Deformation of Kähler metrics and an eigenvalue problem for the Laplacian on a compact Kähler manifold

Abstract

We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the k-th eigenvalue $\lambda _{k}$ as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of $\lambda _{k}$-extremal Kähler metric. We deduce a condition for a Kähler metric to be $\lambda _{k}$-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.

1 Introduction

Let M be a compact manifold of dimension n. Given a Riemannian metric g on M, the volume $\text{ Vol }(M,g)$ and the Laplace-Beltrami operator $\Delta _{g}$ are defined. Let $0 < \lambda _{1}(g) \le \lambda _{2}(g) \le \cdots \lambda _{k}(g) \le \cdots$ be the eigenvalues of $\Delta _{g}$. The quantity $\lambda _{k}(g) \text{ Vol }(M,g)^{2/n}$ is invariant under scaling of the metric g. Hersch [14] proved that on a 2-dimensional sphere $S^{2}$, the scale-invariant quantity $\lambda _{1}(g)\text{ Area }(g)$ is maximized when g is a round metric. Inspired by the work, Berger [3] asked whether

$$\begin{aligned} \Lambda _{1}(M) := \sup _{g} \lambda _{1}(g) \text{ Vol }(M,g)^{2/n} \end{aligned}$$

is finite for a compact manifold M of dimension n. For a surface M, $\Lambda _{1}(M)$ is bounded by a constant depending on the genus [15, 36]. Berger [3] also conjectured that for a 2-dimensional torus $T^{2}$, the flat equilateral torus attains $\Lambda _{1}(T^{2})$. Nadirashvili [24] settled Berger’s conjecture affirmatively. In the same paper, he proved a theorem that a metric g on a given surface M is extremal for the functional $\lambda _{1}: g \mapsto \lambda _{1}(g)$ with respect to all the volume-preserving deformations of the metric if and only if there exists a finite collection of $\lambda _{1}(g)$-eigenfunctions $\{f_{j}\}_{j=1}^{N}$ such that $F:= (f_{1}, \cdots , f_{N}): (M,g) \rightarrow \textbf{R}^{N}$ is an isometric minimal immersion into a sphere in $\textbf{R}^{N}$. Later, El Soufi-Ilias [11] simplified the proof of the theorem and generalized it to a compact manifold M of any dimension. More explicitly, they proved the following:

Theorem 1.1

([11, 24]) Let (M, g) be a compact m-dimensional Riemannian manifold. The metric g is extremal for the functional $\lambda _{1}$ with respect to all the volume-preserving deformations of the metric if and only if there exists a finite collection of $\lambda _{1}(g)$-eigenfunctions $\{f_{1}, \ldots , f_{N} \}$ such that $F:= (f_{1}, \cdots , f_{N}): (M,g) \rightarrow \textbf{R}^{N}$ is an isometric minimal immersion into $S^{N-1}(\sqrt{m/\lambda _{1}(g)}) \subset \textbf{R}^{N}$.

In particular, they showed that the canonical metric on a compact isotropy irreducible homogeneous manifold is extremal for the functional $\lambda _{1}$. For recent remarkable progress in study of $\Lambda _{1}(M)$ for a surface M, see [16, 17, 26,27,28], for example.

On the other hand, on any manifold M with $n \ge 3$, one can construct a 1-parameter family $(g_{t})_{t>0}$ such that the quantity $\lambda _{1}(g_{t}) \text{ Vol }(M,g_{t})^{2/n}$ diverges to infinity as t goes to infinity [8]. (See also [6, 22, 23, 32, 33].) However, for a given Riemannian metric g on M, the restriction of the functional $\lambda _{1}$ to metrics in the conformal class with fixed volume is bounded [10, 19]. El Soufi-Ilias [12] proved that a metric g is extremal for the functional $\lambda _{1}$ among such metrics if and only if there exists a finite collection of eigenfunctions $\{f_{j}\}_{j=1}^{N}$ such that $F:= (f_{1}, \cdots , f_{N}): (M,g) \rightarrow \textbf{R}^{N}$ is a harmonic map into a unit sphere in $\textbf{R}^{N}$ with constant energy density $|dF|^{2} =\lambda _{1}(g)$.

Bourguignon et al. [7] proved the following result:

Theorem 1.2

([7]) Let (M, J) be a compact complex n-dimensional manifold admitting a full holomorphic immersion $\Phi : (M, J) \rightarrow \textbf{C}P^{N}$. Let $\sigma _{FS}$ be the Fubini-Study form on $\textbf{C}P^{N}$ with constant holomorphic sectional curvature 1. Then, for any Kähler form $\omega$ on (M, J), the first eigenvalue $\lambda _{1}(\omega )$ satisfies

$$\begin{aligned} \lambda _{1}(\omega ) \le n\frac{N+1}{N} \frac{\int _{M}\Phi ^{*}\sigma _{FS} \wedge \omega ^{n-1}}{\int _{M} \omega ^{n}}. \end{aligned}$$

The divergence theorem implies that $\lambda _{1}(\omega )$ is bounded by a constant depending on only n, N, $\Phi$ and the Kähler class $[\omega ]$. The above theorem implies that the Fubini-Study metric on $\textbf{C}P^{N}$ is a $\lambda _{1}$-maximizer in its Kähler class. Biliotti-Ghigi [5] generalized the result and showed that the canonical Kähler-Einstein metric on a Hermitian symmetric space of compact type is a $\lambda _{1}$-maximizer in its Kähler class. (See also [2].) Motivated by these results, Apostolov et al. [1] proved that the metric g on a compact Kähler manifold is extremal for the functional $\lambda _{1}$ within its Kähler class if and only if there exists a finite collection of eigenfunctions $\{f_{j}\}_{j=1}^{N}$ such that the equation

$$\begin{aligned} \lambda _{1}(g)^{2}\left( \sum _{j=1}^{N} f_{j}^{2} \right) -2\lambda _{1}(g) \left( \sum _{j=1}^{N} |\nabla f_{j}|^{2} \right) + \sum _{j=1}^{N} |dd^{c}f_{j}|^{2} = 0 \end{aligned}$$

(1.1)

holds. Using this equation, they showed that the metric g of a compact homogeneous Kähler-Einstein manifold $(M, J, g, \omega )$ of positive scalar curvature is extremal for the functional $\lambda _{1}$ within its Kähler class. However, compared to the aforementioned theorems due to Nadirashvili and El Soufi-Ilias, the geometric meaning of the equation (1.1) is not clear.

Let (M, J) be a compact complex manifold satisfying the assumption of Theorem 1.2. Let $H^{1,1}(M,J;\textbf{R}):= H^{1,1}(M,J) \cap H^{2}_{dR}(M)$. Then the map

$$\begin{aligned} H^{1,1}(M,J;\textbf{R}) \rightarrow \textbf{R}, \quad [\omega ] \mapsto \int _{M} \Phi ^{*}\sigma _{FS} \wedge \omega ^{n-1} \end{aligned}$$

is a well-defined continuous function. Thus this is bounded on the compact subset $\{ [\omega ] \in H^{1,1}(M,J;\textbf{R}) \mid \int _{M} \omega ^{n}=1 \}$. In other words, the functional $\lambda _{1}$ is bounded on the set of Kähler metrics with fixed volume on (M, J). However, the property of the $\lambda _{1}$-maximizing Kähler metrics has not been studied.

In this paper, on a compact complex manifold (M, J), we introduce the notion of $\lambda _{k}$-extremal Kähler metric by considering all volume-preserving deformations of the Kähler metric. Be cautioned that we fix the complex structure J and consider only J-compatible Kähler metrics. (See Definition 2.1 for the precise definition of the $\lambda _{k}$-extremality.) The notion of $\lambda _{k}$-extremality in this paper is stronger than that in the above theorem due to Apostolov et al. [1]. We prove that the Kähler metric g is $\lambda _{1}$-extremal if and only if there exists a finite collection of eigenfunctions $\{f_{j}\}_{j=1}^{N}$ such that the equations

$$\begin{aligned} \left\{ \, \begin{aligned}&H \left( \sum _{j=1}^{N} f_{j}dd^{c}f_{j} \right) = -\omega , \\&\lambda _{1}(g)^{2}\left( \sum _{j=1}^{N} f_{j}^{2} \right) -2\lambda _{1}(g) \left( \sum _{j=1}^{N} |\nabla f_{j}|^{2} \right) + \sum _{j=1}^{N} |dd^{c}f_{j}|^{2} = 0 \\ \end{aligned} \right. \end{aligned}$$

(1.2)

hold (Theorem 2.9). Here H is the harmonic projector, which is defined due to the Hodge decomposition on a compact Kähler manifold. It is obvious that the equation (1.2) implies (1.1). In addition, we can obtain a result that is similar to the aforementioned results due to El Soufi-Ilias [11] and Apostolov et al [1]. That is, the metric on a compact isotropy irreducible homogeneous Kähler manifold is $\lambda _{1}$-extremal in our sense (Proposition 2.12). We also give an example of a Kähler metric that is $\lambda _{1}$-extremal within its Kähler class, but not so for all volume-preserving deformations of the Kähler metric (Example 2.17).

In the final section of this paper, we consider flat complex tori. We prove that the metric on a flat complex torus is $\lambda _{1}$-extremal within its Kähler class. Montiel and Ros [21] showed that among all the real 2-dimensional tori, the only square torus $\textbf{R}^{2}/\textbf{Z}^{2}$ admits an isometric minimal immersion into a 3-dimensional Euclidean sphere by the first eigenfunctions. Later, using Theorem 1.1, El Soufi and Ilias [11] improved the result. That is, they proved that a real 2-dimensional torus admits isometric minimal immersion into a Euclidean sphere of some dimension by the first eigenfunctions if and only if the torus is the square torus or the equilateral torus. Very recently, Lü et al. [20] classified all the 3-dimensional flat tori and 4-dimensional flat tori that admit an isometric minimal immersion into a Euclidean sphere of some dimension by the first eigenfunctions. Furthermore, they constructed new examples of flat tori that admit an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions. In Theorem 3.2, we give a necessary and sufficient condition for the metric on a flat complex torus to be $\lambda _{1}$-extremal in our sense. The notion of $\lambda _{1}$-extremality introduced in this paper is stronger than that in Theorem 1.1. Hence Theorem 3.2 gives a necessary condition for a flat complex torus to admit an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions. As far as the author knows, this is the only currently known necessary condition for a flat complex torus to admit an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions. We also show that if the multiplicity of the first eigenvalue of a flat complex torus is 2, then the flat metric on the torus is not $\lambda _{1}$-extremal in our sense (Corollary 3.3). In addition, we show that an m-dimensional flat torus $\textbf{R}^{m}/D_{m}$, where $D_{m}$ is an m-dimensional checkerboard lattice, admits an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions (Proposition 3.6). This fact has been already proved by Lü et al. [20] for $m=3,4$.

2 A $\lambda _{k}$-extremal Kähler metric

Let (M, J, g) be a compact Kähler manifold (without boundary) of complex dimension n. By scaling the metric, we may assume that the volume Vol(M, g) is 1. Let $\omega$ be the Kähler form and $d\mu$ the volume form. It is well known that $d\mu$ is given by $d\mu = \omega ^{n}/n!$. Below we follow the conventions in [1]. We define the exterior differential d by $d = \partial + \overline{\partial }$ and twisted exterior differential $d^{c}$ by $d^{c} = i(\overline{\partial }-\partial )$. Both d and $d^{c}$ are real operators. We clearly have $dd^{c} = 2i\partial \overline{\partial }$. The Kähler metric g induces a pointwise hermitian inner product on (1, 1)-forms, which we also denote by g. Note that g is symmetric for a pair of real (1, 1)-forms. We have

$$\begin{aligned} |\omega |^{2} = g(\omega , \omega ) = n. \end{aligned}$$

(2.1)

Let $\delta$ and $\delta ^{c}$ be the $L^{2}(g)$-adjoints of d and $d^{c}$ respectively. We define the Laplacian $\Delta _{g}$ by $\Delta _{g} =d\delta + \delta d$. Eigenvalues of the Laplacian $\Delta _{g}$ acting on functions are nonnegative and we denote them by $0 < \lambda _{1}(g) \le \lambda _{2}(g) \le \cdots \lambda _{k}(g) \le \cdots$. For any $k \in \textbf{N}$, let $E_{k}(g)$ be the vector space of real-valued eigenfunctions of $\Delta _{g}$ corresponding to $\lambda _{k}(g)$. That is, $E_{k}(g)$ is given by $E_{k}(g) = \text{ Ker }(\Delta _{g}-\lambda _{k}(g)I)$, where I is the identity map acting on functions. We have

$$\begin{aligned} \Delta _{g} \phi = -2 g^{j\overline{k}} \frac{\partial ^{2}\phi }{\partial z^{j} \partial {\overline{z}^{k}}} = -2g(\omega , i\partial \overline{\partial }\phi ) = -g(\omega , dd^{c}\phi ) \end{aligned}$$

(2.2)

for a function $\phi$. We also have

$$\begin{aligned} *\omega = \frac{1}{(n-1)!}\omega ^{n-1}, \end{aligned}$$

where $*$ is the Hodge star operator. This implies that the equation

$$\begin{aligned} \alpha \wedge \omega ^{n-1} = (n-1)!\alpha \wedge * \omega = (n-1)! g(\alpha , \omega )\frac{\omega ^{n}}{n!} = \frac{1}{n} g(\alpha , \omega ) \omega ^{n} \end{aligned}$$

(2.3)

holds for a (1, 1)-form $\alpha$. In particular, the equations (2.2) and (2.3) imply

$$\begin{aligned} ndd^{c}\phi \wedge \omega ^{n-1} = g(dd^{c}\phi , \omega )\omega ^{n} = -(\Delta _{g} \phi ) \omega ^{n}. \end{aligned}$$

(2.4)

Let $Z^{1,1}(M;\textbf{R})$ be the real vector space of d-closed real (1, 1)-forms on M. Let $Z^{1,1}_{0}(M;\textbf{R})$ be the subspace defined by

$$\begin{aligned} Z^{1,1}_{0}(M;\textbf{R}) := \left\{ \alpha \in Z^{1,1}(M;\textbf{R}) \mid \int _{M} g(\alpha , \omega ) d\mu = 0 \right\} . \end{aligned}$$

Fix an arbitrary element $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$. The (1, 1)-form

$$\begin{aligned} \widetilde{\omega }_{t} := \omega + t\alpha \end{aligned}$$

(2.5)

is a Kähler form for a sufficiently small t. In particular, if we consider $\alpha = dd^{c}\psi$ for a real-valued function $\psi$, then $\alpha$ satisfies

$$\begin{aligned} \int _{M} g(\alpha , \omega ) d\mu = -\int _{M} \Delta _{g} \psi d\mu = 0, \end{aligned}$$

and so we have $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$. The 1-parameter family $\widetilde{\omega }_{t} = \omega + tdd^{c}\psi$ is a deformation of $\omega$ in its Kähler class $[\omega ]$, which was studied by Apostolov et al. [1]. Let $\widetilde{g}_{t}$ be the Kähler metric corresponding to the Kähler form $\widetilde{\omega }_{t}$ in (2.5). Set

$$\begin{aligned} g_{t} := \text{ Vol }(M, \tilde{g}_{t})^{-1/n}\widetilde{g_{t}}, \quad \omega _{t} := \text{ Vol }(M, \tilde{g}_{t})^{-1/n}\widetilde{\omega _{t}}. \end{aligned}$$

(2.6)

Then we have $g_{0} = g$ and $\omega _{0} = \omega$. We also see that $(g_{t})_{t}$ is a volume-preserving 1-parameter family of Kähler metrics that depends analytically on t, and $\omega _{t}$ is the Kähler form associated with $g_{t}$. Moreover, we can verify that $\left. \frac{d}{dt} \right| _{t=0} \omega _{t} = \alpha$. (See (2.9) below.)

Definition 2.1

The Kähler metric g on a compact Kähler manifold $(M, J, g, \omega )$ is called $\lambda _{k}$-extremal (for all the volume-preserving deformations of the Kähler metric) if the inequality

$$\begin{aligned} \left( \left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t}) \right) \cdot \left( \left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t}) \right) \le 0 \end{aligned}$$

holds for any 1-parameter family of volume-preserving Kähler metrics $(g_{t})_{t}$ that depends real analytically on t.

Remark 2.2

When we consider whether a Kähler metric g on (M, J) is $\lambda _{k}$-extremal, we may rescale the metric so that $\text{ Vol }(M,g) =1$. Let $(g_{t})_{t}$ be a 1-parameter family of volume-preserving Kähler metrics that depends real analytically on t. Let $\omega _{t}$ be the Kähler form associated with $g_{t}$. El Soufi-Ilias [13] showed that $\left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t})$ and $\left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t})$ depend on only $\omega$ and $\left. \frac{d}{dt}\right| _{t=0}\omega _{t}$. Since $(\omega _{t})_{t}$ is volume-preserving, we have $\left. \frac{d}{dt}\right| _{t=0}\omega _{t} \in Z^{1,1}_{0}(M; \textbf{R})$. Hence it suffices to consider $(\omega _{t})_{t}$ given by (2.6). Thus a Kähler metric g on (M, J) with $\text{ Vol }(M,g) =1$ is $\lambda _{k}$-extremal if and only if for any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$, the associated volume-preserving 1-parameter family of Kähler metrics $(g_{t})_{t}$ defined by (2.6) satisfies

$$\begin{aligned} \left( \left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t}) \right) \cdot \left( \left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t}) \right) \le 0. \end{aligned}$$

We quote the following theorem due to El Soufi-Ilias [13]:

Theorem 2.3

([13]) Let (M, g) be a compact Riemannian manifold and $(g_{t})_{t}$ be a 1-parameter family of Riemannian metrics that depends real-analytically on t with $g_{0}=g$. Let $\Pi _{k}: L^{2}(M,g) \rightarrow E_{k}(g)$ be the orthogonal projection onto $E_{k}(g)$. Define the operator $P_{k}: E_{k}(g) \rightarrow E_{k}(g)$ by

$$\begin{aligned} P_{k}(f) := \Pi _{k} \left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) . \end{aligned}$$

(2.7)

Then the following hold:

• (1) $\left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t})$ and $\left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t})$ are eigenvalues of $P_{k}$.

• (2) If $\lambda _{k}(g) > \lambda _{k-1}(g)$, then $\left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t})$ and $\left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t})$ are the greatest and the least eigenvalues of $P_{k}$.

• (3) If $\lambda _{k}(g) < \lambda _{k+1}(g)$, then $\left. \frac{d}{dt}\right| _{t=0^{-}} \lambda _{k}(g_{t})$ and $\left. \frac{d}{dt}\right| _{t=0^{+}} \lambda _{k}(g_{t})$ are the least and the greatest eigenvalues of $P_{k}$.

For any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$, the associated volume-preserving 1-parameter family of Kähler metrics $(g_{t})_{t}$ given by (2.6) defines the associated operator $P_{k, \alpha }: E_{k}(g) \rightarrow E_{k}(g)$ by (2.7). Since $P_{k, \alpha }$ is symmetric with respect to the $L^{2}(g)$-inner product induced by g, one can consider the corresponding quadratic form on $E_{k}(g)$, given by

$$\begin{aligned} Q_{\alpha }(f) := \int _{M} f P_{k, \alpha }(f) d\mu = \int _{M} f \left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) d\mu . \end{aligned}$$

(2.8)

The following proposition is an immediate consequence of Theorem 2.3 (1):

Proposition 2.4

Let $(M, J, g, \omega )$ be a compact Kähler manifold. If the metric g of $(M, J, g, \omega )$ is $\lambda _{k}$-extremal, then the quadratic form $Q_{\alpha }$, defined in (2.8), is indefinite on $E_{k}(g)$ for any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$.

We also have the following proposition, which follows form Theorem 2.3 (2) and (3):

Proposition 2.5

Let $(M, J, g, \omega )$ be a compact Kähler manifold. Suppose that $\lambda _{k}(g) > \lambda _{k-1}(g)$ or $\lambda _{k}(g) < \lambda _{k+1}(g)$ holds. Then the metric g of (M, J, g) is $\lambda _{k}$-extremal if and only if the quadratic form $Q_{\alpha }$ is indefinite on $E_{k}(g)$ for any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$.

The next corollary immediately follows.

Corollary 2.6

Let $(M, J, g, \omega )$ be a compact Kähler manifold. Then the metric g is $\lambda _{1}$-extremal if and only if the quadratic form $Q_{\alpha }$ is indefinite on $E_{1}(g)$ for any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$.

Theorem 2.7

Let $(M, J, g, \omega )$ be a compact Kähler manifold of complex dimension n with $\text{ Vol }(M,g) =1$. For any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$, the quadratic form $Q_{\alpha }$, given by (2.8), can be expressed as

$$\begin{aligned} Q_{\alpha }(f) = \int _{M}g(fdd^{c}f, \alpha ) d\mu . \end{aligned}$$

Proof

First we calculate $\left. \frac{d}{dt}\right| _{t=0}\text{ Vol }(M, \widetilde{g_{t}})$ and $\left. \frac{d}{dt}\right| _{t=0} \omega _{t}$. The volume form $\widetilde{d\mu }_{t}$, determined by $\widetilde{\omega }_{t}$, can be written as

$$\begin{aligned} \widetilde{d\mu }_{t} = \frac{1}{n!} \widetilde{\omega }_{t}^{n} = \frac{1}{n!}\left( \omega ^{n} + tn\alpha \wedge \omega ^{n-1} \right) + O(t^{2}) = [1+tg(\alpha , \omega )] d\mu + O(t^{2}). \end{aligned}$$

Hence one obtains

$$\begin{aligned} \text{ Vol }(M, \widetilde{g}_{t}) = \int _{M} \widetilde{d\mu }_{t} = 1 + t\int _{M}g(\alpha , \omega ) d\mu +O(t^{2}) = 1+O(t^{2}), \end{aligned}$$

where in the last equality we have used the the assumptions that $\text{ Vol }(M, \widetilde{g}) = 1$ and $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$. Hence this implies $\left. \frac{d}{dt}\right| _{t=0} \text{ Vol }(M, \tilde{g_{t}}) = 0$. Thus one obtains

$$\begin{aligned} \left. \frac{d}{dt} \right| _{t=0} \omega _{t} = \frac{d}{dt} \left( \text{ Vol }(M, \widetilde{g}_{t})^{-1/n}) \tilde{\omega }_{t} \right) = \text{ Vol }(M, \tilde{g}_{0})^{-1/n} \left. \frac{d}{dt} \right| _{t=0} \widetilde{\omega }_{t} = \alpha .\qquad \end{aligned}$$

(2.9)

Next we differentiate

$$\begin{aligned} ndd^{c}f \wedge \omega _{t}^{n-1} = -(\Delta _{g_{t}} f) \omega _{t}^{n}, \end{aligned}$$

which comes from (2.4). Differentiating the left hand side at $t=0$, one obtains

$$\begin{aligned} n(n-1)dd^{c}f \wedge \left( \left. \frac{d}{dt}\right| _{t=0} \omega _{t}\right) \wedge \omega ^{n-2} = n (n-1) dd^{c}f \wedge \alpha \wedge \omega ^{n-2}, \end{aligned}$$

where (2.9) is used. On the other hand, differentiating the right hand side at $t=0$, one obtains

$$\begin{aligned}&-\left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) \omega ^{n} -n(\Delta _{g}f)\left( \left. \frac{d}{dt}\right| _{t=0} \omega _{t} \right) \wedge \omega ^{n-1} \\&\quad = -\left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) \omega ^{n} - n(\Delta _{g}f)\alpha \wedge \omega ^{n-1} \\&\quad = -\left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) \omega ^{n} - (\Delta _{g}f) g(\alpha , \omega ) \omega ^{n}, \end{aligned}$$

where (2.3) is used for the last equality. Thus one obtains

$$\begin{aligned} \left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) \omega ^{n} = -n(n-1)dd^{c}f \wedge \alpha \wedge \omega ^{n-2} - g(\alpha , \omega )\omega ^{n}. \end{aligned}$$

(2.10)

It is known [1] that the equation

$$\begin{aligned} g(\phi , \psi ) \omega ^{n} = g(\phi , \omega ) g(\psi , \omega )\omega ^{n} - n(n-1)\phi \wedge \psi \wedge \omega ^{n-2} \end{aligned}$$

holds for a pair of real (1, 1)-forms $\phi$ and $\psi$. Substituting $\phi = dd^{c}f$ and $\psi = \alpha$ into this equation and combining with (2.10) and (2.2), one obtains

$$\begin{aligned} \left( \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f \right) \omega ^{n} = g(dd^{c}f, \alpha )\omega ^{n}. \end{aligned}$$

Hence one concludes

$$\begin{aligned} \left. \frac{d}{dt}\right| _{t=0} \Delta _{g_{t}}f = g(dd^{c}f, \alpha ) \end{aligned}$$

and thus the assertion follows. $\square$

For a compact Kähler manifold $(M, J, g, \omega )$, $\mathcal {H}^{1,1}(M)$, the vector space of $\textbf{C}$-valued harmonic (1, 1)-forms, is known to be finite dimensional. In particular, $\mathcal {H}^{1,1}(M)$ is a closed subspace of $\Omega ^{1,1}(M)$ and hence we can define the $\textbf{C}$-linear $L^{2}(g)$-orthogonal projection $H: \Omega ^{1,1}(M) \rightarrow \mathcal {H}^{1,1}(M)$. For a real (1, 1)-form $\eta$, $H(\eta )$ is a real harmonic (1, 1)-form. Let $\mathcal {H}^{1,1}(M; \textbf{R})$ be the vector space of real harmonic (1, 1)-forms. Set

$$\begin{aligned} \mathcal {H}^{1,1}_{0}(M; \textbf{R}) := \left\{ \alpha \in \mathcal {H}^{1,1}(M;\textbf{R}) \mid \int _{M} g(\alpha , \omega ) d\mu = 0 \right\} \end{aligned}$$

and

$$\begin{aligned} C^{\infty }_{0}(M; \textbf{R}) := \left\{ \varphi \in C^{\infty }(M;{\textbf {R}}) \mid \int _{M} \varphi d\mu = 0 \right\} . \end{aligned}$$

By the $dd^{c}$-lemma, $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$ can be decomposed as

$$\begin{aligned} \alpha = H(\alpha ) + dd^{c} \varphi . \end{aligned}$$

This decomposition gives an $\textbf{R}$-linear bijection between $Z^{1,1}_{0}(M;\textbf{R})$ and $\mathcal {H}^{1,1}_{0}(M;\textbf{R}) \times C^{\infty }_{0}(M; \textbf{R})$.

Apostolov et al. [1] introduced the fourth order differential operator $L(f):= \delta ^{c}\delta (fdd^{c}f)$, where $\delta$ and $\delta ^{c}$ are the $L^{2}$-adjoints of d and $d^{c}$ respectively. They proved that the equation

$$\begin{aligned} L(f) = \lambda _{k}(g) f^{2} -2\lambda _{k}(g) |\nabla f|^{2} + |dd^{c} f|^{2} \end{aligned}$$

holds for an eigenfunction $f \in E_{k}(g)$.

We have the following theorem:

Theorem 2.8

Let $(M, J, g, \omega )$ be a compact Kähler manifold of complex dimension n. The following are equivalent:

• (1) For any $\alpha \in Z^{1,1}_{0}(M;\textbf{R})$, the quadratic form $Q_{\alpha }$ given by (2.8) is indefinite on the eigenspace $E_{k}(g)$.

• (2) There exists a finite subset $\{ f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ such that the following equations hold:

  $$\begin{aligned} \left\{ \, \begin{aligned}&H \left( \sum _{j=1}^{N} f_{j}dd^{c}f_{j} \right) = -\omega \\&\sum _{j=1}^{N}L(f_{j} ) = \lambda _{k}(g)^{2}\left( \sum _{j=1}^{N} f_{j}^{2} \right) -2\lambda _{k}(g) \left( \sum _{j=1}^{N} |\nabla f_{j}|^{2} \right) + \sum _{j=1}^{N} |dd^{c}f_{j}|^{2} = 0 \\ \end{aligned} \right. \end{aligned}$$

  (2.11)

The proof of this theorem is inspired by that of Lemma 3.1 in [13].

Proof

We may assume that $\text{ Vol }(M,g) =1$. We assume the condition (1). Let K be the convex hull of $\{ \left( H(fdd^{c}f), L(f) \right) \mid f \in E_{k}(g) \}$ in $\mathcal {H}^{1,1}(M; \textbf{R}) \times C^{\infty }(M; \textbf{R})$. Since $E_{k}(g)$ is finite dimensional, K is contained in a finite dimensional subspace of $\mathcal {H}^{1,1}(M; \textbf{R}) \times C^{\infty }(M; \textbf{R})$. We consider the product $L^{2}$-inner metric on the subspace. We assume that $(-\omega , 0) \notin K$. Then the hyperplane separation theorem implies that there exists $\left( \widetilde{\alpha }_{H}, \widetilde{\varphi } \right) \in \mathcal {H}^{1,1}(M; \textbf{R}) \times C^{\infty }(M; \textbf{R})$ such that the inequalities

$$\begin{aligned} \int _{M} g(-\omega , \widetilde{\alpha }_{H}) d\mu < 0 \quad \text{ and } \quad \int _{M} g(\eta , \widetilde{\alpha }_{H}) d\mu + \int _{M}s\varphi d\mu \ge 0 \end{aligned}$$

(2.12)

hold for all $(\eta , s) \in K\setminus {\{0\}}$. Consider $\alpha _{H} \in \mathcal {H}^{1,1}_{0}(M;\textbf{R})$ and $\varphi \in C^{\infty }_{0}(M; \textbf{R})$ respectively defined by

$$\begin{aligned} & \alpha _{H}:= \widetilde{\alpha }_{H}- \frac{1}{n} \left( \int _{M} g(\omega , \widetilde{\alpha }_{H}) d\mu \right) \omega . \end{aligned}$$

(2.13)

$$\begin{aligned} & \varphi := \widetilde{\varphi }- \int _{M} \widetilde{\varphi } d\mu . \end{aligned}$$

(2.14)

Set

$$\begin{aligned} \alpha := \alpha _{H} + dd^{c}\varphi = \alpha _{H} + dd^{c}\widetilde{\varphi }. \end{aligned}$$

Then for any $f \in E_{k}(g)\setminus {\{0\}}$, one has

$$\begin{aligned} Q_{\alpha }(f)&= \int _{M} g(fdd^{c}f, \alpha ) d\mu \\&= \int _{M} g(fdd^{c}f, \alpha _{H}) d\mu + \int _{M}g(fdd^{c}f, dd^{c}\widetilde{\varphi }) d\mu \\&= \int _{M} g(H(fdd^{c}f), \widetilde{\alpha }_{H}) d\mu - \frac{1}{n} \left[ \int _{M}g(fdd^{c}f, \omega ) d\mu \right] \left[ \int _{M} g(\omega , \widetilde{\alpha }_{H} ) d\mu \right] \\&\quad + \int _{M}L(f)\widetilde{\varphi } d\mu \\&= \int _{M} g(H(fdd^{c}f), \widetilde{\alpha }_{H}) d\mu + \int _{M}L(f)\widetilde{\varphi } d\mu + \frac{\lambda _{k}(g) }{n} \left[ \int _{M} f^{2} d\mu \right] \\&\quad \left[ \int _{M}g(\omega , \widetilde{\alpha }_{H}) d\mu \right] \\&>0. \end{aligned}$$

This contradicts the condition (1) and hence one concludes that $(-\omega , 0) \in K$. This implies that $(1) \Rightarrow (2)$.

Conversely, we assume that there exists a finite subset $\{ f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ satisfying (2.11). Take any $\alpha _{H} \in \mathcal {H}^{1,1}_{0}(M; \textbf{R})$ and $\varphi \in C^{\infty }_{0}(M; \textbf{R})$ and consider $\alpha \in Z^{1,1}_{0}(M; \textbf{R})$ defined by $\alpha := \alpha _{H} + dd^{c} \varphi$. Then one has

$$\begin{aligned} \sum _{j=1}^{N}Q_{\alpha }(f_{j})&= \sum _{j=1}^{N} \int _{M} g (f_{j}dd^{c}f_{j}, \alpha ) d\mu \\&= - \int _{M} g\left( \sum _{j=1}^{N}f_{j}dd^{c}f_{j}, \alpha _{H} \right) d\mu + \sum _{j=1}^{N} \int _{M}g(f_{j}dd^{c}f_{j}, dd^{c}\varphi ) d\mu \\&= \int _{M}g(-\omega , \alpha _{H}) d\mu + \sum _{j=1}^{N}\int _{M} L(f_{j}) \varphi d\mu \\&=0. \end{aligned}$$

This implies that $Q_{\alpha }$ is indefinite on $E_{k}(g)$. This completes the proof. $\square$

Combining Corollary 2.6 and Theorem 2.8, we conclude the following:

Theorem 2.9

Let $(M, J, g, \omega )$ be a compact Kähler manifold. Suppose that the Kähler metric g is $\lambda _{k}$-extremal. Then there exists a finite subset $\{ f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ satisfying (2.11). For $k=1$, the existence of such a finite collection of eigenfunctions is a necessary and sufficient condition for the Kähler metric g to be $\lambda _{1}$-extremal.

We remark the relationship between Theorem 2.9 and Theorem 2.1 in [1], which we quote as follows:

Theorem 2.10

([1]) Let (M, J, g) be a compact Kähler manifold. The Kähler metric g is $\lambda _{k}$-extremal for all the deformations of the Kähler metric within its Kähler class. Then there exists a finite subset $\{ f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ such that the equation

$$\begin{aligned} \sum _{j=1}^{N}L(f_{j}) = \lambda _{k}(g)^{2}\left( \sum _{j=1}^{N} f_{j}^{2} \right) -2\lambda _{k}(g) \left( \sum _{j=1}^{N} |\nabla f_{j}|^{2} \right) + \sum _{j=1}^{N} |dd^{c}f_{j}|^{2} = 0 \end{aligned}$$

(2.15)

holds. For $k=1$, the existence of such a finite collection of eigenfunctions is a necessary and sufficient condition for the Kähler metric g to be $\lambda _{1}$-extremal for all the deformations of the Kähler metric within its Kähler class.

Obviously, the condition (2.11) is stronger than the condition (2.15). This fact is natural since our $\lambda _{k}$-extremality is stronger than that in [1]. Using Theorem 2.10 above, one can show that if the Kähler metric g is extremal for the functional $\lambda _{k}$ within its Kähler class, then the eigenvalue $\lambda _{k}(g)$ is multiple [1]. Hence we obtain the same conclusion for our $\lambda _{k}$-extremality:

Corollary 2.11

Let $(M, J, g, \omega )$ be a compact Kähler manifold. Suppose that g is $\lambda _{k}$-extremal for all the volume-preserving deformations of the Kähler metric. Then the eigenvalue $\lambda _{k}(g)$ is multiple.

El Soufi-Ilias[11] showed that the metric on a compact isotropy irreducible homogeneous space is $\lambda _{1}$-extremal with respect to all the volume-preserving deformations of the Riemannian metric. Since their $\lambda _{1}$-exremality is stronger than ours, we have the following proposition:

Proposition 2.12

Let G/K be a compact isotropy irreducible homogeneous Kähler manifold. Then the metric is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

We remark that one can also prove this proposition directly from Theorem 2.9, using a similar discussion to that in [30, Section 3]. It is known that the metric of a compact isotropy irreducible homogeneous Kähler manifold is Einstein[35]. An irreducible Hermitian symmetric space of compact type is a compact isotropy irreducible homogeneous Kähler manifold. In fact, the converse also holds. That is, a compact isotropy irreducible homogeneous Kähler manifold is an irreducible Hermitian symmetric space of compact type [18, 34]. Apotolov et al. [1] proved that the metric on a compact homogeneous Käher-Einstein manifold of positive scalar curvature is $\lambda _{1}$-extremal within its Kähler class.

Before stating the corollaries of Theorem 2.9, we recall basic facts about the first eigenvalue of the Laplacian on a product of Riemannian manifolds. Let (M, g) and $(M', g')$ be Riemannian manifolds. For a function $f \in C^{\infty }(M; \textbf{R})$, we define the function $f \times 1$ on $M \times M'$ by

$$\begin{aligned} (f \times 1)(x, y) := f(x), \quad (x,y) \in M \times M'. \end{aligned}$$

For a function $h \in C^{\infty }(M'; \textbf{R})$, we define the function $1 \times h$ on $M \times M'$ in a similar manner. Suppose that $\lambda _{1}(g) \le \lambda _{1}(g')$. Then the first eigenvalue $\lambda _{1}(g\times g')$ of the product Riemannian manifold $(M \times M', g\times g')$ is equal to $\lambda _{1}(g)$. $E_{1}(g \times g')$, the space of $\lambda _{1}(g \times g')$-eigenfunctions on $M \times M'$ is given by

$$\begin{aligned} E_{1}(g \times g') = \left\{ \begin{array}{ll} \text{ span }\{f \times 1, 1 \times h \mid f \in E_{1}(g), h \in E_{1}(g')\} & (\text{ if } \lambda _{1}(g) = \lambda _{1}(g') )\\ \text{ span }\{f \times 1 \mid f \in E_{1}(g)\} & (\text{ if } \lambda _{1}(g) < \lambda _{1}(g') ).\\ \end{array}\right. \end{aligned}$$

The following corollary follows from Theorem 2.9:

Corollary 2.13

Let $(M, J, g, \omega )$ and $(M', J', g', \omega ')$ be compact Kähler manifolds. Assume that $\lambda _{1}(g) = \lambda _{1}(g')$ and that g and $g'$ are both $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metrics. Then the product Kähler metric $g \times g'$ on $(M, J) \times (M, J)$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

Proof

By hypothesis, there exist finite subsets $\{f_{j}\} \subset E_{1}(g)$ and $\{h_{k} \} \subset E_{1}(g')$ such that

$$\begin{aligned} \left\{ \, \begin{aligned}&\sum _{j}H_{M} \left( f_{j}dd^{c}f_{j} \right) = -\omega , \quad \sum _{j}L_{M}(f_{j}) = 0 \\&\sum _{k}H_{M'} \left( h_{k}dd^{c}h_{k} \right) = -\omega ', \quad \sum _{k}L_{M'}(h_{k}) = 0. \\ \end{aligned} \right. \end{aligned}$$

Then one has

$$\begin{aligned}&\sum _{j}L_{M\times M'} (f_{j} \times 1) + \sum _{k} L_{M\times M'} (1 \times h_{k}) \\&\quad = \sum _{j} L_{M}(f_{j}dd^{c}f_{j}) + \sum _{k} L_{M'}(h_{k}) \\&\quad =0. \end{aligned}$$

We also have

$$\begin{aligned}&\sum _{j}H_{M\times M'} \left( (f_{j}\times 1)d_{M\times M'}d^{c}_{M\times M'}(f_{j}\times 1) \right) \\&\quad = \sum _{j} H_{M\times M'} (f_{j}d_{M}d^{c}_{M}f_{j} \oplus 0) \\&\quad = \sum _{j} H_{M}(f_{j}d_{M}d^{c}_{M}f_{j}) \oplus 0 \\&\quad = -\omega \oplus 0. \end{aligned}$$

Similarly, we obtain

$$\begin{aligned} \sum _{k} H_{M\times M'} \left( (1\times h_{k} )d_{M\times M'}d^{c}_{M\times M'}(1 \times h_{k} ) \right) = 0 \oplus -\omega '. \end{aligned}$$

Hence we obtain

$$\begin{aligned} \sum _{j} H\left( (f_{j}\times 1)dd^{c}(f_{j}\times 1) \right) + \sum _{k} H \left( (1\times h_{k} )dd^{c}(1 \times h_{k} ) \right) = -\omega \oplus -\omega ', \end{aligned}$$

where we omit the subscript $M \times M'$. Thus one concludes that $\{ f_{j} \times 1 \}_{j} \cup \{1 \times h_{k} \}_{k}$ satisfy the equation (2.11). The proof is completed. $\square$

From the above proof, one can immediately obtain the following corollary:

Corollary 2.14

Let $(M, J, g, \omega )$ and $(M', J', g', \omega ')$ be compact Kähler manifolds. Assume that $\lambda _{1}(g) \ne \lambda _{1}(g')$. Then the product Kähler metric $g \times g'$ on $(M, J) \times (M', J')$ is not $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

The above corollaries also correspond to the Corollary 2.3 in [1], which we quote in the following:

Proposition 2.15

([1]) Let $(M, J, g, \omega )$ and $(M', J', g', \omega ')$ be compact Kähler manifolds. Suppose that g is $\lambda _{1}$-extremal for all the deformations of the Kähler metric in its Kähler class and that the inequality $\lambda _{1}(g) \le \lambda _{1}(g')$ holds. Then the product metric $g \times g'$ is $\lambda _{1}$-extremal for all the deformations of the Kähler metric in its Kähler class on $(M, J) \times (M', J')$.

We can see that $\lambda _{1}$-extremality in Corollary 2.13 is stronger than that in Proposition 2.15. To clarify the difference, let us look at the following example:

Example 2.16

Let $(M, J, g, \omega )$, $(M', J', g', \omega ')$ be irreducible Hermitian symmetric spaces of compact type with $\rho = c \, \omega$ and $\rho ' = c'\omega '$ for some $c, c'>0$, where $\rho$ and $\rho '$ are the Ricci forms on M and $M'$ respectively. By Example 2.12, g is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric and so is $g'$. The result due to Nagano [25] shows that $\lambda _{1}(g) = 2c$ and $\lambda _{1}(g') =2c'$ (see also [31]). By Proposition 2.15, the product Kähler metric $g\times g'$ on $(M, J) \times (M', J')$ is $\lambda _{1}$-extremal within its Kähler class. However, Corollary 2.13 and 2.14 imply that the metric $g\times g'$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric if and only if $c=c'$.

The simplest case of this example is the following:

Example 2.17

Let $g_{FS}$ be the Fubini-Study metric on the complex projective space $\textbf{C}P^{n}$. Take any $c>0$. Then the product Kähler metric $g_{FS} \times c g_{FS}$ on $\textbf{C}P^{n} \times \textbf{C}P^{n}$ is $\lambda _{1}$-extremal in its Kähler class. However, $g_{FS} \times c g_{FS}$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric if and only if $c=1$.

El Soufi-Ilias [12] showed that if a metric g of a Riemannian manifold (M, g) is $\lambda _{k}$-extremal for volume-preserving conformal deformations, then there exists a finite subset $\{f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ such that $F:=(f_{1}, \cdots f_{N}): M \rightarrow \textbf{R}^{N}$ is a harmonic map to a sphere with constant energy density. For a Riemann surface $(M, J, g, \omega )$, conformal deformations and deformations within the Kähler class are equivalent, so it is natural to expect the following result due to Apostolov-Jakobson-Kokarev [1]:

Proposition 2.18

([1]) Let $(M, J, g, \omega )$ be a compact Riemann surface without boundary. Let $\{f_{1}, \cdots , f_{N} \} \subset E_{k}(g)$ be eigenfunctions associated with $\lambda _{k}(g)$. The following two conditions are equivalent:

• (1) $F:=(f_{1}, \cdots f_{N}): M \rightarrow \textbf{R}^{N}$ gives a harmonic map to $S^{N-1}(c)$ with $|dF|^{2} = c\lambda _{k}(g)$ for some $c>0$, where $S^{N-1}(c)$ is the unit sphere of radius c.

• (2) $\sum _{j=1}^{N} L(f_{j}) = 0.$

To prove the result, they used the identity $dd^{c}\psi = - (\Delta _{g} \psi ) \omega$ and showed that the equation

$$\begin{aligned} \sum _{j} L(f_{j}) =2\lambda _{k}(g) \left( \lambda _{k}(g)\sum _{j}f_{j}^{2}-\sum _{j}|\nabla f_{j}|^{2}\right) = \lambda _{k}(g) \Delta _{g}\left( \sum _{j}f_{j}^{2}\right) \end{aligned}$$

holds for any finite subset $\{f_{j} \} \subset E_{k}(g)$. Since we have

$$\begin{aligned} H\left( \sum _{j}f_{j}dd^{c}f_{j} \right) = \lambda _{k}(g)H\left( (\sum _{j} f_{j}^{2}) \omega \right) , \end{aligned}$$

one can see that if $\{f_{j}\} \subset E_{k}(g)$ satisfies $\sum _{j}L(f_{j})=0$, then the equation $H(\sum _{j}f_{j}dd^{c}f_{j}) = -\omega$ holds after an appropriate rescaling of $\{f_{j}\}$. Hence Theorem 2.9 and Theorem 2.10 are equivalent for a Riemann surface. This is also a natural result since deformations within the Kähler class and general Kähler deformations are equivalent for a compact Riemann surface.

3 Complex tori

In view of Theorem 2.9, the harmonic projector H and information of the space of eigenfunctions are important. However, it is hard to find an explicit formula for the harmonic projector H on a general Kähler manifold. It is also hard to determine the space of eigenfunctions explicitly on a general Riemannian manifold. However, H and the eigenfunctions can be written explicitly for a complex torus. Since it is hard to compute (2.11) for arbitrary collection of eigenfunctions, we compute it for the $L^{2}$-orthonormal basis of the space of eigenfunctions. From the calculations, we see that the metric on any flat complex torus is $\lambda _{1}$-extremal within its Kähler class. In addition, we deduce a condition for the flat metric to be $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

Let $\gamma _{1}, \cdots , \gamma _{2n}$ be vectors in $\textbf{C}^{n}$ that are linearly independent over $\textbf{R}$. We denote by $\Gamma$ the lattice in $\textbf{C}^{n}$ with basis $\{ \gamma _{1}, \cdots , \gamma _{2n} \}$. Let $z_{1}, \cdots , z_{n}$ be the standard complex coordinates of $\textbf{C}^{n}$ and $x^{1}, \cdots , x^{2n}$ the real coordinates defined by

$$\begin{aligned} z^{k} = x^{2k-1} + ix^{2k} \quad (k=1, \cdots , n). \end{aligned}$$

The lattice $\Gamma$ naturally acts on $\textbf{C}^{n}$ by translation. Then the quotient space $T_{\Gamma }^{n}:= \textbf{C}^{n}/\Gamma$ becomes a complex manifold in a natural way. $T_{\Gamma }^{n}$ is called a complex torus. The standard metric on $\textbf{C}^{n}$ is given by $\sum _{j=1}^{n} dz^{j} \otimes d\overline{z}^{j}$. This defines a Kähler form

$$\begin{aligned} \widetilde{\omega } := \frac{i}{2} \sum _{j=1}^{n} dz^{j} \wedge d\overline{z}^{j} \end{aligned}$$

on $\textbf{C}^{n}$. The canonical holomorphic projection $\textbf{C}^{n} \rightarrow \textbf{C}^{n}/\Gamma = T_{\Gamma }^{n}$ induces a Kähler metric g and a Kähler form $\omega$ on $T_{\Gamma }^{n}$. If we express $w_{1}$, $w_{2} \in \textbf{C}^{n}$ as

$$\begin{aligned} w_{k} = (w_{k}^{1}, \cdots ,w_{k}^{n}), \quad w_{k} = (u_{k}^{1}, \cdots , u_{k}^{2n}), \quad w_{k}^{j} = u_{k}^{2j-1}+iu_{k}^{2j} \quad (k=1, 2) \end{aligned}$$

in the complex coordinates $(z^{1}, \cdots z^{n})$ and the real coordinates $(x^{1}, \cdots , x^{2n})$ respectively, then the standard inner product of $w_{1}$ and $w_{2}$ is given by

$$\begin{aligned} \sum _{j=1}^{2n}u_{1}^{j}u_{2}^{j} = \frac{1}{2}(w_{1}\cdot \overline{w}_{2} + \overline{w}_{1}\cdot w_{2}). \end{aligned}$$

We define the dual lattice of $\Gamma$, which is denoted by $\Gamma ^{*}$, by

$$\begin{aligned} \Gamma ^{*}&= \{ w \in \textbf{C}^{n} \mid \frac{1}{2}(v \cdot \overline{w} + \overline{v} \cdot w) \in \textbf{Z} \quad \text{ for } \text{ all } v \in \Gamma \} \\&= \{ w \in \textbf{C}^{n} \mid \text{ exp }\left( \pi i (v \cdot \overline{w} + \overline{v} \cdot w) \right) = 1 \quad \text{ for } \text{ all } v \in \Gamma \}. \end{aligned}$$

For any $w \in \Gamma ^{*}$, we define a function $\Phi _{w}: T_{\Gamma }^{n} \rightarrow \textbf{C}$ by

$$\begin{aligned} \Phi _{w}(z) : = \text{ exp }\left( \pi i (z \cdot \overline{w} + \overline{z} \cdot w) \right) . \end{aligned}$$

Then $\Phi _{w}$ is actually a well-defined function on the complex torus $T_{\Gamma }^{n}$. It is known that $\lambda$ is an eigenvalue of the Laplacian on $(T_{\Gamma }^{n}, g)$ if and only if there exists $w \in \Gamma ^{*}$ such that $\lambda = 4\pi ^{2}|w|^{2}$, where $|w|^{2} = w \cdot \overline{w}$. We set

$$\begin{aligned} S(\lambda ) := \{ w \in \Gamma ^{*} \mid \lambda = 4\pi ^{2}|w|^{2} \}. \end{aligned}$$

Then the multiplicity of $\lambda$ is given by $\# S(\lambda )$. For $\lambda \ne 0$, the number $\# S(\lambda )$ is an even integer and $S(\lambda )$ can be written as

$$\begin{aligned} S(\lambda ) = \{ \pm w_{1}, \pm w_{2}, \cdots , \pm w_{l(\lambda )} \}, \end{aligned}$$

where each $w_{\nu }$ $(\nu =1, \cdots ,l(\lambda ))$ is an element of $\Gamma ^{*}$ with $\lambda = 4\pi ^{2}|w_{\nu }|^{2}$. Set

$$\begin{aligned} & \varphi _{w}(z):= \sqrt{\frac{2}{\text{ Vol }(T^{n}_{\Gamma })}} \text{ Re }\left( \Phi _{w}(z)\right) = \sqrt{\frac{2}{\text{ Vol }(T^{n}_{\Gamma })}} \cos \left( 2\pi \sum _{k=1}^{2n}x^{k}u^{k} \right) , \\ & \psi _{w}(z):= \sqrt{\frac{2}{\text{ Vol }(T^{n}_{\Gamma })}} \text{ Im }\left( \Phi _{w}(z)\right) = \sqrt{\frac{2}{\text{ Vol }(T^{n}_{\Gamma })}} \sin \left( 2\pi \sum _{k=1}^{2n}x^{k}u^{k}\right) , \end{aligned}$$

where we use the real coordinates $z= (x^{1}, \cdots , x^{2n})$ and $w= (u^{1}, \cdots , u^{2n})$. One can verify that $\{ \varphi _{w_{\nu }}, \psi _{w_{\nu }} \mid \nu =1, \cdots ,l(\lambda ) \}$ is an $L^{2}$-orthonormal basis of the real eigenspace $E(\lambda )$. (For details of the aforementioned spectral property of flat tori, see [29, pp.272-273], for instance.)

Apostolov-Jakobson-Kokarev[1] proved that the metric on a compact homogeneous Kähler-Einstein manifold of positive scalar curvature is $\lambda _{1}$-extremal within its Kähler class. We show that the metric on a flat complex torus is also $\lambda _{1}$-extremal within its Kähler class.

Proposition 3.1

Let $(T^{n}_{\Gamma }, g)$ be a flat complex torus. Then the metric g is $\lambda _{1}$-extremal within its Kähler class.

Proof

In the proof, we use the notations introduced above. By Theorem 2.10, it suffices to show that the $L^{2}$-orthonormal basis $\{ \varphi _{w_{\nu }}, \psi _{w_{\nu }} \mid \nu =1, \cdots ,l(\lambda ) \}$ of $E_{1}(g)$ satisfy

$$\begin{aligned} \sum _{\nu =1}^{l(\lambda _{1}(g) )} L(\varphi _{w_{\nu }}) + L(\psi _{w_{\nu }}) = 0. \end{aligned}$$

(3.1)

By a straightforward calculation, we have

$$\begin{aligned} |\nabla \varphi _{w_{\nu }} |^{2} = \frac{2}{\text{ Vol }(T^{n}_{\Gamma }) } \left[ 4\pi ^{2}\sum _{j=1}^{2n}(u_{\nu }^{j})^{2} \right] \sin ^{2}\left( 2\pi \sum _{k=1}^{2n}x^{k}u_{\nu }^{k}\right) = \lambda _{1}(g)\psi _{w}^{2}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} |\nabla \psi _{w_{\nu }} |^{2} = \lambda _{1}(g)\varphi _{w}^{2}. \end{aligned}$$

On the other hand, it is easy to obtain

$$\begin{aligned} dd^{c}\varphi _{w_{\nu }} = -2\pi ^{2}i \varphi _{w_{\nu }} \sum _{\alpha , \beta =1}^{n} \overline{w}_{\nu }^{\alpha } w_{\nu }^{\beta } dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

(3.2)

Hence we obtain

$$\begin{aligned} |dd^{c}\varphi _{w_{\nu }}|^{2}&= 4\pi ^{4}\varphi _{w_{\nu }}^{2} \sum _{\alpha , \beta , \gamma , \zeta } g^{\alpha \overline{\zeta }} g^{\overline{\beta }\gamma } \overline{w}_{\nu }^{\alpha } w_{\nu }^{\beta } \overline{w}_{\nu }^{\gamma } w_{\nu }^{\zeta } \\&= 16\pi ^{4} \varphi _{w_{\nu }}^{2} \sum _{\alpha , \beta } |w_{\nu }^{\alpha }|^{2}|w_{\nu }^{\beta }|^{2} \\&=\lambda _{1}(g)^{2}\varphi _{w_{\nu }}^{2}. \\ \end{aligned}$$

Similarly, we have

$$\begin{aligned} dd^{c}\psi _{w_{\nu }} = -2\pi ^{2}i \psi _{w_{\nu }} \sum _{\alpha , \beta =1}^{n} \overline{w}_{\nu }^{\alpha } w_{\nu }^{\beta } dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

(3.3)

and

$$\begin{aligned} |dd^{c}\psi _{w_{\nu }}|^{2} =\lambda _{1}(g)^{2}\psi _{w_{\nu }}^{2}. \end{aligned}$$

Thus for each $\nu$, we have

$$\begin{aligned}&L(\varphi _{w_{\nu }}) + L(\psi _{w_{\nu }}) \nonumber \\&\quad = \left( \lambda _{1}(g)^{2}\varphi _{w_{\nu }} - 2\lambda _{1}(g)^{2}\psi _{w_{\nu }}^{2} + \lambda _{1}(g)^{2}\varphi _{w_{\nu }}^{2} \right) \nonumber \\&\qquad + \left( \lambda _{1}(g)^{2}\psi _{w_{\nu }} - 2\lambda _{1}(g)^{2}\varphi _{w_{\nu }}^{2} + \lambda _{1}(g)^{2}\psi _{w_{\nu }}^{2} \right) \nonumber \\&\quad =0. \end{aligned}$$

(3.4)

Thus (3.1) is proved. $\square$

The harmonic projector $H: \Omega ^{1,1}(T_{\Gamma }^{n}) \rightarrow \mathcal {H}^{1,1}(T_{\Gamma }^{n})$ is given by

$$\begin{aligned} H(\phi ) =\frac{1}{\text{ Vol }(T_{\Gamma }^{n})} \sum _{\alpha ,\beta =1}^{n} \left( \int _{T_{\Gamma }^{n}} \phi _{\alpha \overline{\beta }} d\mu \right) dz^{\alpha } \wedge d\overline{z}^{\beta } \end{aligned}$$

for a (1, 1)-form $\phi = \sum _{\alpha , \beta =1}^{n} \phi _{\alpha \overline{\beta }} dz^{\alpha } \wedge d\overline{z}^{\beta }$, where $d\mu$ is the volume form of $(T_{\Gamma }^{n}, g)$. (For details of the Hodge decomposition on a complex torus, see [4, Section 1.4].) Using this fact and Theorem 2.9, we prove the following:

Theorem 3.2

Let $(T_{\Gamma }^{n},g)$ be a flat n-dimensional complex torus. Let $\{w_{\nu } \}_{\nu =1}^{l(\lambda _{k}(g))}$ be linearly independent vectors in $\Gamma ^{*}$ satisfying $\lambda _{k} (g)= 4\pi ^{2}|w_{\nu }|^{2}$. If the flat metric g is $\lambda _{k}$-extremal for all the volume-preserving deformations of the Kähler metric, then there exists $\{R_{\nu } \ge 0\}_{\nu =1}^{l(\lambda _{k}(g))}$ such that the following equations hold:

$$\begin{aligned} \left\{ \, \begin{aligned}&\sum _{\nu =1}^{l(\lambda _{k}(g))} R_{\nu } \overline{w}_{\nu }^{\alpha }w_{\nu }^{\beta } = 0 \quad \text{ for } \quad 1\le \alpha \ne \beta \le n, \\&\sum _{\nu =1}^{l(\lambda _{k}(g))} R_{\nu } |w_{\nu }^{\alpha }|^{2} = 1 \quad \text{ for } \quad 1 \le \alpha \le n.\\ \end{aligned} \right. \end{aligned}$$

(3.5)

For $k=1$, the existence of such $\{R_{\nu } \ge 0\}_{\nu =1}^{l(\lambda _{1}(g))}$ is also a sufficient condition for the metric g to be $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

Proof

First we prove the fist half of the assertion. By theorem 2.9, we must have

$$\begin{aligned} H \left( \sum _{j=1}^{N} f_{j}dd^{c}f_{j} \right) = -\frac{i}{2} \sum _{\alpha =1}^{n} dz^{\alpha } \wedge d\overline{z}^{\alpha } \end{aligned}$$

(3.6)

for some finite collection of eigenfunctions $\{f_{j} \}_{j=1}^{N} \subset E_{k}(g)$. Each eigenfunction $f_{j}$ is of the form

$$\begin{aligned} f_{j}(z) = \sum _{\nu =1}^{l(\lambda _{k}(g))} a_{j\nu } \varphi _{w_{\nu }} + b_{j\nu } \psi _{w_{\nu }} \quad (a_{j\nu }, b_{j\nu } \in \textbf{R}). \end{aligned}$$

Using (3.2) and (3.3), we obtain

$$\begin{aligned} f_{j}dd^{c}f_{j}&= -2\pi ^{2}i \sum _{\alpha , \beta =1}^{n} \sum _{\nu , \tau =1}^{l(\lambda _{k}(g))} \overline{w}_{\nu }^{\alpha }w_{\nu }^{\beta } [ a_{j\nu }a_{j\tau } \varphi _{w_{\nu }} \varphi _{w_{\tau }} + a_{j\nu }b_{j\tau } \varphi _{w_{\nu }} \psi _{w_{\tau }} \\&\quad + a_{j\tau }b_{j\nu } \varphi _{w_{\tau }} \psi _{w_{\nu }} + b_{j\nu } b_{j\tau } \psi _{w_{\nu }} \psi _{w_{\tau }} ] dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

Hence one obtains

$$\begin{aligned} H(f_{j}dd^{c}f_{j}) = \frac{-2\pi ^{2}i}{\text{ Vol }(T_{\Gamma }^{n})} \sum _{\alpha ,\beta =1}^{n} \sum _{\nu =1}^{l(\lambda _{k}(g))} \overline{w}_{\nu }^{\alpha }w_{\nu }^{\beta }( a_{j\nu }^{2} + b_{j\nu }^{2} ) dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

Thus one obtains

$$\begin{aligned} \sum _{j=1}^{N} H (f_{j}dd^{c}f_{j}) = \frac{-2\pi ^{2}i}{\text{ Vol }(T_{\Gamma }^{n})} \sum _{\alpha ,\beta =1}^{n} \sum _{\nu =1}^{l(\lambda _{k}(g))} \overline{w}_{\nu }^{\alpha }w_{\nu }^{\beta }(\sum _{j=1}^{N} a_{j\nu }^{2} + b_{j\nu }^{2} ) dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

(3.7)

Then the equations (3.6) and (3.7) imply that we must have

$$\begin{aligned} \left\{ \, \begin{aligned}&\sum _{\nu =1}^{l(\lambda _{k}(g))} \overline{w}_{\nu }^{\alpha }w_{\nu }^{\beta }(\sum _{j=1}^{N} a_{j\nu }^{2} + b_{j\nu }^{2} ) = 0 \quad \text{ for } \quad 1\le \alpha \ne \beta \le n, \\&\sum _{\nu =1}^{l(\lambda _{k}(g))} |\overline{w}_{\nu }^{\alpha }|^{2}(\sum _{j=1}^{N} a_{j\nu }^{2} + b_{j\nu }^{2} ) = \frac{ \text{ Vol }(T_{\Gamma }^{n}) }{4\pi ^{2}} \quad \text{ for } \quad 1 \le \alpha \le n.\\ \end{aligned} \right. \end{aligned}$$

Setting $R_{\nu }:= 4\pi ^{2}(\sum _{j=1}^{N} a_{j\nu }^{2} + b_{j\nu }^{2} )/\text{Vol }(T_{\Gamma }^{n})$, one concludes the first half of the assertion.

Next we prove the second half. We assume the existence of $\{R_{\nu } \ge 0\}_{\nu =1}^{l(\lambda _{1}(g))}$ satisfying (3.5). We use Theorem 2.9 to prove the proposition. $\{ \varphi _{w_{\nu }}, \psi _{w_{\nu }} \mid \nu =1, \cdots ,l(\lambda _{1}(g)) \}$ is an $L^{2}$-orthonormal basis of $E(\lambda _{1}(g))$. By (3.4), we immediately have

$$\begin{aligned} L(\sqrt{R_{\nu }} \varphi _{w_{\nu }} ) + L(\sqrt{R_{\nu } }\psi _{w_{\nu }}) = 0 \end{aligned}$$

for each $\nu$. Hence we have

$$\begin{aligned} \sum _{\nu =1}^{l(\lambda _{1}(g)} L(\sqrt{R_{\nu }} \varphi _{w_{\nu }} ) + L(\sqrt{R_{\nu } }\psi _{w_{\nu }}) = 0. \end{aligned}$$

Thus it suffices to prove

$$\begin{aligned} \sum _{\nu =1}^{l(\lambda _{1}(g))} R_{\nu } \left[ H(\varphi _{w_{\nu }}dd^{c}\varphi _{w_{\nu }}) + H(\psi _{w_{\nu }}dd^{c}\psi _{w_{\nu }}) \right] = -a\omega \end{aligned}$$

for some $a>0$. Using the equations (3.2) and (3.3), one obtains

$$\begin{aligned}&\sum _{\nu =1}^{l(\lambda _{1}(g))} R_{\nu }\left[ H(\varphi _{w_{\nu }}dd^{c}\varphi _{w_{\nu }}) + H(\psi _{w_{\nu }}dd^{c}\psi _{w_{\nu }}) \right] \\&\quad = \frac{-2\pi ^{2}i}{\text{ Vol }(T_{\Gamma }^{n})} \sum _{\nu =1}^{l(\lambda _{1}(g))} \sum _{\alpha ,\beta =1}^{n} R_{\nu } \left( \int _{T_{\Gamma }^{n}} (\phi _{w_{\nu }}^{2}+\psi _{w_{\nu }}^{2})d\mu \right) \overline{w}_{\nu }^{\alpha } w_{\nu }^{\beta } dz^{\alpha } \wedge d\overline{z}^{\beta }\\&\quad = \frac{-4\pi ^{2}i}{\text{ Vol }(T_{\Gamma }^{n})} \sum _{\nu =1}^{l(\lambda _{1}(g))} \sum _{\alpha ,\beta =1}^{n} R_{\nu } \overline{w}_{\nu }^{\alpha } w_{\nu }^{\beta } dz^{\alpha } \wedge d\overline{z}^{\beta }. \end{aligned}$$

By hypothesis, the proof is completed. $\square$

The implication of this theorem is not clear, so we consider simple cases in what follows. First we consider the case where $\text{ dim }E_{k}(g) =2$, that is, $l(\lambda _{k}(g)) =1$. Then we have the following corollary:

Corollary 3.3

Let $(T_{\Gamma }^{n},g)$ be a flat n-dimensional complex torus. Suppose that $\text{ dim }E_{k}(g) =2$ holds for some k. Then the metric g is not $\lambda _{k}$-extremal.

Proof

We prove the assertion by contradiction. Assume that the metric g is $\lambda _{k}$-extremal. By hypothesis, there exists a pair w, $-w \in \Gamma ^{*} \subset \textbf{C}^{n}$ uniquely up to sign such that $\lambda _{k}(g) = 4\pi ^{2}|w|^{2}$. First we show that the vector w is of the form $w = (0, \cdots , \xi , \cdots , 0)$ for some $\xi \in \textbf{C}$. The first equation in (3.5) implies $\overline{w}^{\alpha }w^{\beta }=0$ for any pair of distinct integers $(\alpha ,\beta )$. Since $w = (w^{1}, \cdots , w^{n})$ is a nonzero vector, we have $w^{\alpha } \ne 0$ for some $\alpha$. Let $w^{j} = u^{2j-1}+iu^{2j}$ for each $1 \le j \le n$. Then for any $\beta \ne \alpha$, we have

$$\begin{aligned} u^{2\alpha -1}u^{2\beta -1} = -u^{2\alpha }u^{2\beta } \end{aligned}$$

(3.8)

and

$$\begin{aligned} u^{2\alpha }u^{2\beta -1} = u^{2\alpha -1} u^{2\beta }. \end{aligned}$$

(3.9)

Assume that $u^{2\alpha } \ne 0$ and $u^{2\beta } \ne 0$. Then by (3.9), there exists $c \in \textbf{R}$ such that $u^{2\alpha -1} = cu^{2\alpha }$ and $u^{2\beta -1} = cu^{2\beta }$. Substituting these for (3.8), one obtains

$$\begin{aligned} c^{2}u^{2\alpha }u^{2\beta } = -u^{2\alpha }u^{2\beta }. \end{aligned}$$

This is a contradiction and so we have $u^{2\alpha }=0$ or $u^{2\beta }=0$.

If we have $u^{2\alpha }=0$, then we must have $u^{2\alpha -1} \ne 0$ since we now assume $w^{\alpha } \ne 0$. Hence by (3.9), we have $u^{2\beta } = 0$. Then (3.8) immediately implies $u^{2\beta -1} = 0$. Thus we have $u^{2\beta -1} = u^{2\beta } = 0$, that is $w^{\beta }=0$.

If we have $u^{2\beta }=0$, then (3.9) implies that we have $u^{2\alpha }=0$ or $u^{2\beta -1}=0$. We have already considered the case where $u^{2\alpha }=0$. Hence we consider the case where $u^{2\beta -1}=0$, but this immediately implies $w^{\beta }=0$.

Thus we conclude that w is of the form $w = (0, \cdots , \xi , \cdots , 0)$. However, this contradicts with the second equation in (3.5). The proof is completed. $\square$

Example 3.4

${{\hbox {The standard lattice: }} \Gamma = \textbf{Z}^{2n}}$. Consider the standard complex torus ${\textbf {C}}^{n}/\textbf{Z}^{2n}$ with the flat metric g. Let $\{e_{j} \}_{j=1}^{n}$ be the standard orthonormal basis of ${\textbf {C}}^{n}$. Set

$$\begin{aligned} w_{2k-1} := e_{k}, \quad w_{2k} := ie_{k} \end{aligned}$$

for every $1 \le k \le n$. Then we have $S(\lambda _{1}(g)) = \{\pm w_{\nu } \}_{\nu =1}^{2n}$ and $l(\lambda _{1}(g)) = 2n$. It is clear that (3.5) is equivalent to the condition where $R_{2k-1}+R_{2k} =1$ for any $1\le k \le n$ and so the torus ${\textbf {C}}^{n}/\textbf{Z}^{2n}$ satisfies the assumption of Proposition 3.2. Hence the metric g is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric. This fact is not new since the metric is $\lambda _{1}$-extremal for all the volume-preserving metric deformations. This can be seen from Theorem 1.1 and the classical fact that the standard torus admits an isometric minimal immersion into a unit sphere by first eigenfunctions as follows:

$$\begin{aligned}&\textbf{C}^{n}/\textbf{Z}^{2n} \rightarrow S^{4n-1}\left( \sqrt{\frac{n}{2\pi ^{2}} }\right) , \\&(x^{1}, \ldots , x^{2n}) \mapsto \\&\quad \left( \frac{1}{2\pi }\cos (2\pi x^{1}), \frac{1}{2\pi }\sin (2\pi x^{1}), \ldots , \frac{1}{2\pi }\cos (2\pi x^{2n}), \frac{1}{2\pi }\sin (2\pi x^{2n}) \right) . \end{aligned}$$

Example 3.5

The checkerboard lattice. First we consider the (real) 4-dimensional checkerboard lattice $D_{4}$, which is defined by

$$\begin{aligned} D_{4} := \{ (x^{1}, \ldots , x^{4}) \in \textbf{Z}^{4} \mid x^{1} + \cdots +x^{4} \in 2\textbf{Z} \}. \end{aligned}$$

$D_{4}$ is self-dual, i.e. $D_{4} \cong D^{*}_{4}$. The dual lattice $D_{4}^{*} (\cong D_{4})$ is known to be the lattice in $\textbf{C}^{2}$ with the basis (1, 0), (0, 1), (i, 0), $\left( \frac{1+i}{2}, \frac{1+i}{2}\right)$. (See [9, pp.117-120], for instance.) Set

$$\begin{aligned}&w_{1} := (1,0), \quad w_{2}:= (0,1), \quad w_{3} := (i, 0), \quad w_{4} := (0, i), \\&w_{5} :=\left( \frac{1+i}{2}, \frac{1+i}{2} \right) , \quad w_{6}:= \left( \frac{1-i}{2}, \frac{1-i}{2} \right) , \\&w_{7}:= \left( \frac{1+i}{2}, -\frac{1+i}{2} \right) , \quad w_{8} := \left( \frac{1-i}{2} , -\frac{1-i}{2} \right) , \\&w_{9} := \left( \frac{1+i}{2}, \frac{1-i}{2} \right) , \quad w_{10} := \left( \frac{1+i}{2}, - \frac{1-i}{2} \right) , \\&w_{11} := \left( \frac{1-i}{2}, \frac{1+i}{2}\right) , \quad w_{12} := \left( \frac{1-i}{2}, -\frac{1+i}{2}\right) . \end{aligned}$$

If we set $R_{1} = \cdots =R_{4} = 1/4$, $R_{5} = \cdots =R_{12} = 1/8$, then it is elementary to check that the equations (3.5) hold for $k=1$. Hence by Theorem 3.2, the flat metric g on the 2-dimensional complex torus $\textbf{C}^{2}/D_{4}$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric. This fact is not new since the metric is $\lambda _{1}$-extremal for all the volume-preserving metric deformations. This follows from the fact that Lü et al. [20] recently found a 2-parameter family of isometric minimal immersion by the first eigenfunctions from $\textbf{C}^{2}/D_{4}$ into the unit sphere $S^{23} \subset \textbf{R}^{24}$. (See Example 1.1 in [20].)

In fact, for any $m \ge 3$, the checkerboard lattice $D_{m}$ is defined as a lattice in $\textbf{R}^{m}$ by

$$\begin{aligned} D_{m} := \{ (x^{1}, \ldots , x^{m}) \in \textbf{Z}^{m} \mid x^{1} + \cdots +x^{m} \in 2\textbf{Z} \}. \end{aligned}$$

We show the following:

Proposition 3.6

For any $m \ge 3$, the flat torus $\textbf{R}^{m}/D_{m}$ admits an isometric minimal immersion into a Euclidean sphere.

Proof

The property of $D_{m}$ should be considered separately for the case $m=3$, $m=4$ and $m \ge 5$. For $m=3$ and $m=4$, the assertion has been proved by Lü et al. [20]. (See Example 4.3 in [20] for $m=3$ and Example 1.1 in [20] for $m=4$.) Hence it suffices to consider the case where $m\ge 5$. For $m \ge 5$, $D_{m}^{*}$ is a lattice with the basis $\{e_{j} \}_{j=1}^{m-1} \cup \{ \frac{1}{2}\sum _{k=1}^{m} e_{k} \}$, where $\{ e_{j} \}_{j=1}^{m}$ is the standard basis in $\textbf{R}^{m}$. (See [9, p.120], for instance.) The shortest vectors are exactly the 2m vectors $\{ \pm e_{j} \}_{j=1}^{m}$. Thus we have $\lambda _{1}(g) = 4\pi ^{2}$ and $E_{1}(g)$ is spanned by $\{ \cos (2\pi x^{j}), \sin (2\pi x^{j}) \}_{j=1}^{m}$, where $\{x^{j} \}_{j=1}^{m}$ is the standard coordinate in $\textbf{R}^{m}$. It is obvious that the map

$$\begin{aligned}&\textbf{R}^{m}/D_{m} \rightarrow S^{2m-1}\left( \frac{ \sqrt{m} }{2\pi }\right) , \\&(x^{1}, \ldots , x^{m}) \mapsto \\&\quad \left( \frac{1}{2\pi }\cos (2\pi x^{1}), \frac{1}{2\pi }\sin (2\pi x^{1}), \ldots , \frac{1}{2\pi }\cos (2\pi x^{m}), \frac{1}{2\pi }\sin (2\pi x^{m}) \right) \end{aligned}$$

is an isometric minimal immersion. $\square$

For a long time, only the standard torus (Example 3.4) had been an example of higher dimensional tori that admit an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions. Very recently, Lü et al. [20] constructed new examples of higher dimensional flat tori that admit an isometric minimal immersion into a Euclidean sphere by the first eigenfunctions. The flat torus $\textbf{R}^{m}/D_{m}$ $(m\ge 3)$ is a new example.

Example 3.7

For $a, b \in [1, \infty )$, consider the lattice $\Gamma _{a,b}$ in $\textbf{C}^{2}$ with the lattice basis (1, 0), $(a^{-1}\sqrt{-1}, 0)$, (0, 1), $(0, b^{-1}\sqrt{-1})$. Let $T^{2}_{a,b}$ be the 2-dimensional complex torus determined by $\Gamma _{a,b}$ with the flat metric $g_{a,b}$. Let $\Gamma _{a} \subset \textbf{C}$ be the lattice with the lattice basis (1, 0), $(a^{-1}\sqrt{-1}, 0)$ and $(T^{1}_{a}, h_{a})$ the flat 1-dimensional complex torus determined by $\Gamma _{a}$. Then $(T^{2}_{a,b}, g_{a,b})$ is the product of $(T^{1}_{a}, h_{a})$ and $(T^{1}_{b}, h_{b})$. We have $\lambda _{1}(T^{1}_{a}, h_{a}) = 1 = \lambda _{1}(T^{1}_{b}, h_{b})$. Hence Proposition 3.1 and Corollary 2.13 imply that the metric $g_{a,b}$ on $T^{2}_{a,b}$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric. However, since we have $E_{1}(g_{a,b}) = \text{ span }\{ \cos (2\pi x^{1}), \sin (2\pi x^{1}), \cos (2\pi x^{3}), \sin (2\pi x^{3}) \}$, the flat torus $(T^{2}_{a,b}, g_{a,b})$ does not admit an isometric minimal immersion into a Euclidean sphere by first eigenfunctions. Thus $g_{a,b}$ is not $\lambda _{1}$-extremal for all the volume-preserving metric deformations.

Example 3.8

For $a, b \in [1, \infty )$, consider the lattice $\widetilde{\Gamma }_{a,b}$ in $\textbf{C}^{2}$ with the lattice basis (1, 0), $(\sqrt{-1}, 0)$, $(0,a^{-1})$, $(0, b^{-1}\sqrt{-1})$. Let $\widetilde{T}^{2}_{a,b}$ be the 2-dimensional complex torus determined by $\widetilde{\Gamma }_{a,b}$ with the flat metric $\widetilde{g}_{a,b}$. Let $(T^{1}_{std}, h_{std})$ be the flat 1-dimensional complex torus determined by the lattice with the lattice basis (1, 0), $(\sqrt{-1}, 0)$. Let $(T^{1}_{a,b}, h_{a,b})$ be the flat 1-dimensional complex torus determined by the lattice with the lattice basis $(0,a^{-1})$, $(0, b^{-1}\sqrt{-1})$. Then $(\widetilde{T}^{2}_{a,b}, \widetilde{g}_{a,b})$ is the product of $(T^{1}_{std}, h_{std})$ and $(T^{1}_{a,b}, h_{a,b})$. We have $\lambda _{1}(T^{1}_{std}, h_{std}) = 1$ and $\lambda _{1}(T^{1}_{a,b}, h_{a,b}) = \text{ min }\{a,b\}$. Hence if we have $a=1$ or $b=1$, then Proposition 3.1 and Corollary 2.13 imply that the metric $\widetilde{g}_{a,b}$ on $\widetilde{T}^{2}_{a,b}$ is $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric. On the other hand, if we have $a>1$ and $b>1$, then by Corollary 2.14, $\widetilde{g}_{a,b}$ is not $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

In Example 3.7 and Example 3.8, if we ignore the complex structure on $\textbf{C}^{2}$ and regard $\textbf{C}^{2}$ as $\textbf{R}^{4}$, then we have $\Gamma _{a,b} \cong \widetilde{\Gamma }_{a,b}$. However, whether the flat metric is $\lambda _{1}$-extremal is different in Example 3.7 and Example 3.8. Hence Example 3.7 and Example 3.8 show that the notion of $\lambda _{1}$-extremality actually depends on the complex structure.

Finally we give a 1-parameter family of 2-dimensional complex tori whose flat metrics are not $\lambda _{1}$-extremal for all the volume-preserving deformations of the Kähler metric.

Example 3.9

For $\pi /3< \theta <\pi /2$, we consider the lattice $\Gamma _{\theta } \subset \textbf{C}^{2}$ with the lattice basis (1, 0), $(\cos \theta , \sin \theta )$, $(\sqrt{-1}, 0)$, $(\sqrt{-1}\cos \theta , \sqrt{-1}\sin \theta )$. Let $g_{\theta }$ be the flat metric on $\textbf{C}^{2}/\Gamma _{\theta }$. It is straightforward to check that the dual lattice $\Gamma _{\theta }^{*}$ is the lattice with the basis $w_{1}:= (1, -\cos \theta /\sin \theta )$, $w_{2}:= (0, 1/\sin \theta )$, $w_{3}:=(\sqrt{-1}, -\cos \theta /\sin \theta )$, $w_{4}:=(0, \sqrt{-1}/\sin \theta )$. Then we have $S(\lambda _{1}(g_{\theta })) = \{ \pm w_{\nu } \}_{\nu =1}^{4}$ and so the multiplicity of $\lambda _{1}(g_{\theta })$ is 8. If g is $\lambda _{1}$-extremal, then Theorem 3.2 implies that there exists $\{ R_{\nu } \}_{\nu =1}^{4}$ such that

$$\begin{aligned} -\frac{\cos \theta }{\sin \theta }(R_{1}+R_{3}) = 0, \quad R_{1}+R_{3}= 1, \quad \frac{1}{\sin ^{2}\theta }(R_{2}+R_{4}) =1. \end{aligned}$$

Since we have $\pi /3< \theta <\pi /2$, this is a contradiction. Hence $g_{\theta }$ is not $\lambda _{1}$-extremal. $\textbf{C}^{2}/\Gamma _{\theta }$ is not a product of 1-dimensional flat complex tori. In fact, assume that $\textbf{C}^{2}/\Gamma _{\theta }$ is a product of $(T_{1},h_{1})$ and $(T_{2}, h_{2})$, where each is a 1-dimensional flat complex torus. If we had $\lambda _{1}(h_{1}) = \lambda _{1}(h_{2})$, then by Proposition 3.1 and Corollary 2.13, $g_{\theta }$ would be $\lambda _{1}$-extremal. Hence we have $\lambda _{1}(h_{1}) \ne \lambda _{1}(h_{2})$. We may assume $\lambda _{1}(h_{1}) < \lambda _{1}(h_{2})$. Then the multiplicity of of $\lambda _{1}(h_{1})$ is equal to that of $\lambda _{1}(g_{\theta })$, that is, 8. This is a contradiction since the multiplicity of the first eigenvalue of a 1-dimensional flat complex torus is at most 6 (see [11], for example). Thus $\textbf{C}^{2}/\Gamma _{\theta }$ is not a product of 1-dimensional complex tori. Let $\widetilde{\Gamma }_{\theta } \subset \textbf{R}^{2}$ be the lattice with the lattice basis (1, 0), $(\cos \theta , \sin \theta )$. Let $(\textbf{R}^{2}/\widetilde{\Gamma }_{\theta }, h_{ \theta })$ be the flat real 2-dimensional torus. If we ignore the complex structure on $\textbf{C}^{2}$ and regard it as $\textbf{R}^{4}$, then $(\textbf{R}^{4}/\Gamma _{\theta }, g_{\theta })$ is a Riemannian product of $(\textbf{R}^{2}/\widetilde{\Gamma }_{\theta }, h_{ \theta })$ and $(\textbf{R}^{2}/\widetilde{\Gamma }_{\theta }, h_{ \theta })$.