Eigenvalues for Laplacian Operator on Submanifolds in Locally Conformal Kaehler Space Forms

Abstract

This paper investigates totally real submanifolds in a locally conformal Kaehler space form. Using the moving-frame method and constant mean curvature, we obtain the upper and lower bounds of the first eigenvalue for totally real submanifolds in a locally conformal Kaehler space form. We discussed the integral inequalities and their properties. Some previous results are generalized from our results.

1. Introduction and Motivations

Let ${\mathrm{F}}^m$ be a compact Riemannian manifold of dimension m and then Laplacian is a quasilinear elliptic map on ${\mathrm{F}}^m$ given as

$$\begin{array}{c}\hfill \Delta \Phi =\mathrm{div}(\nabla \Phi ).\end{array}$$

The Laplacian has some common characteristics with the classic Laplace operator, which means that we can consider the eigenvalue problem of $\Delta$. An eigenvalue equation for the corresponding Laplace operator is defined as a real number $\Lambda$, which is called an eigenvalue if there is a non-zero function $\Phi$ fulfilling the following

$$\begin{array}{c}\hfill \Delta \Phi =-\Lambda \Phi ,\mathrm{on}{\mathrm{F}}^m\end{array}$$

with appropriate boundary conditions. We used [1,2] as general references for the Laplacian operator. Now, let ${\mathrm{F}}^m$ be a Riemannian manifold without boundary. The first nonzero eigenvalue ${\Lambda }_1$ of $\Delta$ is of a Rayleigh type variational characterization (cf. [3]):

$$\begin{array}{c}\hfill {\Lambda }_1=inf\left\{{\displaystyle \frac{{\int }_{\mathrm{F}}{\parallel \nabla \Phi \parallel }^2}{{\int }_{\mathrm{F}}{\parallel \Phi \parallel }^2}}|\Phi \in W^{1,2}\left({\mathrm{F}}^m\right)\setminus \left\{0\right\},{\int }_{\mathrm{F}}\Phi =0\right\}.\end{array}$$

(1)

In recent years, eigenvalue problems of some elliptic operators such as the usual Laplace operator, the p-Laplacian, the drifting Laplacian, the biharmonic operator, and their weighted versions have been investigated for static Riemannian metrics under no boundary condition or various types of boundary conditions such as the Dirichlet boundary condition, Neumann boundary condition, Robin boundary condition, Navier boundary condition, Steklov boundary condition, Wentzell boundary condition, etc. (see [4,5,6] and the references therein). Motivated by the work of Perelman [7] and Cao [8], research on eigenvalues of the Laplace operator and its deformations, such as p-Laplacian, Witten–Laplacian, and weighted p-Laplacian are discussed. Moreover, various geometric flows such as the Ricci flow, the mean curvature flow, and the Yamabe flows have been studied since geometric curvature flows became tools to obtain canonical metrics and describe the topology of the Riemannian manifolds. Some geometric inequalities, including some classical isoperimetric inequalities, were proved (see [9,10], for instance). For eigenvalue problems under the Yamabe flow, Wu [11] established the evolution of the formula for the first eigenvalue of the Laplace operator under Yamabe flow. Afterward, Zhao [12], studied the monotonicity of the first eigenvalue of the Laplace operator under Yamabe flow provided that the initial manifold is homogeneous. Further, Wang and Zheng [13] proved that the first eigenvalue of the p-Laplace operator and Yamabe invariant are both locally Lipschitz along geometric flows under weak assumptions. Guo et al. [14] constructed Perelman’s F and entropy in abstract geometric flows and obtained the monotonicity of the entropies under a technical condition.

On the other hand, the space form geometry is very popular nowadays. Several authors constructed the first eigenvalues for submanifolds in different space forms such as in C-totally real submanifolds in Sasakian space forms [15], Lagrangian submanifolds in complex space forms [16], slant submanifolds of Sasakian space form [17], semi-slant submanifolds of Sasakian space forms [18] and totally real submanifolds in generalized complex space forms [19] that contain a p-laplacian operator. For more references, see [20,21,22,23,24]. It should be noted that little work has been done on estimating eigenvalues in submanifolds in space form geometry. Therefore, we were motivated by some previous work, constructed the first eigenvalue for a totally real submanifold in locally conformal Kaehler space form, and discussed their consequences in the present paper.

2. Preliminaries

Let $({\tilde{F}}^{2n},\mathrm{J},\tilde{g})$ be a complex n-dimensional Hermitian manifold, where $\mathrm{J}$ denotes its complex structure and g its Hermitian metric. Then, $({\tilde{F}}^{2n},\mathrm{J},\tilde{g})$ is a locally conformal Kaehler (l.c.K.) manifold if there is an open cover ${\left\{{\mathrm{U}}_i\right\}}_{i\in I}$ of ${\tilde{F}}^{2n}$ and a family ${\left\{{\rho }_i\right\}}_{i\in I}$ of ${\mathrm{C}}^{\infty }$ functions ${\rho }_i:{\mathrm{U}}_i⟶\mathbb{R}$ so that each local metric

$$\begin{array}{c}\hfill {\tilde{g}}_i=exp(-{\rho }_i){\tilde{g}}_{|{\mathrm{U}}_i}\end{array}$$

is a Kahlerian metric [25,26,27]. In this case, $g_{|{\mathrm{U}}_i}=i_i^{*}\tilde{g}$, where $i_i:{\mathrm{U}}_i⟶{\tilde{F}}^{2n}$ is the inclusion. We define a fundamental 2-form or Kaehlerian form with respect to almost complex structure $\mathrm{J}$, that is

$$\begin{array}{c}\hfill \Omega ({\mathrm{X}}_2,{\mathrm{Y}}_2)=\tilde{g}(\mathrm{J}{\mathrm{X}}_2,{\mathrm{Y}}_2)\end{array}$$

for any ${\mathrm{X}}_2,{\mathrm{Y}}_2\in \Gamma \left(T\tilde{F}\right).$ If $(\tilde{F},\tilde{g})$ is a locally conformal Kaehler manifold, then there exists a closed 1-form ${\rho }_1$, which is globally defined on $\tilde{F}$

$$\begin{array}{cc}\hfill ({\overline{\nabla }}_{{\mathrm{Z}}_2}\Omega )({\mathrm{X}}_2,{\mathrm{Y}}_2)=& {\rho }_1\left({\mathrm{Y}}_2\right)g({\mathrm{X}}_2,{\mathrm{Z}}_2)-{\rho }_1\left({\mathrm{X}}_2\right)g({\mathrm{Z}}_2,{\mathrm{Y}}_2)\hfill \\ \hfill & +{\rho }_2\left({\mathrm{Y}}_2\right)\Omega ({\mathrm{X}}_2,{\mathrm{Z}}_2)-{\rho }_2\left({\mathrm{Y}}_2\right)\Omega ({\mathrm{X}}_2,{\mathrm{Z}}_2)\hfill \end{array}$$

provided that ${\rho }_1\left({\mathrm{X}}_2\right)=-{\rho }_2\left(\mathrm{J}{\mathrm{X}}_2\right)$, where ${\rho }_1$ is a 1-form and $\overline{\nabla }$ is a Levi–Cita connection with respect to g. Moreover, 1-form ${\rho }_2$ is called the Lee form, and its dual vector field is the Lee vector field. Now, we define symmetric $(0,2)$ tensor $\mathrm{P}$ on a locally conformal Kaehler manifold as:

$$\begin{array}{c}\hfill \mathrm{P}({\mathrm{X}}_2,{\mathrm{Y}}_2)=-\left({\overline{\nabla }}_{{\mathrm{X}}_2}{\rho }_2\right){\mathrm{Y}}_2-{\rho }_2\left({\mathrm{X}}_2\right){\rho }_2\left({\mathrm{Y}}_2\right)+{\displaystyle \frac{1}{2}}{\parallel {\rho }_2\parallel }^{2}g({\mathrm{X}}_2,{\mathrm{Y}}_2).\end{array}$$

where $\parallel {\rho }_2\parallel$ is the norm of ${\rho }_2$ with respect to g. Similarly, skew-symmetric $(0,2)$ tensor $\overline{\mathrm{P}}$ is constructed as

$$\begin{array}{c}\hfill \overline{\mathrm{P}}({\mathrm{X}}_2,{\mathrm{Y}}_2)=\mathrm{P}(\mathrm{J}{\mathrm{X}}_2,{\mathrm{Y}}_2).\end{array}$$

If a locally conformal Kaehler manifold $(\tilde{\mathrm{F}},\tilde{g})$ has the holomorphic constant section curvature $\vartheta$, then it is called locally conformal Kaehler space form and it is denoted $\tilde{\mathrm{F}}\left(\vartheta \right)$. Therefore, curvature tensor $\mathrm{R}$ of $\tilde{\mathrm{F}}\left(\vartheta \right)$ is defined as

$$\begin{array}{cc}\hfill \mathrm{R}({\mathrm{X}}_2,{\mathrm{Y}}_2){\mathrm{Z}}_2=& {\displaystyle \frac{\vartheta }{4}}\left\{g({\mathrm{Z}}_2,{\mathrm{Y}}_2){\mathrm{X}}_2-g({\mathrm{X}}_2,{\mathrm{Z}}_2){\mathrm{Y}}_2+\Omega ({\mathrm{Z}}_2,{\mathrm{Y}}_2)\mathrm{J}{\mathrm{X}}_2-\Omega ({\mathrm{Y}}_2,{\mathrm{Z}}_2)\mathrm{J}{\mathrm{Y}}_2-2\Omega ({\mathrm{X}}_2,{\mathrm{Y}}_2)\mathrm{J}{\mathrm{Z}}_2\right\}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}\left\{g({\mathrm{Y}}_2,{\mathrm{Z}}_2){\mathrm{P}}_1{\mathrm{X}}_2-g({\mathrm{X}}_2,{\mathrm{Z}}_2){\mathrm{P}}_1{\mathrm{Y}}_2+\mathrm{P}({\mathrm{Y}}_2,{\mathrm{Z}}_2){\mathrm{X}}_2-\mathrm{P}({\mathrm{X}}_2,{\mathrm{Z}}_2){\mathrm{Y}}_2\right\}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\{\Omega ({\mathrm{Z}}_2,{\mathrm{Y}}_2){\tilde{\mathrm{P}}}_1{\mathrm{X}}_2-\Omega ({\mathrm{Y}}_2,{\mathrm{Z}}_2){\tilde{\mathrm{P}}}_1{\mathrm{Y}}_2+\tilde{\mathrm{P}}({\mathrm{Y}}_2,{\mathrm{Z}}_2)\mathrm{J}{\mathrm{X}}_2-\tilde{\mathrm{P}}({\mathrm{X}}_2,{\mathrm{Z}}_2)\mathrm{J}{\mathrm{Y}}_2\hfill \\ \hfill & -2\tilde{\mathrm{P}}({\mathrm{X}}_2,{\mathrm{Y}}_2)\mathrm{J}{\mathrm{Z}}_2-2\Omega ({\mathrm{X}}_2,{\mathrm{Y}}_2){\tilde{\mathrm{P}}}_1{\mathrm{Z}}_2\}\hfill \end{array}$$

(2)

provided that $g({\mathrm{P}}_1{\mathrm{X}}_2,{\mathrm{Y}}_2)=\mathrm{P}({\mathrm{X}}_2,{\mathrm{Y}}_2)$ and $g({\tilde{\mathrm{P}}}_1{\mathrm{X}}_2,{\mathrm{Y}}_2)=\tilde{\mathrm{P}}({\mathrm{X}}_2,{\mathrm{Y}}_2)$ for any ${\mathrm{X}}_2,{\mathrm{Y}}_2\in \Gamma \left(T\tilde{\mathrm{F}}\left(\vartheta \right)\right)$. Let $\mathrm{F}$ be a submanifold of an almost Hermitian manifold $\tilde{\mathrm{F}}\left(\vartheta \right)$ with induced metric g and if ∇ and ${\nabla }^{\perp }$ are induced connections on the tangent bundle $T\mathrm{F}$ and normal bundle $T^{\perp }\mathrm{F}$ of $\mathrm{F}$, respectively. Then, the Gauss and Weingarten formulas are given by

$$\begin{array}{c}\hfill \left(i\right){\tilde{\nabla }}_{{\mathrm{X}}_2}{\mathrm{Y}}_2={\nabla }_{{\mathrm{X}}_2}{\mathrm{Y}}_2+\sigma ({\mathrm{X}}_2,{\mathrm{Y}}_2),\left(ii\right){\tilde{\nabla }}_{{\mathrm{X}}_2}{\mathrm{Z}}_2^{\perp }=-{\mathbb{A}}_{{\mathrm{Z}}_2^{\perp }}{\mathrm{X}}_2+{\nabla }_{{\mathrm{X}}_2}^{\perp }{\mathrm{Z}}_2^{\perp }\end{array}$$

for each ${\mathrm{X}}_2,{\mathrm{Y}}_2\in \Gamma \left(T\mathrm{F}\right)$ and $N\in \Gamma \left(T^{\perp }\mathrm{F}\right)$, where $\sigma$ and ${\mathbb{A}}_{{\mathrm{Z}}_2^{\perp }}$ are the second fundamental form and shape operator (corresponding to the normal vector field ${\mathrm{Z}}_2^{\perp }$), respectively, for the immersion of $\mathrm{F}$ into $\tilde{\mathrm{F}}$. They are related as:

$$\begin{array}{c}\hfill g(\sigma ({\mathrm{X}}_2,{\mathrm{Y}}_2),{\mathrm{Z}}_2^{\perp })=g({\mathbb{A}}_{{\mathrm{Z}}_2^{\perp }}{\mathrm{X}}_2,{\mathrm{Y}}_2)\end{array}$$

for any ${\mathrm{X}}_2\in \Gamma \left(T\mathrm{F}\right)$. If the almost complex structure $\mathrm{J}$ caries each tangent space of a submanifold $\mathrm{F}$ of a locally conformal Kaehler manifold $\tilde{\mathrm{F}}$ into its normal space, then the submanifold $\mathrm{F}$ is called a totally real submanifold of $\tilde{\mathrm{F}}$ [28]. In this case, $\Omega ({\mathrm{X}}_2,{\mathrm{Y}}_2)=0$ for each ${\mathrm{X}}_2,{\mathrm{Y}}_2$ tangent to $\mathrm{F}$. Then (2) is reduced to the following

$$\begin{array}{cc}\hfill g(\mathrm{R}({\mathrm{X}}_2,{\mathrm{Y}}_2){\mathrm{Z}}_2,{\mathrm{W}}_2)& ={\displaystyle \frac{\vartheta }{4}}\left\{g({\mathrm{Z}}_2,{\mathrm{Y}}_2)g({\mathrm{X}}_2,{\mathrm{W}}_2)-g({\mathrm{X}}_2,{\mathrm{Z}}_2)g({\mathrm{Y}}_2,{\mathrm{W}}_2)\right\}\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}\{g({\mathrm{X}}_2,{\mathrm{W}}_2)\mathrm{P}({\mathrm{Y}}_2,{\mathrm{Z}}_2)-g({\mathrm{Y}}_2,{\mathrm{W}}_2)\mathrm{P}({\mathrm{X}}_2,{\mathrm{Z}}_2)\hfill \\ \hfill & +\mathrm{P}({\mathrm{X}}_2,{\mathrm{W}}_2)g({\mathrm{Y}}_2,{\mathrm{Z}}_2)-\mathrm{P}({\mathrm{Y}}_2,{\mathrm{W}}_2)g({\mathrm{X}}_2,{\mathrm{Z}}_2)\}\hfill \end{array}$$

(3)

for any ${\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2\in \Gamma \left(T\mathrm{F}\right).$ The equation of Gaus is defined for curvature tensors $\tilde{\mathrm{R}}$ and $\mathrm{R}$ of ${\mathrm{F}}^m$ and $\tilde{\mathrm{F}}$, respectively

$$\begin{array}{cc}\hfill \tilde{\mathrm{R}}({\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2)=& \mathrm{R}({\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2)+\tilde{g}(\sigma ({\mathrm{X}}_2,{\mathrm{W}}_2),\sigma ({\mathrm{Y}}_2,{\mathrm{Z}}_2))\hfill \\ \hfill & -\tilde{g}(\sigma ({\mathrm{X}}_2,{\mathrm{Z}}_2),\sigma ({\mathrm{Y}}_2,{\mathrm{W}}_2)).\hfill \end{array}$$

(4)

We considered an orthonormal frame $\{e_1\cdots e_m,e_{m+1}\cdots e_{m+h},e_1^{*}=\mathrm{J}{e}_1\cdots e_m^{*}=\mathrm{J}{e}_m,e_{{(m+1)}^{*}}=\mathrm{J}{e}_{m+1}\cdots e_{{(m+h)}^{*}}=\mathrm{J}{e}_{m+h}\}$ in ${\tilde{\mathrm{F}}}^{m+h}\left(\vartheta \right)$ such that restricted to ${\mathrm{F}}^m$, $e_1\cdots e_m$ is tangent to ${\mathrm{F}}^m$. We provided the indices as follows

$$\begin{array}{cc}\hfill \mathcal{A},\mathcal{B},\mathcal{C}\cdots =& 1,\cdots ,m+h,1^{*}\cdots ,m+h^{*}\hfill \\ \hfill a,b,c\cdots =& 1,\cdots ,m;a^{*},b^{*},c^{*}=m+1^{*},\cdots ,m+h,1^{*},\cdots ,m+h^{*}.\hfill \end{array}$$

Equation (3) expressed in term local coordinates

$$\begin{array}{cc}\hfill {\tilde{\mathrm{K}}}_{\mathcal{A}\mathcal{B}\mathcal{C}\mathcal{D}}& ={\displaystyle \frac{\vartheta }{4}}\left({\delta }_{\mathcal{A}\mathcal{C}}{\delta }_{\mathcal{B}\mathcal{D}}-{\delta }_{\mathcal{A}\mathcal{D}}{\delta }_{\mathcal{B}\mathcal{C}}\right)\hfill \\ \hfill & +{\displaystyle \frac{3}{4}}\left({\delta }_{\mathcal{A}\mathcal{D}}{\mathrm{P}}_{\mathcal{B}\mathcal{C}}-{\delta }_{\mathcal{B}\mathcal{D}}{\mathrm{P}}_{\mathcal{A}\mathcal{C}}+{\mathrm{P}}_{\mathcal{A}\mathcal{D}}{\delta }_{\mathcal{B}\mathcal{C}}-{\mathrm{P}}_{\mathcal{B}\mathcal{D}}{\delta }_{\mathcal{A}\mathcal{C}}\right)\hfill \end{array}$$

(5)

where $\tilde{\mathrm{K}}$ is the sectional curvature of ${\tilde{\mathrm{F}}}^{m+h}\left(\vartheta \right)$. Let $\zeta$ denote the squared length of the second fundamental form $\sigma$ of ${\mathrm{F}}^m$, defined by

$$\begin{array}{c}\hfill \zeta =\sum _{abk}{\left({\sigma }_{ab}^k\right)}^2.\end{array}$$

(6)

Similarly, the mean curvature of ${\mathrm{F}}^m$ is calculated as:

$$\begin{array}{c}\hfill \mathbb{H}={\displaystyle \frac{1}{m}}\sum _{ak}{\sigma }_{aa}^k{e}_k\end{array}$$

(7)

The curvature tensor of indices for submanifold

$$\begin{array}{c}\hfill {\mathrm{R}}_{abcl}={\tilde{\mathrm{K}}}_{abcl}+\sum _{\alpha }\left({\sigma }_{ac}^{\alpha }{\sigma }_{bl}^{\alpha }-{\sigma }_{al}^{\alpha }{\sigma }_{bc}^{\alpha }\right).\end{array}$$

(8)

The Ricci curvature for a totally real submanifold, we define as:

$$\begin{array}{cc}\hfill {\mathrm{R}}_{ab}=& {\displaystyle \frac{\vartheta }{4}}(m-1){\delta }_{ab}+{\displaystyle \frac{3}{4}}(m-2){\mathrm{P}}_{ab}+{\displaystyle \frac{3}{4}}trace(\mathrm{P}/\mathrm{F})\hfill \\ \hfill & +\sum _a\left({\sigma }_{ab}^k\sum _c{\sigma }_{cc}^k-\sum _c{\sigma }_{ac}^k{\sigma }_{cb}^k\right).\hfill \end{array}$$

(9)

From the above, we fix some notations

$$\begin{array}{c}\hfill \Pi ={\parallel \sigma \parallel }^2,\mathbb{H}=\left|\xi \right|,{\mathbb{H}}_{\alpha }={\left({\sigma }_{ab}^{\alpha }\right)}_{m\times m}.\end{array}$$

(10)

Let us assume that $e_{m+1}$ is parallel to $\mathbb{H}$, then we have

$$\begin{array}{c}\hfill tr{\mathbb{H}}_{m+1}=m\mathbb{H},{\mathbb{H}}_{\alpha }=0,\alpha \ne m+1\end{array}$$

(11)

where tr stands for the trace of the matrix ${\mathbb{H}}_{\alpha }=\left({\sigma }_{ab}^{\alpha }\right).$ Taking account of (5), (8) and (11), we have scalar curvature as

$$\begin{array}{c}\hfill \mathrm{R}={\displaystyle \frac{1}{4}}m(m-1)\left(\vartheta +6\mu \right)+m^2{\mathbb{H}}^2-\zeta \end{array}$$

(12)

where $\mathbb{H}$ is the mean curvature vector of ${\mathrm{F}}^m$ and $\mu =\frac{1}{m}{\sum }_{i=1}^m\mathrm{P}(e_i,e_i)$.

Since $\mathbb{H}$ is constant. It can be concluded that the scalar curvature $\mathcal{R}$ is constant if and only if $\zeta$ is constant by (12). Let ${\sigma }_{abc}^k$ denote the second covariant derivative of ${\sigma }_{ab}^k$, we have

$$\begin{array}{c}\hfill \sum _c{\sigma }_{abc}^{\alpha }{\omega }_c=\mathrm{F}{\sigma }_{ab}^{\alpha }-\sum _c{\sigma }_{cb}^{\alpha }{\omega }_{ca}-\sum _c{\sigma }_{ic}^{\alpha }{\omega }_{cb}-\sum _t{\sigma }_{ab}^{\alpha }{\omega }_{tk}\end{array}$$

where $\left\{{\omega }_a\right\}$ is the dual frame of ${\mathrm{F}}^m$. Taking the exterior derivative of the above equation, we obtain

$$\begin{array}{cc}\hfill \sum _l{\sigma }_{abcl}^{\alpha }{\omega }_l=& \mathrm{d}{\sigma }_{abc}^{\alpha }-\sum _l{\sigma }_{abc}^{\alpha }{\omega }_{la}-\sum _t{\sigma }_{abc}^t{\omega }_{tk}\hfill \\ \hfill & -\sum _l{\sigma }_{alc}^{\alpha }{\omega }_{ab}-\sum _l{\sigma }_{abl}^{\alpha }{\omega }_{lc}.\hfill \end{array}$$

Moreover, the Laplacian function $\Phi$ is defined by $\Delta \Phi =-{\sum }_c{\Phi }_{cc}$. Since ${\mathrm{F}}^m$ is minimal, we have the following

$$\begin{array}{cc}\hfill \sum _{a\ast b\ast }\left(tr{\mathbb{H}}_{a\ast }{\mathbb{H}}_{b\ast }\right)=& {\parallel \nabla \sigma \parallel }^2+(n+1)\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)\zeta +\sum _{a\ast b\ast }{\left(tr{\mathbb{H}}_{a\ast }{\mathbb{H}}_{b\ast }-{\mathbb{H}}_{b\ast }{\mathbb{H}}_{a\ast }\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\Delta \mathcal{A}\hfill \end{array}$$

(13)

by the same calculation as given in [29]. We need the following to prove our main result:

Lemma 1 

([30]). Let $\left(f_{ab}\right)$, $a,b=1\cdots m$ be a symmetric $(m\times m)$ matrix and $m\ge 2$. Assume that trace $\left(f_{ab}\right)=T,\zeta ={\sum }_{ab}{\left(h_{ab}\right)}^2$. Then we have

$$\begin{array}{cc}\hfill \sum _a{\left(f_{am}\right)}^2-T{f}_{mn}& \le \{{\displaystyle \frac{(m-1)\zeta }{m}}+{\displaystyle \frac{(m-2)}{m}}\parallel T\parallel \sqrt{{\displaystyle \frac{(m-1)}{m}}\left(\zeta -{\displaystyle \frac{T^2}{m}}\right)}\hfill \\ \hfill & -{\displaystyle \frac{2(m-1)}{m^2}}\parallel T^2\parallel \}.\hfill \end{array}$$

(14)

Now we are in a position to estimate our first main result, which is stated as follows.

Theorem 1. 

Let ${\mathrm{F}}^m$ be an m-dimensional compact totally real submanifold in locally conformal Kaehler space form $\tilde{\mathrm{F}}{\left(\vartheta \right)}^{m+h}$ with constant mean curvature and scalar curvature. Then, we have

$$\begin{array}{c}\hfill {\Lambda }_1\ge \left\{m\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)-\zeta +2m{\mathbb{H}}^2-(m-2)\parallel \mathbb{H}\parallel \sqrt{{\displaystyle \frac{m}{m-1}}(\zeta -m{\mathbb{H}}^2)}\right\}\end{array}$$

(15)

where $\mathbb{H}$ and ζ denote the mean curvature of ${\mathrm{F}}^m$ and the squared of the length of the second fundamental form of ${\mathrm{F}}^m$, respectively.

Proof. 

Assume that $\Phi$ is an eigenfunction of the Laplacian $\Delta$ on totally real submanifold ${\mathrm{F}}^m$ first eigenvalue corresponding to an eigenvalue ${\Lambda }_1$, i.e, $\Delta \Phi ={\Lambda }_1\Phi$. If $\mathrm{d}{\Phi }^{*}={\sum }_a{\Phi }_a{e}_a,{\Phi }_a={\nabla }_a\Phi$, we have the following formula from [31], that is

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\Delta \left(\right|\mathrm{d}\Phi |}^2)=-\sum _{ab}{\left({\Phi }_{ab}\right)}^2+{\Lambda }_1\sum _a{\left({\Phi }_a\right)}^2-\mathrm{Ric}(\mathrm{d}{\Phi }^{*},\mathrm{d}{\Phi }^{*}).\end{array}$$

(16)

Using the same terminology as given in [30] for any unit vector v on submanifold ${\mathrm{F}}^m$ and implementation of Lemma 1, we obtain the following equation

$$\begin{array}{cc}\hfill \mathrm{Ric}(v,v)& \ge (m-1)\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)-\left({\displaystyle \frac{m-1}{m}}\right)\zeta -\left(m-2\right)\parallel \mathbb{H}\parallel \sqrt{\left({\displaystyle \frac{m-1}{m}}\right)\left(\zeta -m{\mathbb{H}}^2\right)}\hfill \\ \hfill & +2\left(m-1\right){\mathbb{H}}^2.\hfill \end{array}$$

If we choose $v=\mathrm{d}{\Phi }^{*}$ in the proceeding equation, we have

$$\begin{array}{cc}\hfill \mathrm{Ric}(\mathrm{F}{\Phi }^{*},\mathrm{F}{\Phi }^{*}) \ge & \{(m-1)\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)-\left({\displaystyle \frac{m-1}{m}}\right)\zeta +2\left(m-1\right){\mathbb{H}}^2\hfill \\ \hfill & -\left(m-2\right)\parallel \mathbb{H}\parallel \sqrt{\left({\displaystyle \frac{m-1}{m}}\right)\left(\zeta -m{\mathbb{H}}^2\right)}|\}\parallel \mathrm{d}{\Phi }^{*}{\parallel }^2.\hfill \end{array}$$

(17)

Since we proceed

$$\begin{array}{c}\hfill \sum _{ab}{\left({\Phi }_{ab}\right)}^2\ge \sum _a{\left({\Phi }_a\right)}^2\ge {\displaystyle \frac{1}{m}}{\left(\sum _{ab}{\left({\Phi }_{ab}\right)}^2\right)}^2={\displaystyle \frac{{\Lambda }_1^2}{m}}{(\Phi )}^2.\end{array}$$

Taking integration along compact boundaries, we have

$$\begin{array}{c}\hfill \int \sum _{ab}{\left({\Phi }_{ab}\right)}^2 \ge {\displaystyle \frac{\Lambda }{m}}\int {\parallel \mathrm{d}\Phi \parallel }^2,\mathit{as}{\displaystyle \frac{{\Lambda }^2}{m}}{(\Phi )}^2\ge {\displaystyle \frac{\Lambda }{m}}\int {\parallel \mathrm{d}\Phi \parallel }^2.\end{array}$$

(18)

Inserting (17) and (18) into (16), we derive

$$\begin{array}{cc}\hfill \int \{\left({\displaystyle \frac{m-1}{m}}\right)\left(\Lambda +\zeta +m\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)\right)& +(m-2)\parallel \mathbb{H}\parallel \sqrt{\left({\displaystyle \frac{m-1}{m}}\right)\left(\zeta -m{\mathbb{H}}^2\right)}\hfill \\ \hfill & \begin{array}{c} \\ \end{array}-2(m-1){\parallel \mathbb{H}\parallel }^2\}{\parallel \mathrm{d}{\Phi }^{*}\parallel }^2\ge 0.\hfill \end{array}$$

(19)

It is concluded that for $m\ge 2$ and from (19) with $\zeta$ is constant, then we have

$$\begin{array}{c}\hfill {\Lambda }_1\ge m\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)-\zeta +2m{\mathbb{H}}^2-(m-2)\parallel \mathbb{H}\parallel \sqrt{{\displaystyle \frac{m}{m-1}}(\zeta -m{\mathbb{H}}^2)}\end{array}$$

where ${\Lambda }_1$ is the first non-zero eigenvalue of ${\mathrm{F}}^m$. This is the complete proof of a theorem. □

If the mean curvature is minimal, then we have the following result from Theorem 1

Corollary 1. 

Let ${\mathrm{F}}^m$ be an m-dimensional compact minimal totally real submanifold in locally conformal Kaehler space form $\tilde{\mathrm{F}}{\left(\vartheta \right)}^{m+h}$ with constant mean curvature and scalar curvature, and then, we obtain

$$\begin{array}{c}\hfill {\Lambda }_1\ge \left\{m\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)-\zeta \right\}.\end{array}$$

In terms of scalar curvature, we have

Corollary 2. 

Let ${\mathrm{F}}^m$ be an m-dimensional compact minimal totally real submanifold in locally conformal Kaehler space form $\tilde{\mathrm{F}}{\left(\vartheta \right)}^{m+h}$ with constant mean curvature and scalar curvature, and then, the following inequality is satisfied

$$\begin{array}{c}\hfill {\Lambda }_1\ge \left\{\mathrm{R}-m(m-1)\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)\right\}.\end{array}$$

For the next result, we need the following lemma to be proved

Lemma 2. 

Let ${\mathrm{F}}^m$ be an m-dimensional submanifold in an $(m+h)$-dimensional manifold ${\tilde{\mathcal{N}}}^{m+h}$ with constant mean curvature, we have

$$\begin{array}{c}\hfill {\parallel \nabla \zeta \parallel }^2\le \left({\displaystyle \frac{4mh}{m+2}}\right)\zeta {\parallel \nabla \sigma \parallel }^2.\end{array}$$

(20)

where h is codimension.

Proof. 

For any $x\in {\mathrm{F}}^m$ and fixed index $\alpha (1\le \alpha \le h)$. Therefore, we can consider of orthonormal frame ${\sigma }_{ab}^{\alpha }=0$ at x, we obtain

$$\begin{array}{cc}\hfill \sum _c{\left(\sum _{ab}{\sigma }_{ab}^k{\sigma }_{abc}^k\right)}^2=& \sum _c{\left(\sum _a{\sigma }_{aa}^k{\sigma }_{aac}^k\right)}^2\hfill \\ \hfill \le & \sum _c\left(\sum _a{\left({\sigma }_{aa}^k\right)}^2\sum _a{\left({\sigma }_{aac}^k\right)}^2\right)\hfill \\ \hfill =& \sum _{ab}{\left({\sigma }_{ab}^k\right)}^2\sum _{ac}{\left({\sigma }_{aac}^k\right)}^2.\hfill \end{array}$$

(21)

Next, we solve the following terms:

$$\begin{array}{cc}\hfill \sum _{abc}{\left({\sigma }_{abc}^k\right)}^2& \ge 3\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2+\sum _a{\left({\sigma }_{aa}^k\right)}^2\hfill \\ \hfill & =2\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2+\sum _{ac}{\left({\sigma }_{aac}^k\right)}^2.\hfill \end{array}$$

(22)

We can obtain the following for constant mean curvature ${\mathrm{F}}^m$ for fixed index c

$$\begin{array}{cc}\hfill \sum _a{\left({\sigma }_{aac}^k\right)}^2& =\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2+{\left(\sum _{a\ne c}\left({\sigma }_{aac}^k\right)\right)}^2\hfill \\ \hfill & \le \sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2+(m-1)\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2\hfill \\ \hfill & =m\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2.\hfill \end{array}$$

Summing for c, we derive

$$\begin{array}{c}\hfill \sum _{ac}{\left({\sigma }_{aac}^k\right)}^2\le m\sum _{a\ne c}{\left({\sigma }_{aac}^k\right)}^2.\end{array}$$

(23)

In view of (22) and (23), we calculate

$$\begin{array}{c}\hfill \sum _{abc}{\left({\sigma }_{abc}^k\right)}^2\ge \left({\displaystyle \frac{m+2}{m}}\right)\sum _{ac}{\left({\sigma }_{aac}^k\right)}^2.\end{array}$$

(24)

Inserting (24) into (21), we arrive at

$$\begin{array}{cc}\hfill \sum _c{\left(\sum _{ab}{\sigma }_{ab}^k{\sigma }_{aac}^k\right)}^2& \le \left({\displaystyle \frac{m+2}{m}}\right)\sum _{ab}{\left({\sigma }_{ab}^k\right)}^2\sum _{abc}{\left({\sigma }_{abc}^k\right)}^2\hfill \\ \hfill & \le \zeta \sum _{abc}{\left({\sigma }_{abc}^k\right)}^2.\hfill \end{array}$$

(25)

Summing for $\alpha$ in (25), we have

$$\begin{array}{c}\hfill \sum _{ck}{\left(\sum _{ab}{\sigma }_{ab}^k{\sigma }_{aac}^k\right)}^2\le \left({\displaystyle \frac{m+2}{m}}\right)\zeta \sum _{abck}{\left({\sigma }_{abc}^k\right)}^2.\end{array}$$

On the other hand, we obtain

$$\begin{array}{cc}\hfill \sum _c{\left(\sum _{abk}{\sigma }_{ab}^k{\sigma }_{aac}^k\right)}^2& \le \sum _c\left\{\sum _k{1}^2.\sum _k{\left(\sum _{ab}{\sigma }_{ab}^k{\sigma }_{aac}^k\right)}^2\right\}\hfill \\ \hfill & =h\sum _{ck}{\left(\sum _{ab}{\sigma }_{ab}^k{\sigma }_{aac}^k\right)}^2.\hfill \end{array}$$

$$\begin{array}{c}\hfill \sum _c{\left(\sum _{abck}{\sigma }_{ab}^k{\sigma }_{abc}^k\right)}^2\le \left({\displaystyle \frac{mh}{m+2}}\right)\zeta \sum _{abck}{\left({\sigma }_{abc}^k\right)}^2\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\parallel \nabla \zeta \parallel }^2\le \left({\displaystyle \frac{4mh}{m+2}}\right)\zeta {\parallel \nabla \sigma \parallel }^2.\end{array}$$

This is the complete proof of lemma. □

Lemma 3. 

Let ${\mathrm{F}}^m$ be an m-dimensional submanifold in an $(m+h)$-dimensional manifold ${\tilde{\mathcal{M}}}^{m+h}$ with constant mean curvature. Then, either the first non-zero eigenvalue ${\Lambda }_1$ of $\Delta$ satisfies the following inequality

$$\begin{array}{c}\hfill {\Lambda }_1\int \zeta \le \left({\displaystyle \frac{mh}{m+2}}\right)\int {\parallel \nabla \sigma \parallel }^2\end{array}$$

(26)

where ζ is the squared norm of the second fundamental form or ${\mathrm{F}}^m$ is totally geodesic.

Proof. 

From the definition of eigenvalue property for $\Phi \ne 0$, we have

$$\begin{array}{c}\hfill {\Lambda }_1\int {\Phi }^2\le \int \Phi \nabla \Phi .\end{array}$$

(27)

Assuming that ${\Phi }_{\xi }={(\mathcal{A}+\xi )}^{1/2}$ for $\xi >0,$ then it is simplified that

$$\begin{array}{cc}\hfill \Delta {\Phi }_{\xi }=& -\sum \nabla \nabla {\Phi }_{\xi }=\sum \nabla \nabla {(\zeta +\xi )}^{1/2}\hfill \\ \hfill =& -{(\zeta +\xi )}^{-1/2}\left(\sum _{abck}{\left({\sigma }_{abc}^k\right)}^2-\sum _{abk}{\sigma }_{ab}^k\Delta {\sigma }_{ab}^k\right)\hfill \\ \hfill & +{(\zeta +\xi )}^{-3/2}\sum _c{\left(\sum _{abk}{\sigma }_{ab}^k{\sigma }_{abc}^k\right)}^2.\hfill \end{array}$$

From Lemma 2, we obtain

$$\begin{array}{c}\hfill \Delta {\Phi }_{\xi }\le {\displaystyle \frac{1}{2}}{(\zeta +\xi )}^{-1/2}\Delta \zeta +{\displaystyle \frac{mh}{m+2}}{(\mathcal{A}+\xi )}^{-3/2}\mathcal{A}{\parallel \nabla \sigma \parallel }^2\\ \hfill \le {\displaystyle \frac{1}{2}}{(\zeta +\xi )}^{-1/2}\Delta \zeta +{\displaystyle \frac{mh}{m+2}}{(\zeta +\xi )}^{-1/2}\zeta {\parallel \nabla \sigma \parallel }^2.\end{array}$$

(28)

Combining (27) and (28), we establish the following

$$\begin{array}{cc}\hfill {\Lambda }_1& \le \left\{{\displaystyle \frac{\int {\Phi }_{\xi }\Delta {\Phi }_{\xi }}{\int {\Phi }_{\xi }^2}}\right\}\hfill \\ \hfill & \le \left\{{\displaystyle \frac{\int \left(\frac{1}{2}\Delta \zeta +\frac{mh}{m+2}{\parallel \nabla \sigma \parallel }^2\right)}{\int (\zeta +\xi )}}\right\}\hfill \\ \hfill & \le \left\{\left({\displaystyle \frac{mh}{m+2}}\right){\displaystyle \frac{{\int \parallel \nabla \sigma \parallel }^2}{\int (\zeta +\xi )}}\right\}.\hfill \end{array}$$

If ${\mathrm{F}}^m$ is not totally geodesic, then it implies that

$$\begin{array}{c}\hfill {\Lambda }_1\le \left\{\left({\displaystyle \frac{mh}{m+2}}\right){\displaystyle \frac{\int {\parallel \nabla \sigma \parallel }^2}{\int (\zeta +\xi )}}\right\}.\end{array}$$

This completes the proof of the lemma. □

Following from the same concept as in [32], we obtain

Lemma 4. 

Let ${\mathrm{F}}^m$ be an m-dimensional submanifold in an $(m+h)$-dimensional manifold ${\tilde{\mathcal{M}}}^{m+h}$ with constant mean curvature; we have inequality

$$\begin{array}{c}\hfill {\parallel \nabla \sigma \parallel }^2\le \left\{{\displaystyle \frac{3}{2}}{\zeta }^2-{\displaystyle \frac{1}{2}}\Delta \zeta -{\displaystyle \frac{(m+1)}{4}}\zeta \right\}.\end{array}$$

(29)

Now, we establish the second most important result:

Theorem 2. 

Let ${\mathrm{F}}^m$ be an m-dimensional compact totally real submanifold with constant mean curvature in locally conformal Kaehler space form $\tilde{\mathrm{F}}{\left(\vartheta \right)}^{m+h}$; then, either the following inequality holds,

$$\begin{array}{c}\hfill {\Lambda }_1\le \left\{{\displaystyle \frac{\zeta }{2(m+2)}}-{\displaystyle \frac{m^2(m+1)}{4(m+2)}}\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)\right\}\end{array}$$

(30)

where $\mathcal{A}$ is squared norm of the second fundamental form or ${\mathrm{F}}^m$ is totally geodesic.

Proof. 

Taking account of Lemmas 3 and 4, we obtain the following inequality

$$\begin{array}{c}\hfill {\Lambda }_1\le \left\{{\displaystyle \frac{\zeta }{2(m+2)}}-{\displaystyle \frac{m^2(m+1)}{4(m+2)}}\left({\displaystyle \frac{\vartheta +6\mu }{4}}\right)\right\}\end{array}$$

(31)

provided that ${\mathrm{F}}^m$ is not totally geodesic submanifold in ${\tilde{\mathrm{F}}}^n({\displaystyle \frac{\vartheta +6\mu }{4}},{\beta }_2)$. This completes the proof of the theorem. □

Remark 1. 

It is concluded that the Theorems 1 and 2 are the generalizations of Theorems 1 and 2 in [33].