Infimum of the spectrum of Laplace-Beltrami operator on Cartan classical domains of type Ⅲ and Ⅳ

1.   Introduction

Let $(M, g)$ be a Kähler manifold of complex dimension $n$ with Kähler metric $g = \sum_{i, j = 1}^n u_{i{\overline{j}}} {\rm d}z_i \otimes {\rm d}{\overline{z}}_j$. The Laplace-Beltrami operator with respect to the Kähler metric $g$ is defined by

where $[u^{i{\overline{j}}}]^t = [u_{i{\overline{j}}}]^{-1}$. Let

Here ${\rm d}V_g$ is the volume measure of $M$ with respect to the Kähler metric $g$.

Spectral theory or eigenvalue estimates for Laplace-Beltrami operators are important subject for mathematics. When $M$ is compact and $\Delta_g$ is uniformly elliptic, $\lambda_1(\Delta_g)$ is the first eigenvalue of $\Delta_g$. Researches on its upper and lower bound estimates have a long history with many results (see ^[1,2,3,4,5] and the reference therein).

Unlike compact manifolds, for the case when $M$ is a non-compact manifold, $\lambda_1(\Delta_g)$ may not be an eigenvalue of $\Delta_g$, but rather the infimum of the positive spectral of $\Delta_g$. This naturally leads to an interesting problem: study of estimate for $\lambda_1(\Delta_g)$ in the complete non-compact case. Quite a bit of research has been done on this problem. For examples, the results on upper bounds and lower bounds estimates obtained by Cheng ^[6], Li and Wang ^[7,8,9,10], Munteanu ^[11], Li and Tran ^[12] are all well known. The rigidity property of manifold may be further obtained when $\lambda_1$ achieves its sharp upper or lower bound estimate. For examples, one may see ^{[7,8,9,13,14]} and references therein.

We recall the following estimates for $\lambda_1(\Delta_g)$, in the Riemannian manifolds, a sharp upper bound estimate for $\lambda_1(\Delta_g)$ is well known from Cheng ^[6]. For the Kähler manifolds, Li and Wang gave an important sharp upper bound estimate in ^[9]. They proved $\lambda_1(\Delta_g)\geq n^2$ with the assumption that the holomorphic bisectional curvature of $M$ is bounded below by $-1$. Later on, Munteanu ^[11] improved Li and Wang's result, and derived another sharp estimate in terms of the Ricci curvature is bounded from below by $-2(n+1)$.

As a continuation of the work of Li and Wang ^[9] and the work of Munteanu ^[11], Li and Tran ^[12] provided many examples of bounded strongly pseudoconvex domains, on which $\lambda_1(\Delta_g)$ can be explicitly formulated. However, those domains were exclusive of most of non-smooth domains, like Cartan classical domains. It would be desirable to proceed the study of $\lambda_1(\Delta_g)$ on the Cartan classical domains.

Suppose ${{\cal R}}_{{\cal A}}(n)$ be the Cartan classical domains of type ${{\cal A}}({{\cal A}} = III, IV)$. Let $D$ be a bounded pseudoconvex domain in ${{\mathbb C}}^n$ and $u(z)\in C^\infty(D)$ be a strictly plurisubharmonic exhaustion function for $D$. Let $g = \sum_{i, j = 1}^n {\partial ^2 u\over \partial z_i \partial {\overline{z}}_j} d z_i \otimes d{\overline{z}}_j$ be the Kähler metric induced by $u$. We set

and

This paper aims to provide an estimate for $\lambda_1(\Delta_g)$ on ${{\cal R}}_{{\cal A}}(n)({{\cal A}} = III, IV)$ with $g$ is the Bergman metric of ${{\cal R}}_{{\cal A}}(n)$. Regarding the estimate for $\lambda_1(\Delta_g)$ on the Cartan classical domains of type Ⅰ and Ⅱ, it has been studied by the first author in ^[15]. We continue to use some ideas in ^[15], but the calculations in the current cases are much more delicate. This is due to the special forms of ${{\cal R}}_{III}(n)$ and ${{\cal R}}_{IV}(n)$.

Our study is motivated by Li and Tran ^[12]. We emphasize that the calculation of $|\partial u|_{g}^2$ and the construction of test functions play important roles in our argument, but both are difficult to solve. We use the Bergman metric $g$ on the Cartan classical domains given by Hua ^[16] and Lu ^[17].

Let $K_{{\cal A}}(z): = K_{{\cal A}}(z, z)$ be the Bergman kernel function of $R_{{\cal A}}(n)$ and let

Our goal is to prove the following results.

Theorem 1.1. Let $\Delta_{_{\mathcal{A}}} = \Delta_g\, (\mathcal{A} = III, IV)$ be the Laplace-Beltrami operator associated to the Bergman metric $g$ on the Cartan classical domains $R_{{\cal A}}(n)$ of type $\mathcal{A}$. Let $u = u_{{\cal A}}$ be the strictly plurisubharmonic exhaustion function for ${{\cal R}}_{\mathcal{A}}(n)$ and $g = g_{\mathcal{A}}$ be the Kähler metric induced by $u_{{\cal A}}$. Assuming that $|\partial u|_{g}^2\le \beta$, one has

(i) $\lambda_1(\Delta_{_{\mathcal{A}}}) \ge N_{{\cal A}} ^2/\beta$ with $N_{III} = n(n-1)/2$ and $N_{IV} = n$;

(ii) $\lambda_1(\Delta_{_{\mathcal{A}}}) \le \beta c_{\mathcal{A}}^2 (1-\alpha_{\mathcal{A}})^2$, where

and

Corollary 1.1. Let the notations and assumptions as in Theorem 1.1. Then

and

The remainder of the paper is organized as follows. In Section 2, we introduce the notion and provide some preliminary results concerning ${{\cal R}}_{_{\mathcal{A}}}(n)$. In Section 3, our main results are stated and proved. Section 4 contains a brief summary of our study.

2.   Cartan classical domains of type Ⅲ and Ⅳ

Let $M^{(n)}$ be the set of all $n\times n$ matrices with entries in ${\mathbb{C}}$. For any $A = [a_{ij}]\in M^{(n)}$, let

We denote by $I_n$ the $n\times n$ identity matrix. The Cartan classical domains of the type Ⅲ and Ⅳ can be represented as follows:

Suppose $A$ and $B$ are two $n\times n$ matrix. Then we define an ${n(n-1)\over 2}\times {n(n-1)\over 2}$ matrix $[A\dot\times B]_{as}$, which consists of the entries $(A\dot\times B)_{(ij) (k\ell)}$ with $i < j$ and $k < \ell$ as follows:

In particular,

The following result can be found in [17, p317–318].

Proposition 2.1. ^[17] Let $A$ and $B$ be two $n\times n$ matrices. Then

and

A straightforward calculation shows:

Proposition 2.2. Let $C = [c_{pq}]$ be a $s\times s$ matrix where $c_{pq}$ is a function of $z = (z_1, \cdots, z_n)\in {\mathbb{C}}^n$ and ${\overline{z}}$. Then

and

where

For $Z\in{{\cal R}}_{_{III}}$, we set

Obviously $2\|z\|^2 = {{\hbox{tr}}} (ZZ^*)$. We know from Hua's book ^[16] or Lu's book [17, Section 3.3] that

and

Here $V({{\cal R}}_{\mathcal{A}})$ is the volume of ${{\cal R}_{\mathcal{A}}}$.

Consider the Bergman kernel function of ${{\cal R}}_{\mathcal{A}}$, we construct $u_{\mathcal{A}}$ as (1.5), hence $u_{\mathcal{A}}$ is strictly plurisubharmonic exhaustion function in ${{\cal R}}_{\mathcal{A}}$. Furthermore, we define a complete Kähler metric $g_{\mathcal{A}}$ which is induced by $u_{\mathcal{A}}$ as follows:

where $N = n(n-1)/2$ when ${{\cal A}} = III$ and $N = n$ when ${{\cal A}} = IV$. Consequently,

where $[{u_{\mathcal{A}}}^{i{\overline{j}}}]^t = [{u_{\mathcal{A}}}_{i{\overline{j}}}]^{-1}$.

Let us mention two important consequences about the complex Hessian matrix for ${u_{\mathcal{A}}}$ on ${{\cal R}}_{\mathcal{A}}$.

Proposition 2.3. The complex Hessian matrix for ${u_{\mathcal{A}}}$ can be stated by

and

where $r(z) = 1+|\sum_{j = 1}^n z_j^2| ^2-2|z|^2$.

Proof. (i) For $Z\in {{\cal R}}_{_{III}}$, by applying (2.7) and (2.8), a straightforward calculation shows that

Hence, we have

As the proof of equality (3.3.45) in ^[17], the transformation property of Bergman kernel function and Möbius transform of $R_{{\cal A}}$, (2.13) follows. By Proposition 2.1, (2.14) obtained.

(ii) For $z = (z_1, z_2\cdots, z_n)\in {{\cal R}}_{_{IV}}$, by equality (3.3.57) in ^[17], we have

This implies (2.15). We claim (2.16) holds from discussion of equality (6.1.29) in ^[17]. □

Proposition 2.4. With the notations above, one has

and

Proof. By Proposition 2.1 and Proposition 2.3, it is easy to deduce that

and

This completes the proof. □

3.   Estimate for $\lambda_1(\Delta_g)$

In this section, in order to prove the main theorems, we begin to establish an estimate for $|\partial u|_{g}^2$. Let

Since

one has

Proposition 3.1. With the notations in (1.5), (2.11) and (2.12), one has the following estimates:

Proof. (i) For $z\in {{\cal R}}_{_{III}}$, according to (3.2), one has

where

Let $D[\lambda_1, \ldots, \lambda_n]$ be $n \times n$ diagonal matrix with all diagonal entries are $\lambda_1, \ldots, \lambda_n$. For $Z\in {{\cal R}}_{_{III}}$, since $ZZ^*$ is the Hermitian matrix, there exists $n\times n$ unitary matrix $U$ such that

and $\lambda_j\in [0, 1), \lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\geq0$. There is no loss of generality in assuming

It follows that

Thus

Therefore

(ii) For $z\in {{\cal R}}_{_{IV}}$, we have

Accordingly,

Notice that

and

Then (3.14) and (3.15) yield

Consequently,

which implies the desired conclusion. The proof of the proposition is complete. □

To prove Theorem 1.1 and for the convenience of the reader, we recall the following proposition from ^[16].

Proposition 3.2. With the notations above, one has,

and

Where $\mu(z) = (1-z\bar z'-\sqrt{(z\bar z')^2-|zz'|^2})$, $\nu(z) = (1-z\bar z'+\sqrt{(z\bar z')^2-|zz'|^2})$.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. By Proposition 2.1 in ^[12], it is evident that statement (i) holds. So we only need to prove statement (ii). Let

By Proposition 2.4, one has

where $C_{ {\cal A}}$ and $c_{ {\cal A}}$ are constants which are dependent on ${{\cal R}}_{{\cal A}}$.

By Faraut and Koranyi ^[18] and Proposition 3.2, there exists $\alpha_{{\cal A}} > 0$ such that

Now we choose $\tau$ such that

Applying the argument of the proof of [12, Theorem 2.2] and Proposition 3.1, one has

Letting $\tau \to {1 \over 2}c_{{\cal A}} (1-\alpha_{{\cal A}})$ we have

Which completes the proof.□

Remark 3.1. Let

and

We conclude from Proposition 3.2 that

and

Finally, we prove Corollary 1.1 here.

Proof of Corollary 1.1. (i) For $Z\in {{\cal R}}_{_{III}}$ with $\beta = 2[{n\over 2}]$, by Theorem 1.1, and Proposition 3.1, (3.28) and (3.29) now lead to

and

Therefore,

and

(ii) For $z\in {{\cal R}}_{_{IV}}$, when $n = 1$, it is evident that

When $n\geq 2$,

The proof of Part (ii) follows. □

4.   Conclusions

In this paper, we investigate estimate for $\lambda_1(\Delta_g)$ on the Cartan classical domains of the last two types ${{\cal R}}_{\cal{A}}(\mathcal{A} = III, IV)$. Based on theories of harmonic analysis in the Cartan classical domains from Hua ^[16] and Lu ^[17]. Firstly, We are dedicated to find the plurisubharmonic exhaustion function $u_{\mathcal{A}}$ under Bergman kernel function of ${{\cal R}}_{\cal{A}}$. Next, we define a complete Kähler metric $g_{\mathcal{A}}$ which induced by $u_{\mathcal{A}}$. Through constructing suitable test function $f_{_{{\cal A}}}(Z) = e^{-\tau u_{_{{\cal A}}}(z)},$ we obtain upper and lower bound estimates for $\lambda_1(\Delta_g)$ on ${{\cal R}}_{\mathcal{A}}$ under the assumption that $|\partial u|_{g}^2 < \beta$. In addition, we provide the value of $\beta$ by establishing an estimate for $|\partial u|_{g}^2$. This brings us to give an explicit range for $\lambda_1(\Delta_g)$. Attributed to the special forms of ${{\cal R}}_{III}$ and ${{\cal R}}_{IV}$, the approach examined in this present study requires complicated but interesting technical work.

As shown in our study, we actually propose an approach which may be adapted to solve the problem of finding estimates for $\lambda_1(\Delta_g)$ on other important domains. It is well known that any bounded symmetric domain may be represented as the topological product of irreducible bounded symmetric domains: the class of irreducible bounded symmetric domains consists of four types of Cartan classical domains and two exceptional ones. So we are encouraged to work on the estimates for $\lambda_1(\Delta_g)$ on bounded symmetric domains. This will be the objective of our future study.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by National Natural Science Foundation of China (Grant Nos. 12001259; 12101288), Natural Science Foundation of Fujian Province (Grant No. 2020J01131142), Natural Science Foundation of Guangxi Province (Grant No. 2016GXNSFDA380031), and the Research Foundation of Minjiang University for the Introduction of Talents (Grant No. MJY17006). The authors wish to thank the referees for their helpful comments and suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.