Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 167-176.

Minimal submanifolds of Kähler-Einstein manifolds without complex directions

and of half dimension^{1}

Giorgio Valli^{2} Isabel M. C. Salavessa

Dipartimento di Matematica, Università di Pavia Via Abbiategrasso 215, 27100 Pavia, Italy e-mail: valli@dimat.unipv.it

Centro de Física das Interacções Fundamentais Instituto Superior Técnico, Edifício Ciência piso 3, 1049-001 Lisboa Codex, Portugal e-mail: isabel@cartan.ist.utl.pt

Abstract. We study minimal real 2n-submanifolds F : M → N without complex directions immersed into a Kähler-Einstein manifold of complex dimension 2n. We find conditions, for example on the curvature of M and N, or equal Kählerangles, or broad-pluriminimality of F, to conclude that F must be a Lagrangian submanifold. Our main tool is a formula of the Laplacian of a certain symmetric function of the Kähler angles of F.

1. Introduction

This paper is a summary of part of research results, [5] and [6], started two years ago, on the study of immersions into Kähler-Einstein manifolds through its Kähler angles, and that is a generalization of part of the work of Wolfson in [7] to higher dimensions.

168 G. Valli and I. M. C. Salavessa: Minimal submanifolds … without complex directions …

Let (N, J, g) be a Kähler manifold of complex dimension 2n, with complex struture J and Riemannian metric g, and let F : M → N be an immersed submanifold of real dimension 2n. We denote by ω the Kähler form of N, ω(X, Y) = g(JX, Y). On M we take the induced metric g_{M} = F^{∗}g. N is Kähler-Einstein if its Ricci tensor is a multiple of the metric, Ricci^{N} = Rg. We start by recalling a result of Wolfson:

1This is a summary of research results which have been submitted to publication elsewhere.

2deceased on October 2nd, 1999

Theorem 1.1 [7] If M is a real compact orientable surface and N is a complex Kähler-Einstein surface with R < 0, and if F is minimal with no complex points, then F is Lagrangian.

The main idea of the proof is the following. Since n = 1, F^{∗}ω is a multiple of the volume element of M, F^{∗}ω = aV ol_{M}, with a : M → [−1, 1] a real smooth map. Then a = cos θ, where θ is called the Kähler angle of the surface. If F is minimal without complex points, that is, | cos θ| ≠ 1, then

△(κ(cos θ)) = −R cos θ

for some convenient real map κ : (−1, 1) → IR. If R < 0, an application of the maximum and minimum principle implies cos θ = 0 everywhere. In some cases (see Sections 2, 3) we generalize this result to higer dimensions.

First we recall the concept of Kälher angles of F, introduced by Chern and Wolfson [1] for surfaces, as some functions that at each point p of M measure the deviation of the tangent plane T_{p}M of M from a complex subspace or a Lagrangian subspace of T_{F (p)}N. At each base point p ∈ M, we identify F^{∗}ω with a skew-symmetric operator of T_{p}M by g_{M}(F^{∗}ω(X), Y) = F^{∗}ω(X, Y). The polar decomposition of F^{∗}ω is given by

F^{∗}ω = J_{ω} g̃,  (1.1)

where J_{ω} : T_{p}M → T_{p}M is a (unique) partial isometry with the same kernel K_{ω} as of F^{∗}w, and where g̃ is the positive semidefinite operator g̃ = |F^{∗}ω| = \sqrt{−(F^{∗}ω)^{2}}. It turns out that J_{ω} : K_{ω}^{⊥} → K_{ω}^{⊥} defines a complex structure on K_{ω}^{⊥}, the orthogonal complement of K_{ω} in T_{p}M. We denote by Ω_{2k}^{0} the set of interior points in M where F^{∗}w has constant rank 2k, 0 ≤ k ≤ n. Then K_{ω}^{⊥} is a smooth sub-vector bundle of T M on Ω_{2k}^{0}. Moreover, g̃ and J_{ω} are both smooth on those open sets, and (K_{ω}^{⊥}, J_{ω}, g_{M}) constitutes a smooth Hermitean vector bundle. We should note that on Ω_{2n}^{0}, the open set where F^{∗}ω is non-degenerate, the almost complex struture J_{ω} is not integrable in general. The positive semidefinite tensor g̃ is in fact continuous in all M and locally Lipschitz. At each point p we may take {X_{α}, Y_{α}}_{1≤α≤n} a g_{M}-orthonormal basis of T_{p}M that diagonalizes F^{∗}ω at p. By this one means that, in that basis, F^{∗}ω is matricially a sum of block matrices of the form

F^{∗}ω = ⨁_{0≤α≤n} [ \begin{matrix}0 & −a_{α} \\ a_{α} & 0 \end{matrix} ] ,

where a_{1}, a_{2}, …, a_{n} are real numbers, also called the singular values of F^{∗}ω. From

|a_{α}| = |F^{∗}ω(X_{α}, Y_{α})| = |g(JdF (X_{α}), dF (Y_{α}))| ≤ 1

we have a_{α} = cos θ_{α}, for some angle θ_{α}. We may assume cos θ_{1} ≥ … ≥ cos θ_{n} ≥ 0, by interchanging X_{α} with Y_{α}, and reordering the basis when necessary.

Definition 1.1 These angles θ_{α}, 1 ≤ α ≤ n, are called the Kähler angles of F at p.

Then, ∀α, F^{∗}ω(X_{α}) = cos θ_{α}Y_{α}, F^{∗}ω(Y_{α}) = − cos θ_{α}X_{α}, and, if 2k is the rank of F^{∗}ω at p with k ≥ 1, J_{ω}X_{α} = Y_{α}, ∀α ≤ k. Since the map A → |A| is a Lipschitz map in the space of normal operators, the Weyl’s pertubation theorem applied to the eigenvalues of the symmetric operator |F^{∗}ω| shows that, ordering the cos θ_{α} in the above decreasing way, for each α the map p → cosθ_{α}(p) is locally Lipschitz on M. A complex direction of F is a real two-plane P of T_{p}M such that dF (P) is a complex line of T_{F (p)}N, that is, JdF (P) ⊂ dF (P). Similarly, P is called a Lagrangian direction of F if ω vanishes on dF (P), that is, JdF (P)⊥dF (P). F has no complex directions iff cos θ_{α} < 1 ∀α. We note that for n = 1 our definition of Kähler angles may be slightly different from the original one in [1], because we demanded cos θ ≥ 0. Our cos θ may not be smooth near a point where it vanishes. We have chosen this definition because in higher dimensions we do not have a preferential orientation assigned to the real planes P_{α} = span{X_{α}, Y_{α}}, even if M is orientable. Let us consider the complex vectors of the complexified tangent space of M at p, T_{p}^{c}(M), where {X_{α}, Y_{α}}_{1≤α≤n} diagonalizes F^{∗}ω at p,

Z_{α} = \frac{X_{α} − iY_{α}}{2} = “ α ”, Zᾱ = \overline{Z_{α}} = \frac{X_{α} + iY_{α}}{2} = “ ᾱ ”.

We extend g_{M} and the curvature tensors of M and N to the complexified tangent spaces by lC-multilinearity. Let NM = (dF (T M))^{⊥} denote the normal bundle of F, and ( )^{⊥} denote the orthogonal projection of F^{−1}T N onto the normal bundle. If F is any immersion of real dimension 2n and no complex directions, then {dF (Z_{α}), dF (Zᾱ), (JdF (Z_{α}))^{⊥}, (JdF (Zᾱ))^{⊥}}_{1≤α≤n} constitutes a complex basis of T_{F (p_{0})}^{c}N. On M the Ricci tensor of N can be described by the following expression, for U, V ∈ T_{F (p)}N,

Ricci^{N}(U, V) = ∑_{1≤μ≤n} \frac{4}{sin θ_{μ}} R^{N}(U, JV, dF (μ), (JdF (μ̄))^{⊥}),

where R^{N} denotes the Riemannian curvature tensor of N. An application of the Codazzi’s equation to the above expression leads the first result:

Proposition 1.1 [5] If F is a totally geodesic map without complex directions and N is Kähler-Einstein with non-zero Ricci tensor, then F is Lagrangian.

We consider the following morphism of vector bundles

\begin{aligned}Φ : & T M & → & NM \\ & X & → & (JdF (X))^{⊥}. \end{aligned}

Both T M and NM are real vector bundles of the same dimension 2n. If F has no complex directions, then Φ is an isomorphism. Moreover, the tensor ĝ given by

ĝ(X, Y) = g_{M}(X, Y) − g(F^{∗}ω(X), F^{∗}ω(Y))

defines a smooth Riemannian metric on M, and ĝ = g_{M} − g̃^{2} = ∑_{α} sin^{2} θ_{α}Z_{∗}^{α} ⊙ Zᾱ ; ∗, where g̃ is given by (1.1). With this metric, Φ : (T M, ĝ) → (NM, g) is an isometry, that is, Φ is an isomorphism of Riemannian vector bundles. Let us denote by

∇  the Levi-Civita connection of (M, g_{M})
∇^{}ˆ  the Levi-Civita connection of (M, ĝ)
∇^{⊥}  the usual connection of NM induced by the Levi-Civita connection of N
∇′  the connection on T M that makes the isomorphism Φ parallel.

We will also denote by ∇the Levi-Civita connection of N. Thus, if U is a smooth section of NM ⊂ F^{−1}T N and X is a vector field on M,

∇_{X}^{⊥}U = (∇_{X}U)^{⊥}

and ∇′= Φ^{−1∗}∇^{⊥}, that is, for X, Y smooth vector fields of M,

Φ(∇′_{X}Y) = ∇_{X}^{⊥}(Φ(Y)).

The connections ∇ and ∇^{}ˆ have no torsion because they are Levi-Civita, but ∇′ may have non-zero torsion T′. Since both ∇^{}ˆ and ∇′ are Riemannian connections of T M for the same Riemannian metric ĝ, T′ = 0 iff ∇^{}ˆ = ∇′. Let us denote by ∇_{X}dF (Y) = ∇dF (X, Y) the second fundamental form of F, which is a symmetric tensor on M that takes values on the normal bundle NM.

Lemma 1.1 [6] If {X_{α}, Y_{α}} is a diagonalizing g_{M}-orthonormal basis of F^{∗}ω at p, then at p

Φ(T′(Z_{α}, Zβ̄)) = i(cos θ_{α} + cos θ_{β})∇_{Z_{α}}dF (Zβ̄)
Φ(T′(Z_{α}, Z_{β})) = i(cos θ_{α} − cos θ_{β})∇_{Z_{α}}dF (Z_{β}).

Proof. Note that, if X ∈ T M, JdF (X) − dF (F^{∗}ω(X)) ∈ NM. Thus

Φ(X) = JdF (X) − dF (F^{∗}ω(X)).

Then

Φ(∇′_{X}Y) = ∇_{X}^{⊥}(Φ(Y)) = (∇_{X}(Φ(Y)))^{⊥}
= (∇_{X}(JdF (Y) − dF (F^{∗}ω(Y)))^{⊥}
= (J∇_{X}dF (Y) + JdF (∇_{X}Y) − ∇_{X}dF (F^{∗}ω(Y)))^{⊥}.

Therefore, using the symmetry of ∇dF and the fact that ∇ is torsionless,

Φ(T′(X, Y)) = Φ(∇′_{X}Y − ∇′_{Y} X − [X, Y])
= −∇_{X}dF (F^{∗}ω(Y)) + ∇_{Y} dF (F^{∗}ω(X)).  (1.2)

The lemma now follows immediately. □

This lemma will be useful in the following. It also shows that T′ vanishes in certain directions for broadly-pluriminimal immersions or for immersions with equal Kähler angles.

2. A formula

For a minimal immersion F with no complex directions, we consider the map

κ = ∑_{1≤α≤n} log (\frac{1 + cos θ_{α}}{1 − cos θ_{α}} ) = \frac{1}{2} log (\frac{det(g_{M} + g̃)}{det(g_{M} − g̃)}) .

This map is non-negative, and vanishes at a point p iff all cos θ_{α} = 0 at that point. It is smooth on each Ω_{2k}^{0}, continuous on all M, and locally a Lipschitz map. We can compute △κ on those open sets where κ is smooth, but this gives a very long computation. In [5], [6] we derive an expression for △κ at p_{0} ∈ Ω_{2k}^{0}, namely

△κ = 4i ∑_{β} Ricci^{N}(JdF (β), dF (β̄))  (2.1)

+ ∑_{β,μ} \frac{32}{sin^{2} θ_{μ}} Im(R^{N}(dF (β), dF (μ), dF (β̄), JdF (μ̄)+i cos θ_{μ}dF (μ̄)))

− ∑_{β,μ,ρ}\frac{32(cos θ_{μ}+cos θ_{ρ})}{sin^{2} θ_{μ} sin^{2} θ_{ρ}}Re(g(∇_{β}dF (μ), JdF (ρ̄))g(∇β̄dF (ρ), JdF (μ̄)))

+ ∑_{β,μ,ρ} 32(\frac{1}{sin^{2} θ_{ρ}} − \frac{1}{sin^{2} θ_{μ}})Im(⟨∇_{β}μ, ρ⟩g(∇β̄dF (ρ̄), JdF (μ̄)))

+ ∑_{β,μ,ρ} 32(\frac{1}{sin^{2} θ_{ρ}} − \frac{1}{sin^{2} θ_{μ}})Im(⟨∇β̄μ, ρ⟩g(∇_{β}dF (ρ̄), JdF (μ̄)))

+ ∑_{β,μ,ρ} \frac{32(cos θ_{μ} + cos θ_{ρ})}{sin^{2} θ_{μ}} (|⟨∇_{β}μ, ρ⟩|^{2} + |⟨∇β̄μ, ρ⟩|^{2}),

where {X_{α}, Y_{α}}_{1≤α≤n} is a g_{M}-orthonormal local frame of M, with Y_{α} = J_{ω}X_{α} for α ≤ k, {X_{α}, Y_{α}}_{α≥k+1} any g_{M}-orthonormal frame of K_{ω}, and which at p_{0} diagonalizes F^{∗}ω. There are two cases where the above equation is simplified. The first one is when F is broadlypluriminimal, and the second one when F has equal Kähler angles. We will treat these cases in Sections 3 and 4, respectively.

3. Broadly-pluriminimal submanifolds

Definition 3.1 [5] F is said to be broadly-pluriminimal if

(i) F is minimal, (ii) For each p ∈ Ω_{2k}^{0}, k ≥ 1, F is pluriharmonic with respect to any g_{M}-orthogonal complex structure J̃ = J_{ω} ⊕ J′ on T_{p}M, where J′ is any g_{M}-orthogonal complex structure of K_{ω}, that is, (∇dF)^{(1,1)} = 0 .

In (ii), the (1,1)-part of ∇dF is just given by

(∇dF)^{(1,1)}(X, Y) = \frac{1}{2}(∇dF (X, Y) + ∇dF (J̃X, Ỹ)) ∀X, Y ∈ T_{p}M.

If K_{ω} = 0, (ii) means that F is pluriharmonic with respect to the almost complex structure J_{ω} (see e.g. [4]). Pluriharmonic maps are trivially minimal. Products of minimal real surfaces of Kähler surfaces, minimal Lagrangian submanifolds, and complex submanifolds are examples of pluriminimal submanifolds. For F with equal Kähler angles Ω_{2k}^{0} is only considered and J̃ = J_{ω}.

If F is broadly-pluriminimal, we get the following very simple final expression for △κ on Ω_{2k}^{0},

△κ = 4i ∑_{1≤β≤k} Ricci^{N}(JdF (β), dF (β̄)).

Consequentely,

Proposition 3.1 [5] If N is Kähler-Einstein and F is broadly-pluriminimal without complex directions, then, on each Ω_{2k}^{0}

△κ = −2R( ∑_{1≤β≤n} cos θ_{β}).

The main problem in applying the maximum principle to this equation at a maximum point p_{0} of κ for n ≥ 2, as Wolfson did in [7] for the case n = 1, is that p_{0} may not lie on a open set where κ is smooth. Since we can write κ = κ_{1} + κ_{2}, where κ_{1} = ∑_{1≤α≤2k} log (\frac{1+cos θ_{α}}{1−cos θ_{α}} ) (with 2k the rank of F^{∗}ω at p_{0}) is the (smooth) piece of κ defined by the angles whose cosine is not zero near p_{0} , and κ_{2} with the remaining angles, κ_{1} and κ_{2} having at p_{0} a maximum and a minimum respectively, we can prove that κ is differentiable at p_{0}, but we do not know if it is C^{2} on a neighbourhood of that point. The following lemma is an immediate conclusion from the above formula and will lead to the next theorem.

Lemma 3.1 [5] If N is Kähler-Einstein with R < 0, and if F is broadly-pluriminimal but not Lagrangian and has no complex directions, then p_{0} is not in Ω_{2k}^{0} ∀0 ≤ k ≤ n. That is, if the rank of F^{∗}ω is 2k at p_{0}, then there exists a sequence p_{m} → p_{0} such that the rank of F^{∗}ω at p_{m} is > 2k.

Theorem 3.1 [5] Assume N is Kähler-Einstein with R < 0, and M is compact.

(i) If F is broadly-pluriminimal, then F either has complex or Lagrangian directions. (ii) If F is broadly-pluriminimal without complex directions, and F^{∗}ω has constant rank or rank ≤ 2, then F is Lagrangian. (iii) If n = 2, that is, M has real dimension 4 and N complex dimension 4, and if M is orientable and F is broadly-pluriminimal without complex directions, then F is Lagrangian.

To prove (iii) we replace the continuous map κ by a smooth map κ̃ which coincides with κ on an open set that has p_{0} at the boundary and allows p_{0} as a maximum as well. Namely,

k̃ = log (\frac{1 + cos θ_{1}}{1 − cos θ_{1}} ) + log (\frac{1 + s_{2}}{1 − s_{2}} ) ,

where s_{2}(p) = ϵ(p) cos θ_{2}(p) for p near p_{0}, with ϵ(p) equal to +1 or −1 according as X_{1}, Y_{1}, X_{2}, Y_{2} is a direct or inverse basis, respectively. We can prove that s_{2} is smooth by using the smoothness of F^{∗}ω ∧ F^{∗}ω. Unfortunately, a similar argument does not seem to work in higher dimensions.

4. Immersions with equal Kähler angles

We say that F has equal Kähler angles if θ_{α} = θ ∀α. In this case

F^{∗}ω = cos θ J_{ω} and ĝ = sin^{2} θ g_{M},

with cos θ a locally Lipschitz map on M, smooth on the open set where it does not vanish, and Ω_{2k}^{0} = ∅ ∀k ≠ 0, n. On the open set Ω_{2n}^{0} = cos θ^{−1}(IR\{0}), J_{ω} defines a smooth almost complex structure g_{M}-orthogonal. On the open set cos θ^{−1}(IR\{1}), ĝ is a smooth metric conformally equivalent to g_{M}. Thus, if n ≥ 2, ∇^{}ˆ = ∇iff θ is constant. Note that in this case any local orthonormal frame of the type {X_{α}, Y_{α} = J_{ω}X_{α}} diagonalizes F^{∗}ω on the whole set it is defined. Moreover, from (1.2) of Lemma 1.1, we get

Φ(T′(X, Y)) = 2 cos θ(∇dF)^{(1,1)}(J_{ω}X, Y).  (4.1)

This shows that

Proposition 4.1 [6] If F is a minimal immersion with equal Kähler angles and without complex directions, then T′ = 0, that is, ∇′= ∇^{}ˆ iff Φ is parallel iff F is pluriminimal.

We may extend Φ : T^{c}M → NM^{c} by lC-linearity to the complexified spaces, and we define Re(u + iv) = u, for u, v ∈ NM.

Proposition 4.2 [6] If F is any immersion with equal Kähler angles and no complex directions, then

Φ(\frac{1−n}{2}∇ log(sin^{2} θ)) = \frac{4 cos θ}{sin^{2} θ}Re(i ∑_{β,μ}(g(∇μ̄dF (μ), JdF (β)) − g(∇μ̄dF (β), JdF (μ)))Φ(β̄)) .

where ∇ log(sin^{2} θ) is the gradient with respect to g_{M}.

Since \{Φ(β), Φ(β̄) = \overline{Φ(β)}\}_{1≤β≤n} multiplied by \frac{\sqrt{2}}{sin θ} constitutes an unitary basis of NM^{c}, we immediately conclude

Corollary 4.1 [6] Let F be an immersion with equal Kähler angles, no complex directions, and n ≥ 2. Then θ is constant iff

∑_{μ} g(∇μ̄dF (μ), JdF (β)) = ∑_{μ} g(∇μ̄dF (β), JdF (μ)) ∀β.  (4.2)

In particular, if F is a pluriminimal immersion, then ∇= ∇^{}ˆ = ∇′ and θ = constant.

To prove Proposition 4.2 we relate the three connections of M, ∇, ∇^{}ˆ , and ∇′. Any local g_{M}-orthonormal frame of the form {e_{1}, …, e_{2n}} = {X_{μ}, Y_{μ} = J_{ω}X_{μ}}_{1≤μ≤n} diagonalizes F^{∗}ω. Let

S′(X, Y) = ∇′_{X}Y − ∇^{}_{ˆ X}Y.

Then S′(X, Y) − S′(Y, X) = T′(X, Y) and so S′ = 0 iff ∇′= ∇^{}ˆ iff S′ is symmetric. In [6] we prove

∑_{i} ĝ(S′(e_{i}, e_{i}), X) = − ∑_{i} ĝ(T′(e_{i}, X), e_{i}).  (4.3)

We may compute

\frac{(1 − n)}{4}∇ log sin^{2} θ = ∑_{μ} ∇^{}ˆμ̄μ − ∇μ̄μ.

Now

Φ(∇′_{X}μ) = ((J − i cos θ)∇_{X}dF (μ))^{⊥} + Φ(∇_{X}μ).

It follows that

Φ(∑_{μ} ∇^{}ˆμ̄μ−∇μ̄μ) = ((J −i cos θ)\frac{H}{4})^{⊥}−∑_{μ} Φ(∇′μ̄μ−∇^{}ˆμ̄μ) = \frac{1}{4}(2n(JH)^{⊥}−Φ(T race_{g_{M}} S′)),

where H = \frac{1}{2n} ∑_{i} ∇dF (e_{i}, e_{i}) = \frac{2}{n} ∑_{μ} ∇dF (μ̄, μ) is the mean curvature of F. On the other hand, using (4.3) and Lemma 1.1, we have

Φ(T race_{g_{M}} S′) = \frac{4}{sin^{2} θ} ∑_{μ,β} 2i cos θ(g(∇μ̄dF (β), JdF (μ))Φ(β̄) − g(∇_{μ}dF (β̄), JdF (μ̄))Φ(β)).

With this, and writing (JH)^{⊥} in terms of Φ(β) and Φ(β̄), we prove Proposition 4.2. □

Note that (4.2) of Corollary 4.1 is a sort of symmetry property (we may commute μ with β, and sum over μ), and the first term is just \frac{n}{2}g(H, JdF (β)). In [6] we compute the divergences of F^{∗}ω and of J_{ω} considering F^{∗}ω as an operator of T M. It is particulary interesting the case n = 2.

Proposition 4.3 Let F be an immersion with equal Kähler angles and ∇ cos θ denote the gradient w.r.t g_{M}.

(i) For n = 1, δJ_{ω} = 0 (obviously!). Moreover, δ(F^{∗}ω) = 0 iff θ is constant. (ii) For n = 2, δ(F^{∗}ω) = 0 and J_{ω}(∇ cos θ) = cos θ(δJ_{ω}). Hence, δJ_{ω} = 0 iff θ is constant. (iii) For n ≠ 1, 2, δ(F^{∗}ω) = (n − 2)J_{ω}(∇ cos θ) = \frac{n−2}{n−1} cos θ(δJ_{ω}). Hence, δ(F^{∗}ω) = 0 iff δJ_{ω} = 0 iff θ is constant iff (4.2) holds.

If N is Kähler-Einstein, expression (2.1) for the Laplacian of κ, for F minimal with equal Kähler angles at a maximum point p_{0} of κ with cos θ(p_{0}) ≠ 0 and cos θ(p_{0}) ≠ 1, can be simplified to

△κ = cos θ( − 2nR + \frac{32}{sin^{2} θ} ∑_{β,μ} R^{M}(β, μ, β̄, μ̄) + A + B),  (4.4)

where R^{M} is the Riemannian curvature of M, and

A = \frac{4}{sin^{2} θ}∥∇J_{ω}∥^{2} B = \frac{8(n − 1)}{sin^{2} θ}∥∇ cos θ∥^{2}.

Then A, B ≥ 0, and A = 0 on an open set of M iff (M, J_{ω}, g) is Kähler on that set. If B = 0 on an open set, then the equal Kähler angle is constant. The curvature term of (4.4)

∑_{β,μ} R^{M}(β, μ, β̄, μ̄)  (4.5)

is a hermitian trace of the curvature of M restricted to a maximal isotropic subspace of T^{c}M. To require it to be ≥ 0 seems to be strictly weaker than the non-negative isotropic sectional curvature defined by Micallef and Moore in [3], namely

K(σ) = \frac{g_{M}(R(z ∧ w), z̄ ∧ w̄)}{||z ∧ w||^{2}} ≥ 0,

where σ = span_{l C}{z, w} is a totally isotropic two-plane in T^{c}M, that is, u ∈ σ ⇒ g_{M}(u, u) = 0, and where g_{M}(R(x ∧ y), u ∧ v) = R^{M}(x, y, u, v). Finally, we present the following result.

Theorem 4.1 [6] Let F be minimal with equal Kähler angles, M compact orientable, and N Kähler-Einstein with R ≤ 0. If n = 2 or θ = constant, and M has non-negative isotropic sectional curvature, then one and only one of the following cases holds:

(i) M is a complex submanifold of N. (ii) M is a Lagrangian submanifold of N. (iii) R = 0, cos θ =constant ≠ 0, 1, and J_{ω} is a complex integrable structure, with (M, J_{ω}, g_{M}) a Kähler manifold.

To prove the theorem for n = 2, we use the Weitzenböck formula applied to F^{∗}ω (see [2]). In this case F^{∗}ω is a harmonic 2-form, because it is closed and co-closed (by Proposition 4.3). Moreover, since F has equal Kähler angles, the curvature term (4.5) can be also expressed as a multiple of < SF^{∗}ω, F^{∗}ω >, where S is the Ricci operator applied to forms. We may then conclude that ∇F^{∗}ω = 0, and so the Kähler angle is constant, and if cos θ ≠ 0, (M, J_{ω}, g_{M}) is a Kähler manifold. If n ≠ 2, we assume the Kähler angle is constant. Then, in both cases, equation (4.4) is valid on all M, with △κ = 0. Of course, we may replace the condition on the curvature of M by the weaker condition (4.5) ≥ 0. This theorem can for instance be applied to flat minimal tori immersed in Calabi-Yau manifolds.

References

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[4] Y. Ohnita, S. Udagawa: Stability, complex analicity and constancy of pluriharmonic maps from compact Kähler manifolds. Math. Z. 205 (1990), 629–644.

[5] I. Salavessa, G. Valli: Broadly-pluriminimal Submanifolds of Kähler-Einstein Manifolds. Preprint, submitted for publication.

[6] I. Salavessa, G. Valli: Minimal submanifolds of Kähler-Einstein manifolds with equal Kähler angles. In preparation.

[7] J.G. Wolfson: Minimal Surfaces in Kähler Surfaces and Ricci Curvature. J. Diff. Geom. 29 (1989), 281–294.