Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 1, 135-143.

Minimal Surfaces in a Complex Projective Space whose Mean Curvature Vectors are Eigenvectors

Shi-jie Li and Qi-bin Chen^{1}

Department of Mathematics, South China Normal University Guangzhou 510631, China

Abstract. B.Y. Chen [Null 2-type surfaces in Euclidean space, Algebra, Analysis and Geometry, Taipei, World Scientific 1989, 1-18] initiated the study of submanifolds whose mean curvature vector H is an eigenvector of the Laplacian Δ. In this article, we study this problem for minimal surfaces in CP^{n}. Two classification theorems are obtained. MSC 1991: 53C42

1. Introduction

Let x̃ : M → CP^{n} be an isometric immersion of a connected Riemannian manifold M into a complex projective space CP^{n}, and Φ : CP^{n} → E^{N} the first standard imbedding of CP^{n} into a Euclidean space E^{N}. Then the composition x = Φ ∘ x̃ : M → E^{N} can be decomposed as x = ∑_{k=0}^{∞} x_{k}, where x_{k} is the k-th eigenfunction of the Laplacion Δ of M. The immersion x is said to be of finite Chen type if

x = x_{0} + x_{1} + ⋯ + x_{k,}  (1)

where x_{0} is a constant vector, x_{1}, ..., x_{k} are non-constant maps satisfying Δx_{i} = λ_{i}x_{i}, i = 1, ..., k. If in particular all eigenvalues λ_{1}, ..., λ_{k} are mutually different, then the immersion x (or the submanifold M) is said to be of k-type and if in particular one of λ_{1}, ..., λ_{k} is zero, then x is said to be of null k-type. It is clear that every submanifold of null k-type is non-compact (for details see [1]).

In the study of finite Chen type submanifolds in CP^{n}, Ros [13] and Dimitric [10] classified 1-type compact CR-minimal submainifolds and 1-type compact submanifolds with parallel mean curvature vector, respectively, and Shen [14] classified 2-type compact non-holomorphic minimal surfaces.

1 Supported by HERF from Guangdong Province, China.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

In [2] Chen initiated the study of submanifolds whose mean curvature vector H is an eigenvector of the Laplacian Δ and proved that such submanfolds are either biharmonic or of 1-type or of null 2-type. The classification of surfaces with ΔH = λH in a Euclidean 3-space was first done by Chen in 1988 [3] and, independently, by W.H. Chen in 1994 [8]. Moreover, in [4] and [5] Chen classified such submanifolds in hyperbolic spaces and de Sitter space-time, respectively.

In this paper, we study minimal surfaces in CP^{n} whose mean curvature vector H in E^{N} satisfies

ΔH = λH,  (2)

where λ is a nonzero constant. According to Chen’s result obtained in [2], (2) implies that such surfaces are of 1-type or null 2-type [2]. Actually we will mainly prove the following.

Theorem 1. Let x̃ : M → CP^{n}(4) be an isometric minimal immersion of a connected (not necessary compact) surface M into a complex projective space CP^{n}(4) with the Fubini-Study metric of constant holomorphic sectional curvature 4 and Φ : CP^{n}(4) → E^{N} the first standard imbedding of CP^{n}(4) into a Euclidean space E^{N}. Then the mean curvature vector H of x = Φ ∘ x̃ satisfies (2) if and only if M is one of the following:

i) a portion of CP^{1}(4);

ii) a totally real isotropic minimal surface of CP^{2}(4).

The authors wish to thank Professor B.Y. Chen and the referee for many valuable comments and suggestions.

2. Preliminaries

Let HM(n + 1) = \{A ∈ GL(n + 1) | \overline{A} = A^{t}\} be the set of all Hermitian matrices of degree (n+1). Equipped with the metric

⟨A, B⟩ = \frac{1}{2}trace(AB), for any A, B ∈ HM(n + 1),  (3)

it becomes the usual Euclidean space E^{N} of dimension N = (n + 1)^{2}. Consider the submanifold CP^{n} = {A ∈ HM(n + 1) | A^{2} = A and trace A = 1}. It is known that CP^{n}, with the metric induced by the one on HM(n + 1), is isometric to the complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature 4. We denote the isometry by Φ : CP^{n}(4) → HM(n + 1).

Let D, ˜∇, σ̃, ˜∇^{⊥}, Λ̃ and H̃ be the Levi-Civita connection of HM(n + 1), the induced connection in CP^{n}, the second fundamental form, the normal connection , the Weingarten map and the mean curvature vector of Φ, respectively. Then we have

σ̃(JX, JY) = σ̃(X, Y), \overline{∇}σ̃ = 0.  (4)

\begin{matrix}⟨σ̃(X, Y), σ̃(V, W)⟩ = & 2 ⟨X, Y ⟩ ⟨V, W ⟩ + ⟨X, W ⟩ ⟨Y, V ⟩ + ⟨X, V ⟩ ⟨Y, W ⟩ \\ + & ⟨X, JW ⟩ ⟨Y, JV ⟩ + ⟨X, JV ⟩ ⟨Y, JW ⟩ , \end{matrix}  (5)

where X, Y, V, W ∈ T (CP^{n}), J is the complex structure of CP^{n} and \overline{∇}σ̃ the covariant derivative of σ̃ (for details see [13]). We recall the curvature tensor R̃ of CP^{n} for later use:

\begin{aligned}R̃(X, Y)Z = & < Y, Z > X− < X, Z > Y + < JY, Z > JX \\ − & < JX, Z > JY + 2 < X, JY > JZ, \end{aligned}  (6)

for any X, Y, Z ∈ T (CP^{n}).

Let x̃ : M → CP^{n} be an immersion of a real surface M into CP^{n}. We choose a local field of orthonormal frames {e_{1}, e_{2}} on M. Define the Kaehler angle α of x̃ by cosα =< Je_{1}, e_{2} >. In particular, if x̃ is minimal, then sin α vanishes identically (then x̃ is ±holomorphic) or sinα has only isolated zeros. For the second case, in the open subset where sinα ≠ 0, we may extend {e_{i}, Je_{i}} to a neighborhood U of CP^{2} such that D_{U} = span{e_{i}, Je_{i}} is a real 4-dimensional distribution on U and put

(sin α)e_{1^{∗}} = −(cos α)e_{1} − Je_{2},
(sin α)e_{2^{∗}} = Je_{1} − (cos α)e_{2}. (7)

It is easy to see that {e_{i}, e_{i^{∗}}} is an orthonormal basis on D_{U}. Take {e_{α}, e_{α^{∗}}} as an orthonormal basis in CP^{n} which extends {e_{i}, e_{i^{∗}}} and satisfies e_{τ^{∗}} = Je_{τ}, for τ ≥ 3 (for details see [9]). Here and in the sequal we employ the following index conventions: i, j, k, ... = 1, 2; τ, ζ, ... = 3, ..., n; α, β, γ, ... = 1, ..., n; and shall agree that repeated indices are summed over the respective ranges.

3. Computation of the Laplacian ΔH

Let M be a connected (not necessary compact) surface and x̃ : M → CP^{n} an isometric minimal immersion. Denote by σ, ∇^{⊥}, Λ and H′ the second fundamental form, normal connection, the Weingarten map and the mean curvature vector of x̃, respectively. Let ∇ be the induced connection on M and H the mean curvature vector of x = Φ ∘ x̃. Then we have

H = H′ + H̃, H′ = 0 and H̃ = \frac{1}{2}Σ σ̃(e_{i}, e_{i}).  (8)

We assume that ∇_{e_{i}}e_{j} |_{A}= 0 for any i, j at a point A ∈ x(M) .By (4) and (5), we have

D_{j}H = \frac{1}{2}ΣD_{j}σ̃(e_{i}, e_{i}) = Σσ̃(σ(e_{i}, e_{j}), e_{i}) − 3e_{j} − Σ < Je_{i}, e_{j} > Je_{i}  (9)

\begin{aligned}ΔH = & − ΣD_{j}D_{j}H = −Σ ˜∇_{j}^{⊥}σ̃(σ(e_{i}, e_{j}), e_{i}) + 6H \\ & + ΣΛ̃_{_{σ}(σ(e_{i,}e_{j}),e_{i})}e_{j} + ΣD_{j}(< Je_{i}, e_{j} > Je_{i}), \end{aligned}  (10)

where we put D_{j} = D_{e_{j}} and ˜∇_{j} = ˜∇_{e_{j}}. And by using the Codazzi equation of x̃ and (4)-(7) the terms of (10) become

ΣΛ̃_{σ̃(σ(e_{i},e_{j}),e_{i})}e_{j} = Σ < Jσ(e_{i}, e_{j}), e_{j} > Je_{i,}  (11)

ΣD_{j}(< Je_{i}, e_{j} > Je_{i}) = Σ < Jσ̃(e_{i}, e_{j}), e_{j} > Je_{i} + 2(cos α)σ̃(Je_{1}, e_{2}),  (12)
Σ ˜∇_{j}^{⊥}σ̃(σ(e_{i}, e_{j}), e_{i}) =Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) − Σσ̃(Λ_{σ(e_{i},e_{j})}e_{j}, e_{i})
−6(cos α)σ̃(Je_{1}, e_{2}) + 6(cos^{2} α)H.

(13)

Since we also have, with the help of the Gauss equation of x̃,

Σ < Jσ(e_{i}, e_{j}), e_{j} > Je_{i} = e_{2}(cos α)Je_{1} − e_{1}(cos α)Je_{2},  (14)
Σσ̃(Λ_{σ(e_{i},e_{j})}e_{j}, e_{i}) = 2[(1 + 3 cos^{2} α) − K]H,  (15)

combining (10)-(15) together, we may obtain

ΔH = − Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) + 8(cos α)σ̃(Je_{1}, e_{2})

+ 2(4 − K)H + 2e_{2}(cos α)Je_{1} − 2e_{1}(cos α)Je_{2}. (16)

Lemma 3.1. Let M be a connected minimal surface in CP^{n}. If the mean curvature vector H of M in E^{n} satisfies (2), then α = const, i.e., M is a slant surface in CP^{n}.

Proof. In (16), we see that the last two terms belong to T (CP^{n}) and (2) implies that they must vanish, which means that α = const.

Lemma 3.2. Let M be a connected ±holomorphic minimal surface in CP^{n}(4). Then the mean curvature vector H of M in E^{N} satisfies (2) if and only if M is a portion of CP^{1}(4).

Proof. Let M be a connected minimal surface in CP^{n}(4). If M is ±holomorphic, then cosα = ±1, it follows that Je_{1} = ±e_{2}, Je_{2} = ∓e_{1} in (7) and hence (16) becomes

ΔH = −Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) + 2(8 − K)H.  (17)

If H satisfies (2), then we have < ΔH, H >= λ < H, H >, which implies from (8) and (17) that λ = 12 − K. On the other hand, we may also have < ΔH, σ̃(σ(e_{1}, e_{1}), σ(e_{1}, e_{1})) >= λ < H, σ̃(σ(e_{1}, e_{1}), σ(e_{1}, e_{1})) >, which gives that λ = 2K. Consequently, we may obtain that K = 4 and hence ∥ σ ∥^{2}= 8−2K = 0, i.e., M is totally geodesic. Thus M is a portion of CP^{1}(4). The converse is obvious.

4. Non ±holomorphic minimal surfaces

In this section, we assume that M is a non ±holomorphic minimal surface in CP^{n}(4). Then, according to Section 2, we may take the basis {e_{α}, e_{a^{∗}}} on M. Thus by (7) we have

(sin^{2} α)Σσ̃(e_{i^{∗},}e_{i^{∗}}) = 2(1 + cos^{2} α)H − 4(cos α)σ̃(Je_{1}, e_{2}).  (18)

Applying it to (16), we obtain

ΔH = − Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) + 2[2(3 + cos^{2} α) − K]H
− 2(sin^{2} α)Σσ̃(e_{i^{∗},}e_{i^{∗}}) + 2e_{2}(cos α)Je_{1} − 2e_{1}(cos α)Je_{2}.

(19)

If H satisfies (2), then from (19) and Lemma 3.1 we have

Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) − [4(3 + cos^{2} α) − 2K − λ]H

+ 2(sin^{2} α)Σσ̃(e_{i^{∗}}, e_{i^{∗}}) = 0. (20)

Lemma 4.1. Let M be a minimal surface of CP^{n}(4). If M is non ±holomorphic, then we have the following relations:

< H, H >= 3 + cos^{2} α,  (21)
< H, Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) >= 2 ∥ σ ∥^{2} +(sin^{2}) ∥ σ_{∗} ∥^{2}  (22)
< H, Σσ̃(e_{i^{∗}}, e_{i^{∗}}) >= 4 + 2(sin^{2} α),  (23)
< H, Σσ̃(e_{k}, e_{k^{∗}}) >= 2(sin α)(cos α),  (24)
< Σσ̃(e_{k^{∗}}, e_{k^{∗}}), Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) >= 4 ∥ σ ∥^{2} +2(1 + cos^{2} α) ∥ σ_{∗} ∥^{2},  (25)
< Σσ̃(e_{k^{∗}}, e_{k^{∗}}), Σσ̃(e_{i^{∗}}, e_{i^{∗}}) >= 4(3 + cos^{2} α),  (26)
< Σσ̃(e_{k}, e_{k^{∗}}), Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) >= −2(sin α cos α) ∥ σ_{∗} ∥^{2},  (27)
< Σσ̃(e_{k}, e_{k^{∗}}), Σσ̃(e_{i^{∗}}, e_{i^{∗}}) >= −4(sin α cos α),  (28)

where ∥ σ ∥^{2}= Σ < σ(e_{i}, e_{j}), σ(e_{i}, e_{j}) > and ∥ σ_{∗} ∥^{2}= Σ < σ(e_{i}, e_{j}), e_{k^{∗}} >^{2}.

The proof can be obtained by using (5) and (7) repeatedly.

Applying < H, − >, < Σσ̃(e_{k^{∗}}, e_{k^{∗}}), − > and < Σσ̃(e_{k}, e_{k^{∗}}), − > to (20) one after another and using Lemma 4.1, we may get the following.

2 ∥ σ ∥^{2} +(sin^{2} α) ∥ σ_{∗} ∥^{2} −[4(3 + cos^{2} α) − 2K − λ](3 + cos^{2} α)

+ 4(sin^{2} α)(2 + sin^{2} α) = 0, (29)

2 ∥ σ ∥^{2} +(1 + cos^{2} α) ∥ σ_{∗} ∥^{2} −[4(3 + cos^{2} α) − 2K − λ](2 + sin^{2} α)

+ 4(sin^{2} α)(3 + cos^{2} α) = 0, (30)

(sin α cos α)[∥ σ_{∗} ∥^{2} +16 − 2K − λ] = 0.  (31)

Since we have assumed that M is non ±holomorphic, (31) implies that either cosα vanishes identically or the following relation:

∥ σ ∥^{2}= 2K + λ − 16  (32)

holds on the open subset of M where cosα ≠ 0. We now claim that the second case can not heppen. Since, otherwise, from the Gauss equation and (6), we have

∥ σ ∥^{2}= 2(1 + 3 cos^{2} α) − 2K,  (33)

adding (29) and (30) together and using (33), we may get

∥ σ_{∗} ∥^{2}= −2K − 3λ + 20 + 12 cos^{2} α.  (34)

Combining (32) and (34), we might deduce the following

K = 9 + 3 cos^{2} α − λ,  (35)
∥ σ_{∗} ∥^{2}= 2 + 6 cos^{2} α − λ.  (36)

Substituting (35) into (33), we would have

∥ σ ∥^{2}= 2λ − 16,  (37)

which implies that λ ≥ 8. Applying this to (36), we would obtain

∥ σ_{∗} ∥^{2}≤ 2 + 6 cos^{2} α − 8 = −6 sin^{2} α < 0,  (38)

which is a contradiction. Thus cosα must vanish identically, i.e., M is totally real and we have proved the following.

Lemma 4.2. Under the same assumption as in Lemma 3.1. If in addition M is non ±holomorphic, then M must be totally real in CP^{n}(4).

In the following we assume that M is totally real. Then we may take

e_{1^{∗}} = −Je_{2}, e_{2^{∗}} = Je_{1.}  (39)

Consequently, (33), (34) and (20) become the following, respectively,

∥ σ ∥^{2}= 2 − 2K,  (40)
∥ σ_{∗} ∥^{2}= 20 − 2K − 3λ,  (41)
Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) − (8 − 2K − λ)H = 0.  (42)

Apply < Σσ̃(e_{τ} , e_{τ}), − > to (42), since we have

< Σσ̃(e_{τ} , e_{τ}), Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) >= 2(n − 1) ∥ σ ∥^{2} −2 ∥ σ_{∗} ∥^{2},  (43)
< Σσ̃(e_{τ} , e_{τ}), H >= 2(n − 2),  (44)

we may get, with the help of (40) and (41),

(n + 1)(λ − 6) = 0,  (45)

which implies that λ = 6 and it follows that

∥ σ ∥^{2}=∥ σ_{∗} ∥^{2}  (46)

Σσ̃(σ(e_{i}, e_{j}), σ(e_{i}, e_{j})) − (2 − 2K)H = 0.  (47)

Proof of Theorem 1. If M is ±holomorphic, the theorem holds by Lemma 3.2. If M is non ±holomorphic, Lemma 4.2 shows that M must be totally real and, moreover, (46) and (47) hold on M. Since (46) implies

< σ(e_{i}, e_{j}), e_{τ} >= 0 and < σ(e_{i}, e_{j}), Je_{τ} >= 0,  (48)

for any i, j, τ, thus by Theorem 2 of [6], M is in a totally geodesic CP^{2}.

Applying < Σσ̃(σ(e_{k}, e_{l}), σ(e_{k}, e_{l})), − > to (47), we get

∥ σ ∥^{4} +Σ < σ(e_{i}, e_{j}), σ(e_{k}, e_{l}) >^{2} −3(1 − K) ∥ σ ∥^{2}= 0.  (49)

Substituting (40) into (49), we may find that

(| σ(e_{1}, e_{1}) |^{2} − | σ(e_{1}, e_{2}) |^{2}) + 4 < σ(e_{1}, e_{1}), σ(e_{1}, e_{2}) >^{2}= 0,  (50)

which implies that

| σ(e_{1}, e_{1}) |^{2}=| σ(e_{2}, e_{2}) |^{2}=| σ(e_{1}, e_{2}) |^{2}  (51)
< σ(e_{1}, e_{1}), σ(e_{1}, e_{2}) >= 0 and < σ(e_{2}, e_{2}), σ(e_{1}, e_{2}) >= 0.  (52)

From these we may conclude that M is isotropic in CP^{2} in the sense of O’Neil [12].

Conversely, if M is a totally real isotropic minimal surface of CP^{2}, then by Theorem 2.8 of [13], M is a minimal surface in a sphere of HM(3) and thus (2) holds on M according to the well known Takahashi theorem, in fact M is of 1-type.

From the proof of Theorem 1, we have the following immediately.

Corollary. There are no minimal surfaces in CP^{n} which is of null 2-type via the first standard imbedding of CP^{n} into E^{N}.

5. Minimal surfaces with constant Gauss curvature

In Theorem 1, if in addition M has constant Gauss curvature, then we will get the result more precisely and thoroughly. Actually we have the following.

Theorem 2. Let x̃ : M → CP^{n}(4) and Φ : CP^{n} → E^{N} be as in Theorem 1. If x̃ is non ±holomorphic and the Gauss curvature K of M is constant, then the mean curvature vector H of x = Φ ∘ x̃ satisfies (2) if and only if M is an open portion of one of the following surfaces:

i) the real projective plane RP^{2}(1) isometrically immersed in CP^{2}(4) as a totally real totally geodesic surface;

ii) the totally real minimal flat surface of CP^{2}(4) defined by {[z_{0}, z_{1}, z_{2}] ∈ CP^{2}(4) : z_{0}\overline{z}_{0} = z_{1}\overline{z}_{1} = z_{2}\overline{z}_{2}\},where [z_{0}, z_{1}, z_{2}] denotes the homogeneous coordinates of CP^{2}(4), and, with respect to a suitable orthonormal coframe field {ω^{1}, ω^{2}, ω^{1^{∗}}, ω^{2^{∗}}} in CP^{2}(4), the connection form (ω_{B}^{A}) of CP^{2}(4), restricted to M, is given by

( \begin{matrix}0 & 0 & −μω^{2} & −μω^{1} \\ 0 & 0 & −μω^{1} & μω^{2} \\ μω^{2} & μω^{1} & 0 & 0 \\ μω^{1} & −μω^{2} & 0 & 0 \end{matrix} ) , μ = \frac{1}{\sqrt{2}}.  (53)

Proof. If K = const., we recall from [7] the following formula for totally real minimal surfaces in CP^{2}(4) :

\frac{1}{2}Δ ∥ σ ∥^{2}=∥ \overline{∇}σ ∥^{2} +2Σtrace(Λ_{i^{∗}}Λ_{j^{∗}})^{2} − 3Σ(traceΛ_{i^{∗}}Λ_{j^{∗}})^{2} + 3 ∥ σ ∥^{2} .  (54)

Since M is isotropic, we may choose a local orthonormal frame field {e_{1}, e_{2}; e_{1^{∗}}, e_{2^{∗}}} on CP^{2}(4) such that ,restricted to M, e_{1}, e_{2} are tangent to M and have

Λ_{1^{∗}} = ( \begin{matrix}μ & 0 \\ 0 & −μ \end{matrix} ) , Λ_{2^{∗}} = ( \begin{matrix}0 & μ \\ μ & 0 \end{matrix} ) .  (55)

Then ∥ σ ∥^{2}= 4μ^{2} and μ^{2} =∥ σ ∥^{2} /4 = (1 − K)/2 = const. Thus from (54) and (55) we have

∥ \overline{∇}σ ∥^{2}= 12μ^{2}(2μ^{2} − 1),  (56)

which implies that either μ = 0 or μ^{2} ≥ \frac{1}{2}. If μ vanishes identically, then ∥ σ ∥^{2}= 0 and K = 1, thus M is a totally geodesic surface of CP^{2}(4) with constant curvature K = 1, i. e., M is an open portion of i). Otherwise, we have that K = 1−2μ^{2} ≤ 0, then by Theorem A of [11], K = 0 and μ^{2} = \frac{1}{2}. Moreover, it is easy to see that (53) follows, which implies that M is an open portion of ii).

The converse is clear.

References

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[3] Chen, B.Y.: Null 2-type surfaces in E^{3} are circular cylinders, Kodai Math. J. 11 (1988), 295-299.

[4] Chen, B.Y.: Some classification theorems for submanifolds in Minkowski space-time, Arch. Math. 62 (1994), 177-182.

[5] Chen, B.Y.: Submanfolds in de Sitter space-time satisfying ΔH = λH, to appear in Iseral J. Math..

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[7] Chen, B.Y. and Ogiue, K.: On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266.

[8] Chen, W.H.: The surfaces whose mean curvature vectors are eigenfunctions, Acta Scientiarum Naturelium, Universitatis Pekinensis, 30 (1994), 257-263.

[9] Chern, S.S. and Wolfson, J.G.: Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), 59-83.

[10] Dimitric, I.: 1-type submanifolds of the complex projective space, Kodai Math. J. 14 (1991), 281-295.

[11] Ohnita, Y.: Minimal sufaces with constant curvature and Kaehler angle in complex space forms, Tsukuba J. Math., 13 (1989), 191-207.

[12] O’Neil, B.: Isotropic and Kaehler immersions, Canadian J. Math. 17 (1965), 907-915.

[13] Ros, A.: Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J. 6 (1983), 88-99.

[14] Shen, Y.B.: On spectral geometry of minimal surfaces in CP^{n}, preprint.

Received October 30, 1994; revised version February 21, 1995.