General existence of minimal surfaces with prescribed flux II

Shin Kato^{1},_{Osaka City Univ.} Masaaki Umehara^{1}_{Osaka Univ.} and Kotaro Yamada^{1, 2}_{Kumamoto Univ.}

Introduction.

The present paper is a continuation of the papers [KUY1] and [KUY2] by the same authors. In the previous papers, we showed the existence of n-end catenoids (n ≥ 3) in R^{3} for “almost all” flux data, without no upper restrictions of the number n of ends. However, the above result does not work for specified case, for example, the case that all the flux vector lies in the same plane. We say that such minimal surfaces are of Type II. In this paper, we shall show that the method as in [KUY2] works even for such specified case, and prove the general existence of n-end catenoids (n ≤ 8) of Type II. Here we describe the details:

Consider a minimal immersion x : M^{2} → R^{3}, whose induced Riemannian metric is complete and of finite total curvature. The conformal structure of M^{2} is induced from the metric ds^{2}. So if we assume the surface is orientable, it can be considered as a Riemann surface and the immersion is a conformal map. Furthermore, since x is complete and of finite total curvature, there exists a compact Riemann surface \overline{M}^{2} such that M^{2} is biholomorphic to \overline{M}^{2} \ \{q_{1}, …, q_{n}\}, where q_{1}, …, q_{n} are a finite number of points on \overline{M}^{2}. The each point q_{j} is called an end of the surface. We set

g := ∂x^{3}/(∂x^{1} − \sqrt{−1} ∂x^{2}),
ω := ∂x^{1} − \sqrt{−1} ∂x^{2}.

Then g is a meromorphic function and ω is a holomorphic 1-form. We have the relation

∂x = \frac{1}{2} ((1 − g^{2}) ω, \sqrt{−1} (1 + g^{2}) ω, 2gω) .

This leads us to the well known Weierstrass representation

x = Re (∫_{z_{0}}^{z}(1 − g^{2}) ω, ∫_{z_{0}}^{z} \sqrt{−1} (1 + g^{2}) ω, ∫_{z_{0}}^{z} 2gω) ,

1Supported by Volkswagen-Stiftung (RiP Program in Mathematisches Forschungsinstitut Oberwolfach).

2Supported by Inamori Foundation.

where z_{0} is a fixed point on M^{2}. In this expression, the meromorphic function g is identified with the Gauss map by the stereographic projection from the north pole. Conversely, if M^{2} is simply connected and

ds^{2} = (1 + |g|^{2})^{2}ω · \overline{ω}

is positive definite, x becomes a conformal minimal immersion, where “·” means the symmetric tensor product. A degenerate point of the metric ds^{2} is called a branch point. We now define a flux vector φ_{j} around each end q_{j} by taking the imaginary part of the above expression:

φ_{j} := − Im (∮_{γ_{j}}(1 − g^{2}) ω, ∮_{γ_{j}} \sqrt{−1} (1 + g^{2}) ω, ∮_{γ_{j}} 2gω) (j = 1, …, n),

where γ_{j} is a loop surrounding the end q_{j} from the left. There is an alternative definition of the flux vector: Let n_{j}(s) be the conormal of γ_{j}(s), where s is the length parameter. Then the flux vector v_{j} is expressed by

φ_{j} = ∮_{γ_{j}} n_{j}(s) ds (j = 1, …, n).

Now we denote by ν(q_{j}) := lim_{z→q_{j}} ν(z) the limit of the Gauss map at the end. Then

• φ_{j} is proportional to the limit normal vector ν(q_{j}).

• The proportional constant w_{j} := φ_{j}/4πν(q_{j}) is called the weight at the end. It can take a negative value.

It seems quite natural that we restrict our attention to the special class of minimal surface, so-called n-end catenoid, which is a conformal minimal immersion x : C ∪ {∞} \ {q_{1}, …, q_{n}} → R^{3} whose ends are all asymptotic to catenoid ends. Recently, new examples of n-end catenoids have been found by [Kar], [L], [X], [R1], [R2], [Kat] and [UY].

For n-end catenoids, the weight is considered as the size of catenoid end relative to the standard catenoid. Now from the definition of the flux vector, we have

∑_{j=1}^{n} φ_{j} = ∑_{j=1}^{n} 4πw_{j}ν(q_{j}) = 0, (0.1)

which is called the flux formula. The inverse problem of the flux formula is now stated as follows:

Problem. For given unit vectors v_{1}, …, v_{n} and given real numbers a_{1}, …, a_{n} ≠ 0 satisfying the balancing condition ∑_{j=1}^{n} a_{j}v_{j} = 0, is there an n-end catenoid

x : C ∪ {∞} \ {q_{1}, …, q_{n}} → R^{3}

with ν(q_{j}) = v_{j} and w_{j} = a_{j}? (Is it unique?)

Now, we classify arrangements of v = {v_{1}, …, v_{n}} to the following three types according as [KUY1]:

Type I v = {v_{1}, …, v_{n}} satisfies dim⟨v_{1}, …, v_{n}⟩ = 1,
Type II v = {v_{1}, …, v_{n}} satisfies dim⟨v_{1}, …, v_{n}⟩ = 2,
Type III v = {v_{1}, …, v_{n}} satisfies dim⟨v_{1}, …, v_{n}⟩ = 3.

We call a (branched) n-end catenoid is of Type I (resp. II, III), if v is of Type I (resp. II, III). The following facts are already known (e.g. [M], [B], [L], [Kat]):

(1) There are no 1-end catenoids. Any 2-end catenoid is the catenoid.

(2) There are no 3-end catenoids of Type I. Consequently, any 3-end catenoid is of Type II. Moreover, for any unit vectors v_{1}, v_{2}, v_{3} of Type II and nonzero real numbers a_{1}, a_{2}, a_{3} satisfying ∑_{j=1}^{3} a_{j}v_{j} = 0, there exists an essentially unique 3-end catenoid x : C \ {q_{1}, q_{2}, q_{3}} → R^{3} which satisfies ν(q_{j}) = v_{j} and the weight w(q_{j}) = a_{j} (j = 1, 2, 3).

From these results, for n ≤ 3, the set of n-end catenoids is understood completely. In [KUY1] and [KUY2], we showed that for almost all flux data v_{1}, …, v_{n} ∈ S^{2}(1) (n ≥ 4) (of Type III ) and a_{1}, …, a_{n} ∈ R\{0} satisfying ∑_{j=1}^{n} a_{j}v_{j} = 0, there exists a (non-branched) n-end catenoid

x : C ∪ {∞} \ {q_{1}, ..., q_{n}} → R^{3}

with ν(q_{j}) = v_{j} and w_{j} = a_{j}. In the above statement, there is no upper restriction of n. We also remark that the uniqueness of the inverse problem is no longer true for n ≥ 4. (See [KUY1].) Moreover, the “almost all” in the assertion is essential. In fact, there are several obstruction for the inverse problem. (See [KUY1] and also [P].) However, the class of n-end catenoids of Type II is not included in the above statement, because such flux data is not generic. In this paper, we shall prove the following

Theorem. Suppose that 3 ≤ n ≤ 8. For almost all flux data v_{1}, …, v_{n} ∈ S^{2}(1) of Type II and a_{1}, …, a_{n} ∈ R\{0} satisfying ∑_{j=1}^{n} a_{j}v_{j} = 0, there exists a (nonbranched) n-end catenoid x : C ∪ {∞} \ {q_{1}, …, q_{n}} → R^{3} with ν(q_{j}) = v_{j} and w_{j} = a_{j}.

The method of the proof is based on that in [KUY2]. By virtue of it, the theorem is proved by finding a sampling point where the flux map attains non-vanishing Jacobian. Though an explicit proof is given only for n ≤ 8, our numerical (not rigorous) calculations indicates that the Jacobian of the flux map at Jorge-Meeks’ surface does not vanish at least n ≤ 13. So it is expected that the theorem holds for any n ≥ 3. To prove the theorem, we have to distinguish the cases of n into even and odd. In the case of odd n, the proof is rather easier, however in the case of n even, we need a computation of Pfaffians of certain skew-symmetric matrices instead of their determinants. In Section 1, we reduce the problem to seeking a sampling point satisfying certain non-degeneracy condition. In Section 2, we prove the theorem. In Section 3, we give several new concrete examples of n-end catenoids of Type II.

For n-end catenoids of Type I, the general existence can not be expected. (See [KUY1].) Recently the moduli space of minimal surfaces with flat ends is investigated in [KS], where a construction of minimal surfaces is explained by using the term of spin structure of Riemann surfaces. It should be remarked that our approaches in [KUY1] and [KUY2] can also be interpreted in terms of spin structure. (See Remark 1.1.)

1 Outline of the proof.

We reduce the problem to solving a certain system of algebraic equations. Let v_{1}, …, v_{n} ∈ S^{2}(1) be a set of unit vectors, and set p_{j} = π(v_{j}), where π : S^{2}(1) → C ∪ {∞} is the stereographic projection. We may assume p_{1}, …, p_{n} ∈ C by a suitable rotation of the surface.

Now we consider a system of equations (cf. [KUY1] and [KUY2]):

\begin{matrix}b_{j} ∑_{k≠j} b_{k} \frac{p_{j} − p_{k}}{q_{j} − q_{k}} = a_{j} \\ b_{j} ∑_{k≠j} b_{k} \frac{\overline{p_{j}}p_{k} + 1}{q_{j} − q_{k}} = 0 \end{matrix} (j = 1, …, n). (1.1)

Here p = (p_{1}, …, p_{n}) and a = (a_{1}, …, a_{n}) are flux data as given parameters, and q = (q_{1}, …, q_{n}) and b = (b_{1}, …, b_{n}) are unknown parameters. If we find a solution q_{j} and b_{j} of the above equations, we get a branched n-end catenoid as follows: Set

P (z) := ∑_{j=1}^{n} p_{j}b_{j} ∏_{k≠j}(z − q_{k}), Q(z) := ∑_{j=1}^{n} b_{j} ∏_{k≠j}(z − q_{k}), (1.2)

and define Weierstrass data (g, ω) by

g := \frac{P (z)}{Q(z)}, ω := − \{∑_{j=1}^{n} \frac{b_{j}}{z − q_{j}} \}^{2} dz. (1.3)

Then the minimal immersion

x = Re (∫_{z_{0}}^{z}(1 − g^{2}) ω, ∫_{z_{0}}^{z} \sqrt{−1} (1 + g^{2}) ω, ∫_{z_{0}}^{z} 2gω)

is a branched n-end catenoid with the flux data (p, a). Now ds^{2} > 0 if and only if both of the following two conditions are satisfied:

• P (z) and Q(z) are mutually prime,

• Max(deg P, deg Q) = n − 1.

Remark 1.1 The constructin of n-end catenoids mentioned above is related to the spinor representation of minimal surfaces (cf. [KS]);

x = Re (∫_{z_{0}}^{z}(s_{1}^{2} − s_{2}^{2}), ∫_{z_{0}}^{z} \sqrt{−1} (s_{1}^{2} + s_{2}^{2}), ∫_{z_{0}}^{z} 2s_{1} s_{2}) ,

where (s_{1}, s_{2}) is a pair of meromorphic sectons of the half-canonical bundle on C ∪ {∞}. In fact, s_{1} and s_{2} have the following explicit expressions in this case:

s_{1} = \frac{Q(z)}{R(z)}\sqrt{−dz}, s_{2} = \frac{P (z)}{R(z)}\sqrt{−dz},

where we set

R(z) = ∏_{k=1}^{n}(z − q_{k}).

We assume n ≥ 4 hereafter. To prove our theorem, we may restrict our attention to the flux data of Type II. So we may assume

|p_{j}| = 1 (j = 1, ..., n). (1.4)

This condition is rewritten as p = (p_{1}, …, p_{n}) ∈ C_{II}^{n}, where

C_{II}^{n} := {p ∈ C^{n}; |p_{j}| = 1 (j = 1, ..., n)} .

First, we fix our attention to the second equation of (1.1). Consider a skewsymmetric matrix

A_{p} := (\frac{p_{j} + p_{k}}{q_{j} − q_{k}} )_{j,k=1,...,n} , (1.5)

where the diagonal components are interpreted as 0. By (1.4), we have

\overline{p_{j}}p_{k} + 1 = \frac{p_{k}}{p_{j}} + 1 = \frac{p_{j} + p_{k}}{p_{j}} .

Thus we have

diag(1/p_{1}, ..., 1/p_{n}) · A_{p} = (\frac{\overline{p_{j}}p_{k} + 1}{q_{j} − q_{k}})_{j,k=1,...,n} ,

where diag(1/p_{1}, ..., 1/p_{n}) denotes the diagonal matrix with the diagonal components 1/p_{1}, ..., 1/p_{n}. Then the vector ^{t}(b_{1}, …, b_{n}) belongs to the kernel of the matrix A_{p}.

The case of odd n.

When n is odd, det A_{p} is identically zero. Moreover, it is reasonable to expect that the rank of the matrix A_{p} is generically n − 1. (In fact, as shown in Lemma 2.1, Jorge-Meeks’ (2m + 1)-noid have such a property for any m ≥ 1.) In this case, ^{t}(b_{1}, …, b_{n}) should be proportional to any column vector of the cofactor matrix Ã_{p} of A_{p}. (By the definition, A_{p} · Ã_{p} = Ã_{p} · A_{p} = (det A_{p})I holds.) So we set

b_{p}(q) = ^{t} (b_{p}^{1}(q), …, b_{p}^{n}(q)) := the first column of Ã_{p}(q).

For fixed p := (p_{1}, …, p_{n}) ∈ C_{II}, define a rational map between two complex projective spaces

Fℓ_{p} = [f_{p}^{1}, …, f_{p}^{n}] : P^{n−1} → P^{n−1}

by

f_{p}^{j}(q_{1}, …, q_{n}) := b_{p}^{j}(q) ∑_{k≠j} b_{p}^{k}(q)\frac{p_{j} − p_{k}}{q_{j} − q_{k}} (j = 1, …, n). (1.6)

The map Fℓ_{p} is called a flux map. We set

fℓ_{p}^{j}(q) := Δ(q)^{5} · f_{p}^{j}(q),

where Δ(q) is the difference product defined by

Δ(q_{1}, …, q_{n}) := ∏_{j>k}(q_{j} − q_{k}). (1.7)

It is easily seen that each fℓ_{p}^{j} is a homogeneous polynomial in q_{1}, …, q_{n} and Fℓ_{p} has another expression

Fℓ_{p} = [fℓ_{p}^{1}, …, fℓ_{p}^{n}].

Since p_{j} is the stereographic image of v_{j}, the balancing condition (0.1) is rewritten as

∑_{j=1}^{n} \frac{\overline{p_{j}}}{|p_{j}|^{2} + 1}a_{j} = 0,  ∑_{j=1}^{n} \frac{p_{j}}{|p_{j}|^{2} + 1}a_{j} = 0. (1.8)

We define a subspace W_{p}^{n−3} in P^{n−1} by

W_{p}^{n−3} := \{[a_{1}, …, a_{n}] ∈ P^{n−1} ; ∑_{j=1}^{n} \frac{\overline{p_{j}}}{|p_{j}|^{2} + 1}a_{j} = 0, ∑_{j=1}^{n} \frac{p_{j}}{|p_{j}|^{2} + 1}a_{j} = 0\} .

It should be remarked that the dimension of the space of flux data W_{p}^{n−3} is one up comparing the case of flux data of Type III. We set

J_{p} = (\frac{∂f_{p}^{k}}{∂q_{j}} · f_{p}^{n} − f_{p}^{k} · \frac{∂f_{p}^{n}}{∂q_{j}} )_{k,j=1,...,n−1} . (1.9)

The following lemma holds.

Lemma 1.2 Suppose that there exist u_{0} ∈ C_{II}^{n} (n = 2m + 1) and a point c = [c_{1}, …, c_{n}] ∈ P^{n−1} satisfying the following conditions:

(1) c_{1}, …, c_{n} are all distinct;

(2) The rank of the matrix A_{u_{0}}(c) is n − 1;

(3) The rank of the matrix J_{u_{0}}(c) is n − 3;

(4) Two polynomials P (z) and Q(z) defined in (1.2) associated with the data (q, p) = (c, u_{0}) and b = b_{u_{0}}(c) are mutually prime and one of them has degree n − 1;

(5) f_{u_{0}}^{j}(c) ≠ 0 (j = 1, …, n);

(6) c_{j} ≠ 0 (j = 1, …, n).

Then there exists an open dense subset U ⊂ C_{II}^{n} and an open dense subset Ω_{p} of the totally real set W_{R} = {[a] ∈ W_{p}^{n−3} ; a_{j} ∈ R} such that, for any p ∈ U and [a] ∈ Ω_{p}, there exists an (non-branched) n-end catenoid with the flux data (p, a).

Proof. One can easily prove the lemma modifying the proof of Proposition 1.6 in [KUY2]. By condition (4), at least one (n−3)-submatrix S_{u_{0}} of J_{u_{0}} is of rank n − 3. Let 1 ≤ j_{1} < j_{2} < ⋯ < j_{n−3} ≤ n be the indices of the columns of the submatrix S_{u_{0}}, and {m_{1}, m_{2}, m_{3}} their complement, namely {m_{1}, m_{2}, m_{3}} = {1, …, n} \ {j_{1}, …, j_{n−3}}. The position of the ends {q_{1}, …, q_{n}} in C ∪ {∞} has the freedom of Möbius transformations. To kill these ambiguities, we may restrict the flux map into the following subspace of P^{n−1} containing the sampling point c:

V^{n−3} := {[q_{1}, …, q_{n}] ∈ P^{n−1} ; c_{m_{2}}q_{m_{1}} − c_{m_{1}}q_{m_{2}} = 0, c_{m_{3}}q_{m_{1}} − c_{m_{1}}q_{m_{3}} = 0} .

Now we define a homogeneous polynomial in q_{1}, …, q_{n} by

H_{p}(q) := det (Δ(q)^{l}S_{p}(q)) · R_{p}(q) · ∏_{j=1}^{n} fℓ_{p}^{j}(q) · ∏_{k=1}^{n} q_{k},

where l is chosen sufficiently large so that det(Δ(q)^{l}S_{p}(q)) is a homogeneous polynomial in q_{1}, …, q_{n}, and R_{p} is the resultant of the two polynomials P (z) and Q(z) of degree n − 1 defined by (1.2). (It can be easily shown that R_{p} is also a homogeneous polynomial with respect to q. Or one may replace R_{p} by the resultant of P (q_{1}z) and Q(q_{1}z).) Then by the conditions (1)–(6), c ∈ V^{n−3} satisfies H_{u_{0}}(c) ≠ 0.

Since det A_{p} = 0 for any [q] ∈ P^{n−1}, we do not need to consider the subvariety X^{n−4} as in [KUY2]. So the arguments in the proof of Proposition 1.6 in [KUY2] can be easily modified in this case, only ignoring the parts concerning the differentials of det A_{p}. (q. e. d.)

The case of even n.

When n is even, det A_{p} is not identically zero on P^{n−1}. But the number of zero eigenvalues is even. So, we expect that the rank of the matrix A_{p} is generically n − 2 (not n − 1) for n-end catenoid of Type II. In this case, it is convenient to consider the Pfaffian pf A_{p} of the matrix A_{p}, which satisfies

det A_{p} = (pf A_{p})^{2}. (1.10)

Moreover, if we define a skew-symmetric matrix

Â_{p} = (b_{ij}) , b_{ij} = (−1)^{i+j} pf A_{ij} if i < j, (1.11)

where A_{ij} is a submatrix of A_{p} which is obtained after deleting i, j-th colummns and rows from A_{p}, it holds that

A_{p} · Â_{p} = Â_{p} · A_{p} = (pf A_{p})I. (1.12)

We call Â_{p} the Pfaffian cofactor matrix of A_{p}. In particular it is expressed as a homogeneous polynomial in components of A_{p}.

We set

b_{p}^{(1)}(q) = ^{t} (b_{p}^{(1),1}(q), …, b_{p}^{(1),n}(q)) := the first column of Â_{p}(q),
b_{p}^{(2)}(q) = ^{t} (b_{p}^{(2),1}(q), …, b_{p}^{(2),n}(q)) := the second column of Â_{p}(q).

For a fixed p := (p_{1}, …, p_{n}) ∈ C_{II}^{n}, define a rational map

Fℓ_{p} = [f_{p}^{1}, …, f_{p}^{n}] : P^{n−1} × P^{1} → P^{n−1}

by

f_{p}^{j}(q_{1}, …, q_{n}, s_{1}, s_{2}) := b_{p}^{j}(q, s) ∑_{k≠j} b_{p}^{k}(q, s)\frac{p_{j} − p_{k}}{q_{j} − q_{k}} (j = 1, …, n), (1.13)

where [q] = [q_{1}, …, q_{n}] ∈ P^{n−1}, [s] = [s_{1}, s_{2}] ∈ P^{1}, and

b_{p} ([q], [s]) := s_{1}b_{p}^{(1)}(q) + s_{2}b_{p}^{(2)}(q).

We define an (n − 1)-matrix J_{p} by

J_{p} := ((f_{p}^{n})^{2} \{\frac{∂(pf A_{p})}{∂q_{n}} · \frac{∂ f_{p}^{k}^{∘}}{∂q_{j}} − \frac{∂(pf A_{p})}{∂q_{j}} · \frac{∂ f_{p}^{k}^{∘}}{∂q_{n}} \})_{k,j=1,...,n−1} ,

where

f_{p}^{j}^{∘}:= \frac{f_{p}^{j}}{f_{p}^{n}} (j = 1, …, n − 1).

The matrix J_{p} has a direct expression

\begin{matrix}J_{p} = (\frac{∂(pf A_{p})}{∂q_{n}} · \{\frac{∂f_{p}^{k}}{∂q_{j}} · f_{p}^{n} − f_{p}^{k} · \frac{∂f_{p}^{n}}{∂q_{j}} \} \\ − \frac{∂(pf A_{p})}{∂q_{j}} · \{\frac{∂f_{p}^{k}}{∂q_{n}} · f_{p}^{n} − f_{p}^{k} · \frac{∂f_{p}^{n}}{∂q_{n}} \})_{k,j=1,...,n−1} . \end{matrix} (1.14)

Define an (n − 1) × n matrix J_{p} as

• The first column of J_{p} is given by

(\frac{∂(pf A_{p})}{∂q_{n}} \{\frac{∂f_{p}^{k}}{∂s_{2}} · f_{p}^{n} − f_{p}^{k} · \frac{∂f_{p}^{n}}{∂s_{2}} \})_{k=1,...,n−1} .

• The second to n-th column are equal to J_{p}. Lemma 1.3 Suppose that there exist u_{0} ∈ C_{II}^{n} (n = 2m) and a point (c, t) = ([c_{1}, …, c_{n}], [t_{1}, t_{2}]) ∈ P^{n−1} × P^{1} satisfying the following conditions:

(1) c_{1}, …, c_{n} are all distinct ;

(2) The rank of the matrix A_{u_{0}}(c) is n − 2;

(3) ∂(pf A_{u_{0}})/∂q_{n} does not vanish at q = c;

(4) The rank of the matrix J_{u_{0}}(c, t) is n − 3;

(5) Two polynomials P (z) and Q(z) defined in (1.2) associated with the data (q, s, p) = (c, t, u_{0}) and b = b_{u_{0}}(c, t) are mutually prime and one of them has degree n − 1;

(6) f_{u_{0}}^{j}(c, t) ≠ 0 (j = 1, …, n);

(7) c_{j} ≠ 0 (j = 1, …, n) and t_{j} ≠ 0 (j = 1, 2). Then there exists an open dense subset U ⊂ C_{II}^{n} and an open dense subset Ω_{p} of the totally real set W_{R} = {[a] ∈ W_{p}^{n−3} ; a_{j} ∈ R} such that, for any p ∈ U and [a] ∈ Ω_{p}, there exists an (non-branched) n-end catenoid with the flux data (p, a). Proof. The lemma can also be obtained modifying the proof of Proposition 1.6 in [KUY2]. We restrict the flux map into the following subspace of P^{n−1} × P^{1} containing the sampling point c: V^{n−2} := {[q_{1}, …, q_{n}] ∈ P^{n−1} ; c_{m_{2}}q_{m_{1}} −c_{m_{1}}q_{m_{2}} = 0, c_{m_{3}}q_{m_{1}} −c_{m_{1}}q_{m_{3}} = 0}×P^{1}.

Now we define a homogeneous polynomial in (q_{1}, …, q_{n}) and in (s_{1}, s_{2}) by

\begin{matrix}H_{p}(q, s) := \\ Δ(q)^{2} \frac{∂(pf A_{p})}{∂q_{n}}(q) · det (Δ(q)^{l}S_{p}(q)) · R_{p}(q, s) · ∏_{j=1}^{n} fℓ_{p}^{j}(q, s) · ∏_{k=1}^{n} q_{k} ∏_{k=1}^{2} s_{k}. \end{matrix}

Then by the conditions (1)–(7), (c, t) ∈ V^{n−2} satisfies H_{u_{0}}(c, t) ≠ 0. The arguments in the proof of Proposition 1.6 in [KUY2] can be easily modified in this case, only replacing det A_{p} by pf A_{p} and X^{n−4} by X^{n−3}. (q. e. d.)

2 Computations of the rank at a sampling point.

In the previous section, we reduced our problem to finding a regular point of the flux map (Lemmas 1.2 and 1.3). In this section, we take Jorge-Meeks’ n-noid [JM] as a sampling point, and show that it satisfies the assumptions of Lemma 1.2 or 1.3 for n ≤ 8.

Let n ≥ 4 be an integer. Jorge-Meeks’ n-noid is an n-end catenoid of Type II with the data

p_{j} = q_{j} = q_{j}^{0} = ζ^{j},  a^{j} = n − 1,  b^{j} = 1 (j = 1, …, n), (2.1)

where ζ := exp(2π\sqrt{−1}/n). For p in (2.1), the matrix defined in (1.5) is given by

A_{p}(q) = (\frac{ζ^{j} + ζ^{k}}{q_{j} − q_{k}} )_{j,k=1,...,n} , (2.2)

where the diagonal components of A_{p}(q) are zero. At the inital point

q^{0} = (q_{1}^{0}, …, q_{n}^{0}) = (ζ, …, ζ^{n}) ,

we have

A_{p}(q^{0}) = ( \begin{matrix}0 & α_{1} & α_{2} & … & α_{n−1} \\ α_{n−1} & 0 & α_{1} & … & α_{n−2} \\ α_{n−2} & α_{n−1} & 0 & … & α_{n−3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ α_{1} & α_{2} & α_{3} & … & 0 \end{matrix} ) , where α_{k} = \frac{1 + ζ^{k}}{1 − ζ^{k}} .

Since α_{n−j} = −α_{j}, A_{p}(q^{0}) is a skew-symmetric matrix.

Lemma 2.1 The rank of A_{p}(q^{0}) is n − 1 if n is odd, and n − 2 if n is even.

Proof. Since A_{p}(q^{0}) is a cyclic matrix, the eivengalues λ_{0}, …, λ_{n−1} are given by

λ_{l} = ∑_{j=0}^{n−1} ζ^{lj}α_{j} = \{ \begin{matrix}0 & (l = 0) \\ −n + 2l & (1 ≤ l ≤ n − 1) \end{matrix} ,

where α_{0} is considered as 0. Thus, if n is odd, only λ_{0} is a zero-eigenvalue of A_{p}(q^{0}), and if n is even, zero-eigenvalues are λ_{0} and λ_{n/2}. (q. e. d.)

In the rest of this section, we calculate the rank of the Jacobian matrix J_{p} in (1.9) for odd n and J_{p} for even n at the point q_{j} = q_{j}^{0}.

The case of odd n.

Lemma 2.2 The derivaties of b_{p}(q) at q = q^{0} are given by

\frac{∂}{∂q_{j}} ( b_{p}^{1} ; b_{p}^{2} ; ⋮ ; b_{p}^{n} ) (q^{0}) = β \{Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y ) I − Y · \frac{∂A_{p}}{∂q_{j}} (q^{0})\} · ( 1 1 ⋮ 1 ) ,

where

Y = ( \begin{matrix}0 & 0 & … & 0 \\ 0 \\ ⋮ & & A_{1}^{−1} \\ 0 \end{matrix} ) , A_{1} = [the square (n−1)-submatrix of A_{p}(q^{0}) excluded the first column and the first row.] ,

and β = det A_{1} ≠ 0.

Proof. Let B(q) = Ã_{p}(q) be the cofactor matrix. It is easily checked that t(1, 1, …, 1) is a zero-eigenvector of A_{p}(q^{0}) and ^{t}A_{p}(q^{0}) simultaneously. Since rank of A_{p}(q^{0}) is n − 1, we have

B(q^{0}) = β ( \begin{matrix}1 & 1 & … & 1 \\ 1 & 1 & … & 1 \\ ⋮ & ⋮ & ... & ⋮ \\ 1 & 1 & … & 1 \end{matrix} ) ,

where β = det A_{1} ≠ 0.

Take an n × n matrix X as

X = ( \begin{matrix}1 & 0 & … & 0 \\ 0 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 0 \end{matrix} )

and let A_{t}(q) = A_{p}(q) + tX for any real number t. Since

\frac{∂}{∂t}|_{t=0} det A_{t}(q^{0}) = Tr (\frac{∂A_{t}}{∂t} (q^{0}) · B(q^{0})) = Tr (X · B(q^{0}) ) = β ≠ 0,

A_{t}(q) is non-singular for sufficiently small t and q which is sufficiently close to q^{0}. Let B_{t}(q) be the cofactor matrix of A_{t}(q). Differentiating A_{t} ·B_{t} = B_{t} ·A_{t} = (det A_{t})I with respect to q_{j}, we have

\frac{∂B_{t}}{∂q_{j}} = A_{t}^{−1} · (\frac{∂ det A_{t}}{∂q_{j}} I − \frac{∂A_{t}}{∂q_{j}} · B_{t})
= \frac{1}{det A_{t}} (B_{t} · \frac{∂ det A_{t}}{∂q_{j}} I − B_{t} · \frac{∂A_{t}}{∂q_{j}} · B_{t}) .

On letting t → 0, both the denominator and the numerator of the right-hand side of the above equation tending to zero. So we can apply de l’Hospital’s rule. Thus, since det A_{p}(q) ≡ 0 and ∂(det A_{t})/∂t = Tr(∂A_{t}/∂t) · B_{t}, we have

\begin{matrix}\frac{∂B}{∂q_{j}} (q^{0}) = \frac{1}{β} \{Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y̌ ) B(q^{0}) \\ −Y̌ · \frac{∂A_{p}}{∂q_{j}} (q^{0}) · B(q^{0}) − B(q^{0}) · \frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y̌ \} , \end{matrix} (2.3)

where Y̌ = ∂B_{t}(q^{0})/∂t|_{t=0}. Since

A_{t}(q^{0}) = ( \begin{matrix}t & α_{1} & α_{2} & … & α_{n−1} \\ α_{n−1} & 0 & α_{1} & … & α_{n−2} \\ α_{n−2} & α_{n−1} & 0 & … & α_{n−3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ α_{1} & α_{2} & α_{3} & … & 0 \end{matrix} ) ,

the first row and the first column of the cofactor matrix B_{t}(q^{0}) do not depend on t, and one can easily see that

Y̌ = \frac{∂B_{t}(q^{0})}{∂t}|_{t=0} = ( \begin{matrix}0 & 0 & … & 0 \\ 0 \\ ⋮ & & Ã_{1} \\ 0 \end{matrix} ) = ( \begin{matrix}0 & 0 & … & 0 \\ 0 \\ ⋮ & & βA_{1}^{−1} \\ 0 \end{matrix} ) .

Substituting

( b_{p}^{1}(q) ; ⋮ ; b_{p}^{n}(q) ) = [the first column of B(q)] = B(q) · ( 1 0 ⋮ 0 )

into (2.3), we have the conclusion, since Y ·^{t}(1, …, 0) = 0. (q. e. d.)

Let

b̌_{p}^{j}(q) = \frac{1}{β} b_{p}^{j}(q), and f̌_{p}^{j}(q) = \frac{1}{β^{2}} f_{p}^{j}(q).

Then

\frac{∂}{∂q_{j}} ( b̌_{p}^{1} ; ⋮ ; b̌_{p}^{n} ) (q^{0}) = \{Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y ) I − Y · \frac{∂A_{p}}{∂q_{j}} (q^{0})\} · ( 1 ⋮ 1 ) ,

and differentiating (1.6), we have

\frac{1}{β^{2}} \frac{∂f_{p}^{j}}{∂q_{l}} = \frac{∂f̌_{p}^{j}}{∂q_{l}} = ∑_{k=1}^{n} \frac{∂b̌_{p}^{k}}{∂q_{l}} + (n − 2)\frac{∂b̌_{p}^{j}}{∂q_{l}} + \{ \begin{matrix}−\frac{1}{2}ζ^{−l}(n − 1) & (if j = l) \\ \frac{1}{ζ^{j} − ζ^{l}} & (if j ≠ l), \end{matrix}

because b_{p}^{j}(q^{0}) = β. Since f^{j}(q^{0}) = (n − 1)β^{2}, the Jacobian matrix (1.9) becomes

J_{p} = (n − 1)β^{4}J̌_{p}, where J̌_{p} = (\frac{∂f̌_{p}^{k}}{∂q_{j}} (q^{0}) − \frac{∂f̌_{p}^{n}}{∂q_{j}} (q^{0}))_{k,j=1,...,n−1} . (2.4)

Lemma 2.3 If n = 5 or 7, the rank of J̌_{p} is n − 3.

Proof. Let Δ_{n} be an (n − 3)-matrix which consits of the first (n − 3) rows and (n − 3) columns of J̌_{p}.

By direct calculations, we have

| det Δ_{5}| = 20 ≠ 0.

Hence the rank of J_{p} is n − 3 if n = 5.

In the case n = 7, we can also make a direct calculation, and have det Δ_{7} as a rational function of ζ. Since it holds that 1 + ζ + … + ζ^{6} = 0, det Δ_{7} is written as

det Δ_{7} = \frac{a_{10} + a_{11}ζ + a_{12}ζ^{2} + a_{13}ζ^{3} + a_{14}ζ^{4} + a_{15}ζ^{5}}{a_{20} + a_{21}ζ + a_{22}ζ^{2} + a_{23}ζ^{3} + a_{24}ζ^{4} + a_{25}ζ^{5}} ,

where a_{ij}’s are integers whose values are apploximately 10^{440}. More precisely

\{ \begin{matrix}a_{10} ≡ 0, & a_{11} ≡ 4, & a_{12} ≡ 4, & a_{13} ≡ 1, & a_{14} ≡ 2, & a_{15} ≡ 3 \\ a_{20} ≡ 0, & a_{21} ≡ 3, & a_{22} ≡ 4, & a_{23} ≡ 0, & a_{24} ≡ 3, & a_{25} ≡ 4 \end{matrix} (mod 5)

holds. Thus, det Δ_{7} is not zero, and we have the conclusion. ^{1} (By a numerical calculation, we know that the value | det Δ_{7}| is approximately 5773.) (q. e. d.)

Thus, by Lemma 1.2, we have proved the Theorem mentioned in Introduction for n = 5 and 7.

1To calculate det Δ_{n} we use Mathematica Version 2.2, a system of symboic calculation

The case of even n

Let B(q) = Â_{p}(q) be the Pfaffian cofactor matrix and b_{p}(s, q) as in the previous seciton. At q = q^{0}, one can easily see that ^{t}(0, 1, 0, 1, …, 1) and ^{t}(1, 0, 1, …, 0) are elements of the kernel of A_{p}(q^{0}). Thus, since rank of A_{p}(q^{0}) = n − 2 and B(q^{0}) is skew-symmetric, we have

B(q^{0}) = β ( \begin{matrix}0 & −1 & 0 & −1 & … & −1 \\ 1 & 0 & 1 & 0 & … & 0 \\ 0 & −1 & 0 & −1 & … & −1 \\ 1 & 0 & 1 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & 0 & 1 & 0 & … & 0 \end{matrix} ) ,

where

(2.5) β = pf A_{12} ≠ 0, A_{12} = [the square (n−2)-submatrix of A_{p}(q^{0}) excluded the first and second columns and the first and second rows.] .

On the other hand, at the initial point (Jorge-Meeks’ n-noid) b^{1} = b^{2} = … = b^{n}. Thus we can set s_{1}^{0} = 1 and s_{2}^{0} = −1. (2.6)

Lemma 2.4 The derivatives of pf A_{p} are given by

\frac{∂(pf A_{p})}{∂q_{j}}(q^{0}) = ζ^{−j}(−1)^{j+1} \frac{1}{2}β m(m − 1) (n = 2m).

In particular ∂(pf A_{p})/∂q_{j} ≠ 0.

Proof. In general,

\frac{∂(pf A)}{∂q} = \frac{1}{2} Tr (\frac{∂A}{∂q} · Â)

holds, where Â is the Pfaffian cofactor matrix of A. Using this, the conclusion is obtained by a direct calculation. (q. e. d.)

Lemma 2.5 The derivatives of b_{p}(q, s) at (q, s) = (q^{0}, s^{0}) are given by

\frac{∂}{∂q_{j}} ( b_{p}^{1} ; b_{p}^{2} ; ⋮ ; b_{p}^{n} ) (q^{0}, s^{0}) = β \{\frac{1}{2} Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y ) I − Y · \frac{∂A_{p}}{∂q_{j}} (q^{0})\} · ( 1 1 ⋮ 1 ) ,

where

Y = ( \begin{matrix}0 & 0 & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & & A_{12}^{−1} \\ 0 & 0 \end{matrix} ) (A_{12} is as in (2.5) ). (2.7)

Proof. Take an n × n skew-symmetric matrix X as

X = ( \begin{matrix}0 & 1 & … & 0 \\ −1 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 0 \end{matrix} )

and let A_{t}(q) = A_{p}(q) + tX for any real number t. Since

\frac{∂}{∂t}|_{t=0} pf A_{t}(q^{0}) = \frac{1}{2} Tr (\frac{∂A_{t}}{∂t} (q^{0}) · B(q^{0})) = \frac{1}{2} Tr (X · B(q^{0}) ) = β ≠ 0,

A_{t}(q) is non-singular for sufficiently small t and q which is sufficiently close to q^{0}. Let B_{t}(q) be the Pfaffian cofactor matrix of A_{t}(q). Differentiating A_{t} · B_{t} = B_{t} · A_{t} = (pf A_{t})I with respect to q_{j}, we have

\frac{∂B_{t}}{∂q_{j}} = A_{t}^{−1} · (\frac{∂(pf A_{t})}{∂q_{j}} I − \frac{∂A_{t}}{∂q_{j}} · B_{t})
= \frac{1}{pf A_{t}} (B_{t} · \frac{∂(pf A_{t})}{∂q_{j}} I − B_{t} · \frac{∂A_{t}}{∂q_{j}} · B_{t}) .

On letting t → 0, both the denominator and the numerator of the right-hand side of the above equation tend to zero. So we can apply de l’Hospital’s rule. Thus we have

\begin{matrix}\frac{∂B}{∂q_{j}} (q^{0}) = \frac{1}{β} \{\frac{1}{2} Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y̌ ) B(q^{0}) \\ +\frac{1}{2} Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · B(q^{0})) Y̌ \\ −Y̌ · \frac{∂A_{p}}{∂q_{j}} (q^{0}) · B(q^{0}) − B(q^{0}) · \frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y̌ \} , \end{matrix} (2.8)

where Y̌ = ∂B_{t}(q^{0})/∂t|_{t=0}. Since

A_{t}(q^{0}) = ( \begin{matrix}0 & α_{1} + t & α_{2} & … & α_{n−1} \\ α_{n−1} − t & 0 & α_{1} & … & α_{n−2} \\ α_{n−2} & α_{n−1} & 0 & … & α_{n−3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ α_{1} & α_{2} & α_{3} & … & 0 \end{matrix} ) ,

the first and second rows, and the first and second columns of the Pffafian cofactor matrix B_{t}(q^{0}) do not depend on t, and one can easily see that

Y̌ = \frac{∂B_{t}(q^{0})}{∂t}|_{t=0} = ( \begin{matrix}0 & 0 & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & & Â_{12} \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 0 & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & βA_{12}^{−1} \\ 0 & 0 \end{matrix} ) ,

where A_{12} is the matrix obtaind by excluding the first and second columns and the first and second rows from A_{p}(q^{0}).

Substituting

( b_{p}^{1}(q, s^{0}) ; b_{p}^{2}(q, s^{0}) ; b_{p}^{3}(q, s^{0}) ; ⋮ ; b_{p}^{n}(q, s^{0}) ) = [ \begin{matrix}s_{1} · (the first column of B(q))+ \\ s_{2} · (the second column of B(q)) \end{matrix} ]_{(s_{1},s_{2})=(1,−1)}

= B(q) ·^{t} (1, −1, 0, …, 0)

into (2.8), we have the conclusion, since Y ·^{t}(1, −1, 0, …, 0) = 0. (q. e. d.)

Let

b̌_{p}^{j}(q, s) = \frac{1}{β} b_{p}^{j}(q, s), and f̌_{p}^{j}(q, s) = \frac{1}{β^{2}} f_{p}^{j}(q, s).

Then

\frac{∂}{∂q_{j}} ( b̌_{p}^{1} ; b̌_{p}^{2} ; ⋮ ; b̌_{p}^{n} ) (q^{0}, s^{0}) = \{\frac{1}{2} Tr (\frac{∂A_{p}}{∂q_{j}} (q^{0}) · Y ) I − Y · \frac{∂A_{p}}{∂q_{j}} (q^{0})\} · (1 1 ⋮ 1) , (2.9)

and differentiating (1.13), we have

\frac{1}{β^{2}} \frac{∂f_{p}^{j}}{∂q_{l}} = \frac{∂f̌_{p}^{j}}{∂q_{l}} = ∑_{k=1}^{n} \frac{∂b̌_{p}^{k}}{∂q_{l}} + (n − 2)\frac{∂b̌_{p}^{j}}{∂q_{l}} + \{ \begin{matrix}−\frac{1}{2}ζ^{−l}(n − 1) & (if j = l) \\ \frac{1}{ζ^{j} − ζ^{l}} & (if j ≠ l), \end{matrix}

because b_{p}^{j}(q^{0}, s^{0}) = β. Since f^{j}(q^{0}, s^{0}) = (n − 1)β^{2} and by Lemma 2.4, the Jacobian matrix J_{p} in (1.14) becomes

J_{p} = \frac{1}{2}m(m − 1)(2m − 1)β^{5}J̌_{p} (n = 2m),

J̌_{p} = (− \{\frac{∂f̌_{p}^{k}}{∂q_{j}} (q^{0}, s^{0}) − \frac{∂f̌_{p}^{n}}{∂q_{j}} (q^{0}, s^{0})\}

+ (−1)^{j}ζ^{−j} \{\frac{∂f̌_{p}^{k}}{∂q_{n}} (q^{0}, s^{0}) − \frac{∂f̌_{p}^{n}}{∂q_{n}} (q^{0}, s^{0})\})_{k,j=1,...,n−1} .

On the other hand, since

f^{j}(q^{0}, s) = (s_{1}b_{p}^{(1),j}(q^{0}) + s_{2}b_{p}^{(2),j}(q^{0})) ∑_{k≠j} (s_{1}b_{p}^{(1),k}(q^{0}) + s_{2}b_{p}^{(2),k}(q^{0}))

and b_{p}^{(2)}(q^{0}) = β ·^{t}(0, −1, 0, …, −1), we have

\frac{∂f^{j}}{∂s_{2}} (q^{0}, s^{0}) = \{ \begin{matrix}(−\frac{3}{2}n + 2) β^{2} & (if j is even) \\ (−\frac{n}{2}) β^{2} & (if j is odd) \end{matrix} .

We set an (n − 1) × n matrix J̌_{p} as follows:

• The first column is given by

(\frac{∂f̌^{k}}{∂s_{2}} − \frac{∂f̌^{n}}{∂s_{2}} )_{k=1,...,n−1} .

• The second to n-th columns is equal to J̌_{p}

Thus by definition, the rank of J_{p} is equal to those of J̌_{p}.

Lemma 2.6 If n = 4, 6 or 8, the rank of J̌_{p} is n − 3.

Proof. Let Δ_{n} be an (n − 3)-matrix which consits of the first (n − 3) rows and (n − 3) columns of J̌_{p}.

By direct calculations, we have

det Δ_{4} = 2
det Δ_{6} = 2187 = 3^{7}
det Δ_{8} = \sqrt{−1} · 2^{15} · 3^{6} · 5^{2}(\sqrt{2} − 2) ∼ −3 · 10^{8}\sqrt{−1}.

Thus we have the conclusion. ^{2} (q. e. d.)

Thus, by Lemma 2.6, we have proved the Theorem mentioned in Introduction for n = 4, 6 and 8.

Remark 2.7 For calculations in this section, we must compute the inverse matrix of A_{1} or A_{12}. For large n, it is difficult to do that by computer as symbolic calculations. However, we can calculate | det Δ_{n}| numerically for large n. As a conclusion of such numerical calculations, we may say that the assertion of the theorem is true if n ≤ 13.

2To calculate det Δ_{n} we use Mathematica Version 2.2, a system of symboic calculation

3 Examples

In [KUY1], [KUY2] and this paper, we showed general existence of n-end catenoids with prescribed flux of Type II and Type III. However, the following problems are still remaining:

• Are there futher obstructions for the inverse problem of the flux formula except for the ones listed in [KUY1], especially for Type II and III? (In fact, any obstruction for the existence of n-end catenoids of Type III have not been found yet.)

• Is the set of n-end catenoids connected?

• Can any n-end catenoids of Type II be deformed into n-end catenoids of Type III?

Before finding answers to these questions, maybe we should examine many concrete examples. In this section, we will observe two examples of deformations from Jorge-Meeks’ 2m-noid, one of them are related to the third question. First, we see a family of D_{m}-invariant 2m-end catenoids of Type II.

Example 3.1 Let m be an integer greater than 1, and set ζ_{m} := exp (2π \sqrt{−1}/m). Let p be a unit complex number such that 0 ≤ arg p < π. Then, for any flux data

\{ \begin{matrix}p_{j} = π(v_{j}) := \{ \begin{matrix}pζ_{m}^{j−1} & (j = 1, …, m), \\ p^{−1}ζ_{m}^{j−1} & (j = m + 1, …, 2m), \end{matrix} \\ a_{1} = … = a_{2m} = 1, \end{matrix}

the equation (1.1) has a solution as follows:

\{ \begin{matrix}q_{j} = \{ \begin{matrix}qζ_{m}^{j−1} & (j = 1, …, m), \\ q^{−1}ζ_{m}^{j−1} & (j = m + 1, …, 2m), \end{matrix} \\ b_{j} = \{ \begin{matrix}qr & (j = 1, …, m), \\ pr & (j = m + 1, …, 2m), \end{matrix} \end{matrix}

where q is a unit complex number satisfying

q^{2−2m} = −p^{2}, q^{2m} ≠ 1,

and r is given by

r := \sqrt{\frac{q^{2m} − 1}{q\{(m − 1)p(q^{2m} − 1) + mq(p^{2}q^{2m−2} − 1)\}}}.

Define an immersion

x_{(m,q)} : C \ {q, qζ_{m}, …, qζ_{m}^{m−1}, q^{−1}, q^{−1}ζ_{m}, …, q^{−1}ζ_{m}^{m−1}} → R^{3}

by these data, or equivalently by

g(z) := \frac{(pq^{m} + q^{1−m})z^{m} − (p + q)}{z^{m−1}\{(p + q)z^{m} − (pq^{m} + q^{1−m})\}},
ω := −r^{2} [\frac{mz^{m−1}\{(p + q)z^{m} − (pq^{m} + q^{1−m})\}}{(z^{m} − q^{m})(z^{m} − q^{−m})}]^{2} dz.

Then x_{(m,q)} is a conformal minimal immersion satisfying

\{ \begin{matrix}g(qζ_{m}^{j−1}) = pζ_{m}^{j−1}, g(q^{−1}ζ_{m}^{j−1}) = p^{−1}ζ_{m}^{j−1} \\ w(qζ_{m}^{j−1}) = w(q^{−1}ζ_{m}^{j−1}) = 1 \end{matrix} (j = 1, …, m),

where w(qζ_{m}^{j−1}) means the weight of the end qζ_{m}^{j−1}. It is easy to see that

(1) x_{(m,q)} is branched if and only if q^{2m} = −1 and p = −q;

(2) In the other cases, x_{(m,q)} is a D_{m}-invariant 2m-end catenoid of Type II.

In particular,

(3) x_{(m,ζ_{4m})} is Jorge-Meeks’ 2m-noid.

In the case when Re(q/p) > 0, these surfaces are constructed also by Rossman [R1].

Clearly, for any unit complex number p such that 0 ≤ arg p ≤ π/2m, complex numbers

\{q^{(l)} = p^{−1/(m−1)} exp (\frac{2l − 1}{2m − 2}π\sqrt{−1}) : l = 1, …, 2m − 2\}

satisfy the same equality (q^{(l)})^{2−2m} = −p^{2}. Hence there are (2m − 2)-tuples of 2m-end catenoids x_{(m,q^{(1)})}, …, x_{(m,q^{(2m−2)})} which have the same flux data. It can be easily observed that if 0 < arg p < π/2m, then these 2m − 2 surfaces are not congruent to each other. When arg p = 0 or π/2m, some of the surfaces are branched or not realized, and some two of them are congruent to each other. However, if arg p = 0 (resp. π/2m), then 2[(m−1)/2] (resp. 2[m/2]−1) surfaces of them are non-branched and not congruent to each other.

Next, we observe a typical example of a deformation of 2m-end catenoids from Type II to Type III. Through this, we can see that Jorge-Meeks’ 4noid (resp. 6-noid) and the tetrahedrally (resp. octahedrally) symmetric one are included in the same connected component of the moduli of 4-end (resp. 6end) catenoids. (The case of n = 4 has been already observed in [KUY1].)

Example 3.2 Let m be an integer greater than 1, and set ζ_{2m} := exp (2π\sqrt{−1} /2m), ζ_{m} := ζ_{2m}^{2}. Let p be a positive real number. Then there is a real number c_{m} > 1 depending only on m (e.g. c_{2} = (\sqrt{6} + \sqrt{2})/2, c_{3} = 2, c_{4} ≈ 2.0503) such that, if c_{m}^{−1} ≤ p ≤ c_{m}, then, for any flux data

\{ \begin{matrix}p_{j} = σ(v_{j}) := \{ \begin{matrix}pζ_{m}^{j−1} & (j = 1, …, m), \\ p^{−1}ζ_{2m}^{2j−1} & (j = m + 1, …, 2m), \end{matrix} \\ a_{1} = ⋯ = a_{2m} = 1, \end{matrix}

the equation (1.1) has a solution as follows:

\{ \begin{matrix}q_{j} = \{ \begin{matrix}qζ_{m}^{j−1} & (j = 1, …, m), \\ q^{−1}ζ_{2m}^{2j−1} & (j = m + 1, …, 2m), \end{matrix} \\ b_{j} = \{ \begin{matrix}qr & (j = 1, …, m), \\ pr & (j = m + 1, …, 2m), \end{matrix} \end{matrix}

where q is a non-zero real numbers satisfying

\frac{mq(q^{2m−2} − 1)}{(m − 1)(q^{2m} + 1)} = \frac{p^{2} − 1}{2p},

and r is given by

r := \sqrt{\frac{q^{2m} + 1}{q\{(m − 1)p(q^{2m} + 1) + mq(p^{2}q^{2m−2} + 1)\}}}.

Define an immersion

x_{(m,q)} : C \ {q, qζ_{m}, …, qζ_{m}^{m−1}, q^{−1}ζ_{2m}, q^{−1}ζ_{2m}^{3}, …, q^{−1}ζ_{2m}^{2m−1}} → R^{3}

by these data, or equivalently by

g(z) := \frac{(pq^{m} − q^{1−m})z^{m} + (p + q)}{z^{m−1}\{(p + q)z^{m} − (pq^{m} − q^{1−m})\}},
ω := −r^{2} [\frac{mz^{m−1}\{(p + q)z^{m} − (pq^{m} − q^{1−m})\}}{(z^{m} − q^{m})(z^{m} + q^{−m})}]^{2} dz.

Then x_{(m,q)} is a branched conformal minimal immersion such that

\{ \begin{matrix}g(qζ_{m}^{j−1}) = pζ_{m}^{j−1}, g(q^{−1}ζ_{2m}^{2j−1}) = p^{−1}ζ_{2m}^{2j−1} \\ w(qζ_{m}^{j−1}) = w(q^{−1}ζ_{2m}^{2j−1}) = 1 \end{matrix} (j = 1, …, m)

except for the case when q = − ^{m} \sqrt{(m ± \sqrt{2m − 1})/(m − 1)} (i.e. r = ∞). It is easy to see that

(1) x_{(m,q)} is branched if and only if q = −1 (p = 1); (2) In the other cases, x_{(m,q)} is a 2m-end catenoid with the symmetry of a

regular m-angular antiprism.

In particular, by a direct calculation ([Kat], [Xu]), we have

(3) x_{(m,1)} is Jorge-Meeks’ 2m-noid;

(4) x_{(2,(\sqrt{6}+\sqrt{2})/2)} is the tetrahedrally symmetric 4-end catenoid, and x_{(3,(\sqrt{6}+\sqrt{2})/2)} is the octahedrally symmetric 6-end catenoid.

By (2), (3) and (4), we see that the family

\{x_{(2,q)} (resp. x_{(3,q)}) ; 1 ≤ q ≤ (\sqrt{6} + \sqrt{2})/2\}

gives a deformation from Jorge-Meeks’ 4-noid (resp. 6-noid) to the tetrahedrally (resp. octahedrally) symmetric one.

References

[B] E. L. Barbanel: Complete minimal surface in R^{3} of low total curvature, Ph.D Thesis, (1987) University of Massachusetts.

[JM] L. P. M. Jorge and W. H. Meeks, III: The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, 22 (1983), 203–221.

[Kar] H. Karcher: Construction of minimal surfaces, Surveys in Geometry 1989/1990, University of Tokyo.

[Kat] S. Kato: Construction of n-end catenoids with prescribed flux, Kodai Math. J., 18 (1995), 86–98.

[KUY1] S. Kato, M. Umehara and K. Yamada: An inverse problem of the flux for minimal surfaces, preprint.

[KUY2] S. Kato, M. Umehara and K. Yamada: General existence of minimal surface with prescribed flux, preprint.

[KS] R. Kusner and N. Schmitt: The spinor representation of minimal surfaces, Preprint.

[L] F. J. Lopez: The classification of complete minimal surfaces with total curvature greater than −12π, Trans. Amer. Math. Soc., 334 (1992), 49–74.

[LR] F. J. Lopez and A. Ros: On embedded complete minimal surfaces of genus zero, J. Differential Geometry 33 (1991), 293–300.

[M] W. H. Meeks III: The classification of complete minimal surfaces in R^{3} with total curvature greater than −8π, Duke Math. J., 48 (1981), 523–535.

[P] J. Perez: Balancing formulae for minimal surfaces in flat three-manifolds, preprint.

[RT] H. Rosenberg and E. Toubiana: Complete minimal surfaces and minimal herissons, J. Differential Geometry, 28 (1988), 115–132.

[R1] W. Rossman: Minimal surfaces in R^{3} with dihedral symmetry, Tohoku Math. J., 47 (1995), 31–54.

[R2] W. Rossman: On embeddedness of area-minimizing disks, and an application to constructing complete minimal surfaces, preprint.

[UY] M. Umehara and K. Yamada: Surfaces of constant mean curvature c in H^{3}(−c^{2}) with prescribed hyperbolic Gauss map, Math. Ann., 304 (1996), 203–224.

[X] Y. Xu: Symmetric minimal surfaces in R^{3}, Pacific J. Math., 171 (1995), 275–296.

Shin Kato: Department of Mathematics, Faculty of Science, Osaka City University, Osaka 558, JAPAN E-mail: shinkato@sci.osaka-cu.ac.jp

Masaaki Umehara: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560, JAPAN E-mail: umehara@math.wani.osaka-u.ac.jp Current Address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, GERMANY

Kotaro Yamada: Department of Mathematics, Faculty of General Education, Kumamoto University, Kumamoto 860, JAPAN E-mail: kotaro@gpo.kumamoto-u.ac.jp