrass data is given by (g, ), the following assertions hold: () X(I (z)) t (X (z), X (z), X 3 (z)) holds up to parallel translations if and only if (g, ) satisfies (6.5 ) g(i (z)) g(z), I. () X(I (z)) t (cos(ln)x (z) sin(ln)x (z), sin(ln)x (z) cos(ln)x (z), X 3 (z)) holds up to parallel translations if and only if (g, ) satisfies (6.6 ) g(i (z)) N L g(z), I N L. Since the Weierstrass data of X Neu satisfies both of the conditions (6.5 ) and (6.6 ), it is symmetric with respect to both x x 3 -plane and the plane {(x, x, x 3 ) ¾ R 3 x Ô x ¾ N L R} up to parallel translations. Since G Ô (NL)(N L) Æ g Ô (NL)(N L) holds that also have the same symmetry as X Neu, if we denote it by G, then it X Neu (I (z)), G(I (z)) X Neu (I (z)), G(I (z)) X Neu (z), G(z). Hence we see that X Neu, G is an eigenfunction of the Laplacian ½ with respect to the metric G (ds ) on ÆM S C Ç which satisfies the Neumann boundary condition as an eigenfunction on the fundamental closed domain of the symmetry Đ Ï {z ¾ Ç C 0 arg z N}. On the other hand, since the Weierstrass data of X Dir satisfies both of the conditions (6.5 ) and (6.6 ), it holds that X Dir (I (z)), G(I (z)) X Dir (I (z)), G(I (z)) X Dir (z), G(z).

28 8 S. KATO AND K. TATEMICHI g(đ) g(đ) «0 «L N Fig. 6.. The case (N, L) (, ). Hence we see that X Dir, G is an eigenfunction of ½ which satisfies the Dirichlet boundary condition as an eigenfunction on Đ. Now, the pushforwards of these functions by G can be regarded as eigenfunctions of the Laplacian with respect to the standard metric ds on the closed domain S G(Đ) Æ g(đ) (Û) ¾ S ¾ Û C, N L N arg Û 0 { (½)}, which satisfy the Neumann or Dirichlet boundary condition if we regard G(Đ) Æ g(đ) { (x) ¾ S x ¾ R, x «or 0 x} { ( x L N ) ¾ S x ¾ R, «x} { (½)} as its boundary, where we set «Ï g s (t N ) (st ){t LN (t s)} (see Fig. 6.). Since t s, «0 holds. Moreover, since g s ¼ (t N ) 0, we see that d«ds d ds g s(t N ) g s s sz N z L (z N s) s (t N )g s ¼ (t N ) d ds (t N ) ztn 0 d ds (t N ) zn z L (z N s) zt N t t LN (t s) 0, that is, «is monotonically increasing with respect to s, and hence the boundary G(Đ) is monotonically increasing and the domain G(Đ) is monotonically decreasing. In this situation, we can show that each Neumann (resp. Dirichlet) eigenvalue is monotonically non-increasing (resp. non-decreasing) with respect to s by the same way as [0, Lemmas (b) and 6]. Since Ind(g s ) d (N L) and Nul(g s ) 3 hold for

29 INDEX, NULLITY AND FLUX OF n-noids 9 any s enough close to Ô (N L)(N L), and since Nul(g Ô (NL)(N L)) 5, it must hold that Ind(g Ô (NL)(N L)) d (N L). Theorem 6.. Let g s be the rational function given by (6.). If N and s S N,L (N ) (N L)(N L) 0, then Ind(g s ) d (N L) and Nul(g s ) 5 hold. 7. Index and nullity of Z N -invariant n-noids In this section, we give examples of n-noids with nontrivial nullity by applying the computations in 6. EXAMPLE 7. (n-noids with parallel flux). As we have already mentioned in introduction, any n-noid with parallel ends has nontrivial nullity. Here we determine their indices for a typical case. Let N be an integer such that N. For the data j,, N N N p j 0 0 ½ a j a a(n ) a(n ) with a ¾ RÒ{0}, by solving the equation (.4), we get the following Weierstrass data: g(z) t f (z), ta f (z) dz, where f is a rational function given by f (z) (N )z N (N ), z(z N ) and t ¾ R Ò {0} is a parameter of so-called López Ros deformation. This data realizes a family of Z N -invariant (N )-noids of TYPE I for each N (cf. [, Example 4.9]). Now, since f ( z r ) (N )r N r N z z(z N r N ), r N N, N the above g is equivalent with g s with s Ô (N L)(N L) and (N, L) (N, ) in 6. Hence, by Theorem 6., we have Ind(g) (N ) (N ) 4 and Nul(g) 5.

30 30 S. KATO AND K. TATEMICHI EXAMPLE 7. (N-noids with antiprismatic flux). that N, M N and (N, M). For the data j,, N N,, N p j M( j ) p N p M( j ) N a j a a Let N and M be integers such with p ¾ R Ò {0} and a ¾ R, by solving the equation (.3), we get the following Weierstrass data: (7.) g(z) sz N z N M (z N s), t z N M (z N s) (z N q N ) (z N q N ) dz, where q ¾ CÒ{0} satisfies s pqn M q N M (p q M ), p (q)p 0, (q) N qn M q M, N M q N and t ¾ RÒ{0} is a parameter of homothety chosen to satisfy t an(p )(q N )(p q M ) (p )q M (p q N M ) if p q N M 0. In particular, in the case q ¾ R, any N-noid given by one of these data has the symmetry of a regular N-gonal antiprism, which has no branch point if M and q. Here we regard q ¾ R Ò {0} as the parameter of deformation, and consider the case that p (q) Ô (q) 0. In this case, by direct computation, we have p q where we set p Ô (q) ¼ (q), N M N ¼ (q) q M (q N ) ³(q ), ³(t) Ï Mt N M (N M)t N (N M)t N M M.

31 INDEX, NULLITY AND FLUX OF n-noids 3 Hence pq 0 holds if and only if ³(q ) M(q 4N M ) (N M)q N M (q M ) 0. On the other hand, s S N, N M (N M)M holds if and only if 0 M(pq N M ) (N M){q N M (p q M )} ³(q )(p q N M ). q N M Namely, if pq 0, that is, q is a double solution of the equation (q) (p )(p), then, by Theorem 6. again, we have Ind(g) (N M) and Nul(g) 5. p q N M 0 with q N M 0 is the case of flat-ended ones (cf. [4, Remarks and ] for M ). In particular, in the case that N, M and q p Ô ( 6 Ô ), the data (7.) realizes a tetrahedrally symmetric 4-noid. Since s 3 S,, this is a special case of both the consideration above and Example 3.3, and hence its nullity is 5. On the other hand, in the case that N 3, M and q p Ô ( 6 Ô ), the data (7.) realizes an octahedrally symmetric 6-noid. Since s 5 5 S 3,, by Theorem 6. and the fact for the indices of generic surfaces by Ejiri Kotani [4] we introduced in the introduction, we get the following: Theorem 7.3. Let X be the octahedrally symmetric 6-noid as above. Then Ind(X) and Nul(X) 3 hold. Namely, symmetries of platonic solids do not always induce nontrivial bounded Jacobi functions. 8. Nullity and a flux map In this section, we study the correspondence between nullity and a flux map. In the case of n-noids of TYPE I and positive genus, Pérez Ros [3] considered a map from the moduli space of such n-noids to the space of the weights and the heights of the ends, and defined the nondegeneracy of such n-noids mainly by the property that any bounded Jacobi function is a trivial one, that is, the nullity of the surface is 3 4. By using these concepts, they analyzed the real analytic structure of the moduli space of such n-noids. On the other hand, Umehara, Yamada and the first author [,, 3] considered a flux map defined as a map from the parameter space of n-noids of genus zero with common limit normals to the space of the weights of the ends for each suit of limit normals, and proved that, for a generic flux data of TYPE III (or TYPE II with n 8), there exists an n-noid of genus zero which realizes the given flux data, by showing that the rank of the Jacobian matrix of the flux map is maximal for generic parameters and limit normals.

32 3 S. KATO AND K. TATEMICHI For the case of TYPE II, genus zero and Alexandrov-embedded, Cosín Ros [3] considered a flux map defined as a map from the moduli space of such n-noids to the space of flux polygons, that is the ordered flux vectors of the ends, and defined the nondenegeracy of such n-noids in the same way as in Pérez Ros [3] with the condition that the nullity is 3. They proved that the flux map is a real analytic diffeomorphism on to the space of flux polygons each of which bounds an immersed disc in the plane. From these points of view, it seems natural to expect, as the contraposition to some generalization of these results, that if an n-noid X is degenerate or a critical point of a flux map in some sense, then there exists a nontrivial bounded Jacobi function and Nul(X) 3. Indeed, we can show this for n-noids of arbitrary genus. Although its proof is given by a quite natural calculation, we show its detail here to observe the correspondence between Jacobi functions and flux precisely. Here we define a flux map in an essentially similar way as in [,, 3]. Since we treat a situation different from that of [3], our consequence also takes a somewhat different form from that in [3]. Let U be an open subset of C, and I an open interval in R. Let q(t) (t ¾ I ) be a smooth curve in U, and XÏ (U I )Ò{(q(t),t)t ¾ I} R 3 a smooth -parameter family of conformal minimal immersions in the sense that X(, t) is a conformal minimal immersion for each t ¾ I and that both Æ g and Æ (dz) are smooth with respect to (z, t) as maps from U I to S, where (g, ) is the Weierstrass data of X(, t). Assume that each of X(, t) has a catenoidal or planar end at q(t). Then the Taylor or Laurent expansions of (g, ) around q q(t) is of the following form: g p (z q)(z q) g (z), B b (z q) z q f 0(z) where p,, B and b are smooth functions depending only on the parameter t ¾ I, and g and f 0 are holomorphic functions on U both of which are smooth on I. By these expansions, it follows that g pb pb B (z q) z q g p B p bp B (z q) z q dz, f (z) dz, f (z) for some holomorphic functions f and f, from which it also follows that Ï ( g ) ( p )B ( p )b p B ( f dz (z q) 0 (z) f (z)), z q dz

33 INDEX, NULLITY AND FLUX OF n-noids 33 Ï Ô (g ) dz Ô ( p )B Ô {( p )bp B} Ô ( f (z q) 0 (z) f (z)), z q 3 Ï g pb (pb B) f dz (z q) (z). z q Let (Ú, a) be the flux data of X(, t). Then we have and hence we get ( p )b p B aú p a (p p), Ô {( p )bp B} aú p a Ô ( )(p p), (pb B) aú 3 p a (p ), a B, By integrate the -forms above, we have Ï z Ï z 3 Ï z b p a p. dz ( p )B z q aú log(z q)(f 0 F ), dz Ô ( p )B z q aú log(z q) Ô (F 0 F ), 3 dz pb z q aú 3 log(z q)f, where F 0, F and F are holomorphic functions on U each of which is smooth on I. Henceforth we denote the derivative with respect to the parameter t by t or ( ) t. By differentiate, and 3 by t, we get ( ) t q t( p )B pp t B ( p )B t q t aú (z q) z q {a t Ú a(ú ) t } log(z q)(f 0 F ) t, ( ) t tô q ( Ô p )B {pp t B ( p )B t } q t aú (z q) z q {a t Ú a(ú ) t } log(z q) Ô (F 0 F ) t, ( 3 ) t q t pb (z q) (p t B pb t ) q t aú 3 z q {a t Ú 3 a(ú 3 ) t } log(z q)(f ) t.

34 34 S. KATO AND K. TATEMICHI On the other hand, since the Gauss map G(, t) of X(, t) is given by G t (G, G, G 3 ) Æ g g t (g Ô g, (g g), g ), by the expansion g p (z q)(z q) g (z), we have G g {(p p) (z q) (z q) O(z q )}, G g Ô Ô {( )(p p)( ) (z q) Ô (z q) O(z q )}, G 3 g ) {(p p (z q) p (z q) O(z q )}. Note here that (p p, Ô (p p),p )(p )(Ú,Ú,Ú 3 ), Ú Ú Ú 3 and (Ú Ú Ú 3 ) t 0. By direct computation, we have (g ) t, G (g ){( ) t G ( ) t G ( 3 ) t G 3 } (z q) [ q t B{( p )(p p) Ô ( p )( Ô )(p p)p(p )}] z q [ q t B{( p ) Ô ( p )( Ô )p p} p t B{ p(p p) Ô p( Ô )(p p)(p )} B t {( p )(p p) Ô ( p )( Ô )(p p)p(p )} q t a{ú (p p)ú ( Ô )(p p)ú 3 (p )}] z q (z q) [ q t B{( p ) Ô ( p ) Ô p p}] log(z q)[a t {(p p)ú Ô ( )(p p)ú (p )Ú 3 } a{(p p)(ú ) t Ô ( )(p p)(ú ) t (p )(Ú 3 ) t }] O() (z q) ( q t B 0) z q [ q t B (p )p t B (p ) B t 0 q t a{ú Ú (p )Ú Ú (p )Ú 3 Ú 3 (p )}] z q (z q) ( q t B 0) log(z q)[a t {Ú (p ) Ú Ú (p ) Ú Ú 3 (p ) Ú 3 } a{ú (p ) (Ú ) t Ú (p ) (Ú ) t Ú 3 (p ) (Ú 3 ) t }] O()