Bounds for Eigenvalues of q-Laplacian on Contact Submanifolds of Sasakian Space Forms

Abstract

This study establishes new upper bounds for the mean curvature and constant sectional curvature on Riemannian manifolds for the first positive eigenvalue of the q-Laplacian. In particular, various estimates are provided for the first eigenvalue of the q-Laplace operator on closed orientated $(l+1)$-dimensional special contact slant submanifolds in a Sasakian space form, ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$, with a constant ${\psi }_1$-sectional curvature, $ϵ$. From our main results, we recovered the Reilly-type inequalities, which were proven before this study.

1. Introduction and Statement of Main Results

Let $\mathit{M}$ be a compact Riemannian manifold. Then, the spectrum of the Dirichlet problem for $\mathit{M}$ is the set of real numbers $\Lambda$ such that

$$\begin{array}{c}\Delta \sigma =-\Lambda \sigma ,\hfill \end{array}$$

where the boundary condition ${\sigma }_{\partial \mathit{M}}=0$ has a non-trivial solution. The constant $\Lambda$ is called an eigenvalue and $\sigma$ is called an eigenfunction corresponding to $\Lambda$ (for more details, see [1]). The set of eigenvalues of the Laplacian is an increasing sequence, i.e., ${\Lambda }_1<{\Lambda }_2<{\Lambda }_3\cdots {\Lambda }_{n+1}$, where ${\Lambda }_1,{\Lambda }_2,\cdots ,{\Lambda }_{n+1}$ are the eigenvalues of $\Delta$. A significant part of the spectral differential geometry is estimating the first eigenvalue. One of the primary tools in the study of the Dirichlet eigenvalues is the max–min principle, and the first eigenvalue minimizes the Dirichlet energy. Let $\mathit{M}$ be an l-dimensional Riemannian manifold; then, the first eigenvalue of $\Delta$, including its variational properties, is characterized by

$$\begin{array}{c}{\lambda }_1=inf\left\{\frac{{\int }_{\mathit{M}}{\parallel \nabla \sigma \parallel }^{2}dV}{{\int }_{\mathit{M}}{\parallel \sigma \parallel }^{2}dV}:\sigma \in W^1,2\left(\mathit{M}\right),{\int }_{\mathit{M}}\sigma dV=0\right\}.\hfill \end{array}$$

It is crucial to determine the different bounds for the Laplacian eigenvalue on a particular manifold in Riemannian geometry. We are interested in examining eigenvalues that show up as solutions to the Dirichlet or Neumann boundary value problems for curvature functions. Dirichlet’s perspective on the Laplacian shows that determining the upper bound of the eigenvalue is a method used to locate the proper bound for a particular manifold when there are a variety of boundary conditions. It has become more popular in recent years to obtain an eigenvalue for the q-Laplace and the Laplace operations. If the first eigenvalue on $\Sigma$ with a Dirichlet boundary is given by ${\Lambda }_1(\Sigma )>0$, where $\Sigma$ is a compact domain in the complete non-compact Riemannian manifold, ${\mathit{M}}^l$, then we have

$$\begin{array}{c}\Delta \sigma +\Lambda \sigma =0,in\Sigma \mathrm{and}\sigma =0on\partial \Sigma .\hfill \end{array}$$

(1)

Here, $\Delta$ stands for the Laplacian on ${\mathit{M}}^l$. Then, ${\Lambda }_1\left(\mathit{M}\right)$ can be expressed as ${\Lambda }_1\left(\mathit{M}\right)={inf}_{\Sigma }{\Lambda }_1(\Sigma ).$ The Reilly formula only applies to the manifold’s inherent geometry and definitely to the particular PDE being examined. With the help of the example below, you can easily comprehend this. If ${\Lambda }_1\ne 0$ denotes the first eigenvalue with respect to the Neumann boundary condition on a compact l-dimensional Riemannian manifold $({\mathit{M}}^l,g)$, we have

$$\begin{array}{c}\Delta \sigma +{\Lambda }_1\sigma =0\mathrm{on}\mathrm{M}\mathrm{and}\frac{\partial \sigma }{\partial N}=0\mathrm{on}\partial \mathrm{M},\hfill \end{array}$$

(2)

where N stands for the outward unit normal to $\partial {\mathit{M}}^l$. A result of Reilly [2] proved the following famous upper bound inequality for the Laplacian operator associated with the first non-zero eigenvalue, ${\Lambda }_1^{\nabla }$.

$$\begin{array}{c}{\Lambda }_1^{\nabla }\le \frac{l}{Vol\left({\mathit{M}}^l\right)}{\int }_{{\mathit{M}}^l}{|\mathbb{H}|}^{2}dV,\hfill \end{array}$$

(3)

for the Riemannian submanifold ${\mathit{M}}^l$ isometrically embedded in the Euclidean space ${\mathbb{R}}^m$, including the mean curvature, $\mathbb{H}$, with dimensions denoted by l and m, respectively. In this case, the submanifold ${\mathit{M}}^l$ is connected, closed, and oriented, and the boundary satisfies $\partial \mathit{M}=0$.

After the discovery of inequality (3), many authors were inspired to create such problems for various ambient settings, as can be observed in the literature, for example, for Minkowski spaces [3]; for the integral curvature conditions on the closed Riemannian manifold [4] of the q-Laplacian; in hyperbolic spaces [5], with some integral conditions imposed on mean curvature; for product manifolds [6] for the Hodge Laplacian operator; for Euclidean space, a unit sphere, or even a projective space [7]; for the q-Laplacian operator that generalized (3) on Kaehler manifolds [8]; and for the Wentzel–Laplace operator in Euclidean space [9]. Motivated by the literature, the following upper bound of the first eigenvalue, ${\Lambda }_1^{\nabla }>0$, of the Laplacian is established in [2,10] for the simply connected space form ${\mathit{M}}^m\left(ϵ\right)$, which contains a constant curvature, $ϵ$:

$$\begin{array}{c}{\Lambda }_1^{\nabla }\le \frac{l}{Vol\left(\mathit{M}\right)}{\int }_{{\mathit{M}}^l}\left({|\mathbb{H}|}^2+ϵ\right)dV,\hfill \end{array}$$

(4)

where ${\mathit{M}}^l$ stands for a closed and orientable submanifold of dimension l in ${\mathbb{M}}^m\left(ϵ\right).$ It is easy to study that the inequality (4) is generalized for the Euclidean space ${\mathbb{R}}^m$ with $ϵ=0$, the unit sphere ${\mathbb{S}}^m\left(1\right)$ with $ϵ=1$, and the hyperbolic space $\mathbb{H}{(-1)}^m$$ϵ=-1$, respectively. The equality case holds in (4) ⟺${\mathit{M}}^l$ is minimal in a geodesic sphere of radius $r_{ϵ}$ of ${\tilde{\mathbb{M}}}^m\left(ϵ\right)$ with $r_0={(l/{\Lambda }_1^{\Delta })}^{1/2},r_1=arcsin{r}_0$, and $r_{-1}=arcsinh{r}_0$.

Next, the inequality (4) is extended for the p-Laplacian operator in [11,12], as given by the results in [2], where expanded applications were assumed. Similar results can be found in [13,14,15,16,17,18,19,20,21,22,23,24,25,26] through the work of [2]. We have the following definition:

$$\begin{array}{c}{\Delta }_q\sigma ={div(|\nabla \sigma |}^{q-2}\nabla f\sigma ).\hfill \end{array}$$

(5)

The q-Laplacian operator for $q>1$ satisfies the above equation. If we substitute $q=2$ in (5), then it becomes the usual Laplacian. Similarly, the eigenvalue $\Lambda$ of ${\Delta }_q$ is as follows:

$$\begin{array}{c}{\Delta }_q\sigma =-\Lambda {|\sigma |}^{q-2}\sigma ,\hfill \end{array}$$

(6)

for the Dirichlet boundary problem (1) (or Neumann problem (2)). The first non-zero eigenvalue, ${\Lambda }_{1,q}$, of ${\Delta }_q$ for a Riemannian manifold, ${\mathit{M}}^l$, with no boundary demonstrates the variational characteristic of the Rayleigh-type manifold [27]:

$$\begin{array}{c}{\Lambda }_{1,q}=inf\left\{{\displaystyle \frac{{\int }_{\mathit{M}}{|\nabla \sigma |}^q}{{\int }_{\mathit{M}}{\parallel \sigma \parallel }^q}}|\sigma \in W^{1,q}\left({\mathit{M}}^l\right)\left\{0\right\},{\int }_{\mathit{M}}{|\sigma |}^{q-2}\sigma =0\right\}.\hfill \end{array}$$

(7)

Therefore, influenced by the studies of [9,11,12,28], we will give a precise estimate of the q-Laplacian’s first eigenvalue on the special contact slant submanifold of the Sasakian space form ${\mathbb{M}}^{2k+l}\left(ϵ\right)$, given as:

Theorem 1. 

Let ${\mathit{M}}^{l+1}$ be an $(l+1)(\ge 2)$-dimensional closed orientated special contact slant submanifold in a Sasakian space form, ${\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right)$. The first non-zero eigenvalue, ${\Lambda }_{1,q}$, of the q-Laplacian satisfies

$$\begin{array}{cc}\hfill {\Lambda }_{1,q}\le & \frac{2^{(1-\frac{q}{2})}{(k+1)}^{(1-\frac{q}{2})}{(l+1)}^{\frac{q}{2}}}{{\left(Vol\left(\mathit{M}\right)\right)}^{q/2}}\left\{{\int }_{{\mathit{M}}^{l+1}}\right(\left(\frac{ϵ+3}{4}\right)\hfill \\ \hfill & +\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2)dV{\}}^{q/2}\hfill \\ for1<q\le 2\hfill \\ \hfill and\end{array}$$

(8)

$$\begin{array}{cc}\hfill {\Lambda }_{1,q}\le & \frac{2^{(1-\frac{q}{2})}{(k+1)}^{(\frac{q}{2}-1)}{(l+1)}^{\frac{q}{2}}}{Vol\left(\mathit{M}\right)}{\int }_{{\mathit{M}}^{l+1}}\{\left(\frac{ϵ+3}{4}\right)\hfill \\ \hfill & +\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2{\}}^{q/2}dV\hfill \\ for2<q\le \frac{l+1}{2}+1.\hfill \end{array}$$

(9)

Furthermore, this equality holds if and only if $q=2$ and ${\mathit{M}}^l$ is minimally immersed in a geodesic sphere of radius $r_{ϵ}$ of ${\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right)$ with $r_0={(l+1/{\Lambda }_1^{\Delta })}^{1/2},r_1=arcsin{r}_0$, and $r_{-1}=arcsinh{r}_0$.

The Reilly-like inequality (3) will be generalized for the case of the q-Laplacian for all values of $ϵ$ in this paper.

The manuscript’s plan is organized in the following pattern. Section 2 shows the structural equations for a special contact slant submanifold, ${\mathit{M}}^{l+1}$, in ${\mathbb{M}}^{2l+l}\left(ϵ\right)$. This section also indicates the change in geometric quantities as the metric on ${\mathbb{M}}^{2k+l}\left(ϵ\right)$ changes under a conformal transformation. The proof of Theorem 1 is given in Section 3, where we find suitable test functions through a conformal transformation to a unit sphere.

2. Preliminaries and Notations

An almost contact manifolds are odd-dimensional, $C^{\infty }$-manifold $({\tilde{\mathbb{M}}}^{2m+1},g)$, with an almost contact structure $({\psi }_1,\xi ,\eta )$ that satisfies the succeeding properties:

$$\begin{array}{c}{\psi }_1^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1,{\psi }_1\left(\xi \right)=0,\eta \circ {\psi }_1=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}g({\psi }_1{U}_0,{\psi }_{1}V)=g(U_0,V_0)-\eta \left(U_0\right)\eta \left(V_0\right),\eta \left(U_0\right)=g(U_0,\xi )\hfill \end{array}$$

(11)

for any $U_0$ and $V_0$ that belong to ${\tilde{\mathbb{M}}}^{2m+1}$. Under the above structure conditions, the almost contact manifold is a Sasakian manifold [29,30] if

$$\begin{array}{c}\left({\tilde{\nabla }}_{U_0}{\psi }_1\right)V_2=g(U_0,V_2)\xi -\eta \left(V_2\right)U_0.\hfill \end{array}$$

(12)

This indicates that

$$\begin{array}{c}{\tilde{\nabla }}_{U_0}\xi =-{\psi }_1{U}_0,\hfill \end{array}$$

(13)

where ∇ indicates the Riemannian connection in regard to g and $U_2$ and $V_2$ are any vector fields on ${\tilde{\mathbb{M}}}^{2k+1}$. Therefore, the curvature tensor $\tilde{R}$ for Sasakian Space forms ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$ with a ${\psi }_1$-sectional constant curvature $ϵ$ as follows:

$$\begin{array}{cc}\tilde{R}(X_0,Y_0,Z_0,W_0)=\frac{ϵ+3}{4}\hfill & \{g(Y_0,Z_0)g(X_0,W_0)-g(X_0,Z_0)g(Y_0,W_0)\}\hfill \\ \hfill +\frac{ϵ-1}{4}& \{\eta \left(X_0\right)\eta \left(Z_0\right)g(Y_0,W_0)+\eta \left(W_0\right)\eta \left(Y_0\right)g(X_0,Z_0)\hfill \\ \hfill & -\eta \left(Y_0\right)\eta \left(Z_0\right)g(X_0,W_0)-\eta \left(X_0\right)g(Y_0,Z_0)\eta \left(W_0\right)\hfill \\ \hfill & +g({\psi }_1{Y}_0,Z_0)g({\psi }_1{X}_0,W_0)-g({\psi }_1{X}_0,Z_0)g({\psi }_1{Y}_0,W_0)\hfill \\ \hfill & +2g(X_0,{\psi }_1{Y}_0)g({\psi }_1{Z}_0,W_0)\},\hfill \end{array}$$

(14)

for any arbitrary $X_0,Y_0,Z_0$ and $W_0$ belonging to ${\tilde{\mathbb{M}}}^{2k+1}$. For more detail, go to [29,30,31,32,33]. The Gauss equation is defined as:

$$\begin{array}{cc}\tilde{R}(X_0,Y_0,Z_0,W_0)=\hfill & R(X_0,Y_0,Z_0,W_0)+g(h(X_0,W_0),h(Y_0,Z_0))\hfill \\ \hfill & -g(h(X_0,Z_0),h(Y_0,W_0)).\hfill \end{array}$$

(15)

Assuming that $\mathit{M}$ is an $(l+1)$-dimensional special contact slant submanifold of a $(2k+1)$-dimensional Sasakian space form, $\tilde{\mathit{M}}\left(ϵ\right)$, we have written ${\psi }_1{X}_0=T{X}_0+N{X}_0$ for any $X_0\in \Gamma \left(T\mathit{M}\right)$ such that $T{X}_0$ and $N{X}_0$ are tangential and normal parts of $\psi X_0$. If the angle $\theta \left(U_0\right)$ between ${\psi }_1{U}_0$ and $T\mathit{M}$ is a constant angle for any point $x\in \mathit{M}$ tangent to $\xi$ and any vector field $U_0\in \Gamma \left(T\mathit{M}\right)$ that is linearly independent, then the submanifold ${\mathit{M}}^l$ is said to be $\theta$-slant, and $\theta \left(U_0\right)$ has a position between 0 and $\frac{\pi }{2}$ [34]. A proper contact $\theta$-slant submanifold is a special contact $\theta$-slant submanifold if the following equation is satisfied:

$$\begin{array}{c}\left({\nabla }_{X_0}T\right)Y_0=cos\theta \left\{g(X_0,Y_0)\xi -\eta \left(Y_0\right)\xi \right\}\hfill \end{array}$$

(16)

The orthonormal basis of $T_x\mathit{M}$ is given as $\{E_0=\xi ,E_1\cdots E_l\}$ and the orthonormal basis of $T_x^{\perp }\mathit{M}$ is defined as $\{E_1^{*}\cdots E_l^{*}\}$ with $E_t^{*}=\frac{N{E}_t}{sin\theta }$ for $t\in \{1\cdots n\}$. We write $h_{ij}^t=g(h(E_i,E_j),E_t^{*})$ for $i,j\in \{0\cdots l\}$ and $t\in \{1\cdots l\}$. For a special contact slant submanifold, one can define $h_{ij}^t=h_{kj}^i=h_{ik}^j(h_{ji}^t=h_{jt}^i=h_{ti}^j)$ for all $i,j,t\in \{1\cdots l\}$. We can write the tangential component as follows:

$$\begin{array}{c}{\parallel T\parallel }^2={\displaystyle \sum _{i,j=1}^l}{g}^2(T{E}_i,E_j).\hfill \end{array}$$

(17)

Next, we quickly provide some clarification on the curvature tensor $\tilde{R}$ for a slant submanifold in a Sasakian space form, ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$, which is given by:

$$\begin{array}{cc}{\displaystyle \sum _{i,j=1}^l}\tilde{R}(E_i,E_j,E_i,E_j)=\hfill & \left(\frac{ϵ+3}{4}\right)l(l+1)\hfill \\ \hfill & +\left(\frac{ϵ-1}{4}\right)\left\{3{\displaystyle \sum _{i,j=1}^l}{g}^2(T{E}_i,E_j)-2l\right\}.\hfill \end{array}$$

(18)

An analysis of special contact slant submanifolds and conformal geometry is presented. Convection has been applied to this particular range of indices. However, it should be excluded in a manner that ensures

$$\begin{array}{c}1\le i,j,t,\cdots \le l+1;l+2\le \alpha ,\beta ,\gamma ,\cdots \le 2k+11\le a,b,c,\cdots \le 2k+1.\hfill \end{array}$$

2.1. Structure Equations for Special Contact Slant Submanifolds

From ${\mathit{M}}^{l+1}$ to a Riemannian manifold $(\tilde{\mathbb{M}},\tilde{g})$, the totally real immersion is denoted by x. An induced metric on $\mathit{M}$ is given as $g_{\mathit{M}}=x^{*}\tilde{g}$. Here, ∇ and $\tilde{\nabla }$ specify the Levi-Civitas connections on ${\mathit{M}}^{l+1}$ and ${\tilde{\mathbb{M}}}^{2k+1}$. We consider an orthogonal frame ${\left\{E_a\right\}}_{a=1}^{2k+1}$ on ${\tilde{\mathbb{M}}}^{2k+1}$ in a way that ${\left\{E_i\right\}}_{i=1}^{l+1}$ and ${\left\{E_{\alpha }\right\}}_{\alpha =l+2}^{2k+2}$ are tangent and normal to ${\mathit{M}}^{l+1}$, respectively. The structural equation of ${\tilde{\mathbb{M}}}^{2k+1}$, with ${\left\{{\Pi }_a\right\}}_{a=1}^{2k+1}$ as dual co-frame of ${\left\{E_a\right\}}_{a=1}^{2k+1}$, is given as the following

$$\begin{array}{c}\left\{\begin{array}{c}d{\Pi }_a={\displaystyle \sum _b}{\Pi }_{ab}\wedge {\Pi }_b,{\Pi }_{ab}+{\Pi }_{ba}=0,\hfill \\ \\ d{\Pi }_{ab}-{\displaystyle \sum _c}{\Pi }_{ac}\wedge {\Pi }_{cb}=-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{cd}}{\tilde{R}}_{abcd}{\Pi }_c\wedge {\Pi }_d,\hfill \end{array}\right.\hfill \end{array}$$

(19)

where $\left\{{\Pi }_{ab}\right\}$ depicts the connection forms on ${\tilde{\mathbb{M}}}^{2k+1}$, whereas ${\tilde{R}}_{abcd}$ are the required apparatuses of the curvature tensor of ${\tilde{\mathbb{M}}}^{2k+1}$. We can denote the following:

$$\begin{array}{c}{x}^{*}{\Pi }_a{\Xi }_a,x^{*}{\Pi }_{ab}={\Xi }_{ab}\hfill \end{array}$$

(20)

Then, restricted to ${\mathit{M}}^{l+1}$, we have

$$\begin{array}{c}{\Xi }_{\alpha }=0,{\Xi }_{i\alpha }={\displaystyle \sum _j}{h}_{ij}^{\alpha }{\Xi }_j\hfill \end{array}$$

(21)

$$\begin{array}{c}\left\{\begin{array}{c}d{\Xi }_i={\displaystyle \sum _j}{\Xi }_{ij}\wedge {\Xi }_j,{\Xi }_{ij}+{\Xi }_{ji}=0,\hfill \\ \\ d{\Xi }_{ij}-{\displaystyle \sum _k}{\Xi }_{ik}\wedge {\Xi }_{kj}=-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,n}}{R}_{ijkn}{\Xi }_k\wedge {\Xi }_l\hfill \end{array}\right.\hfill \end{array}$$

(22)

where $R_{ijkn}$ are the components of the curvature tensor of ${\mathit{M}}^{l+1}$ and $h_{ij}^{\alpha }$ are the components of the second fundamental form of ${\mathit{M}}^{l+1}$ in ${\tilde{\mathbb{M}}}^{2k+1}$. By considering x and using (21) and (22), we obtain the Gauss equations for a special contact slant submanifold in a Sasakian space form, ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$, taking into account (18):

$$\begin{array}{cc}{\displaystyle \sum _{i,j=1}^l}R(E_i,E_j,E_i,E_j)=\hfill & \left(\frac{ϵ+3}{4}\right)l(l+1)+\left(\frac{ϵ-1}{4}\right)\left\{{3\parallel T\parallel }^2-2l\right\}\hfill \\ \hfill & +{\displaystyle \sum _{\alpha }}\left(h_{ii}^{\alpha }{h}_{jj}^{\alpha }-{\left(h_{ij}^{\alpha }\right)}^2\right).\hfill \end{array}$$

(23)

For the $\theta$-slant submanifold ${\sum }_{j=2}^l{g}^2(T{E}_1,E_j)={cos}^2\theta$, we have, from the above,

$$\begin{array}{cc}{\displaystyle \sum _{i,j=1}^l}R(E_i,E_j,E_i,E_j)=\hfill & \left(\frac{ϵ+3}{4}\right)l(l+1)+\left(\frac{ϵ-1}{4}\right)\left\{3l{cos}^2\theta -2l\right\}\hfill \\ \hfill & +{\displaystyle \sum _{\alpha }}\left(h_{ii}^{\alpha }{h}_{jj}^{\alpha }-{\left(h_{ij}^{\alpha }\right)}^2\right).\hfill \end{array}$$

(24)

As we know, $E_0=\xi$ and $h_{00}^r=g(h(E_0,E_0),E_r^{*})=g(h(\xi ,\xi ),E_r^{*})=g({\tilde{\nabla }}_{\xi }\xi ,E_r^{*})=0$. On the other hand, ${\sum }_{r=1}^l{\left(h_{10}\right)}^2={\sum }_{r=1}^l{g}^2(h(E_1,E_0),E_r^{*})={\sum }_{r=1}^l{g}^2({\tilde{\nabla }}_{E_1}{E}_0,E_r^{*})=$${\sum }_{r=1}^l{g}^2({\psi }_1{E}_1,E_r^{*})={sin}^2\theta$. Also, since ${\mathit{M}}^{l+1}$ is a special slant submanifold, $h_{ij}^t=h_{tj}^i$, so we use all the above conclusions and find the following by taking the trace of Equation (24)

$$\begin{array}{cc}R=\hfill & \left(\frac{ϵ+3}{4}\right)l(l+1)+\left(\frac{ϵ-1}{4}\right)\left\{3{cos}^2\theta -2\right\}l\hfill \\ \hfill & +l{sin}^2\theta +{(l+1)}^2{|\mathbb{H}|}^2-S\hfill \end{array}$$

(25)

where R is the scalar curvature of ${\mathit{M}}^{l+1}$, $S={\displaystyle \sum _{\alpha ,i,j}}{\left(h_{ij}^{\alpha }\right)}^2$ is the squared norm of the second fundamental form, and $\mathbb{H}={\displaystyle \sum _{\alpha }}{\mathbb{H}}^{\alpha }{E}_{\alpha }={\displaystyle \frac{1}{l+1}}{\displaystyle \sum _{\alpha }}\left({\displaystyle \sum _i}{h}_{ii}^{\alpha }\right)E_{\alpha }$ is the mean curvature vector of ${\mathit{M}}^l$.

2.2. Conformal Relations

This section discusses how curvature and the second fundamental form change with respect to a conformal transformation. These relations are well known (cf. [35,36]), and a brief proof using the method of moving frames is presented here for the readers’ convenience.

Suppose that ${\tilde{\mathbb{M}}}^{2k+1}$ is endowed with a new metric $\tilde{\tilde{g}}=E^{2\nu }\tilde{g}$, which is conformal to $\tilde{g}$ for $\nu \in C^{\infty }\left(\tilde{\mathbb{M}}\right)$. Then, $\{{\tilde{E}}_a=E^{-\nu }{E}_a\}$ is an orthogonal frame of $(\tilde{\mathbb{M}},\tilde{\tilde{g}})$, and ${\tilde{\Pi }}_a=E^{\nu }{\Pi }_a$ is the dual co-frame of $\left\{{\tilde{E}}_a\right\}$. The structure equations of $(\tilde{\mathbb{M}},\tilde{g})$ are given by

$$\begin{array}{c}\left\{\begin{array}{c}d{\tilde{\Pi }}_a={\displaystyle \sum _b}{\tilde{\Pi }}_{ab}\wedge {\tilde{\Pi }}_b,\hfill \\ \\ {\tilde{\Pi }}_{ab}+{\tilde{\Pi }}_{ba}=0\hfill \end{array}\right.\hfill \end{array}$$

(26)

where $\left\{{\tilde{\Pi }}_{ab}\right\}$ are the connection forms on $(\tilde{\mathbb{M}},\tilde{\tilde{g}})$. We denote the following

$$\begin{array}{c}{\tilde{g}}_{\mathit{M}}=x^{*}\tilde{\tilde{g}},x^{*}{\tilde{\Pi }}_a={\tilde{\Xi }}_a,\&x^{*}{\tilde{\Pi }}_{ab}={\tilde{\Xi }}_{ab}.\hfill \end{array}$$

(27)

Then, restricted to $({\mathit{M}}^l,{\tilde{g}}_{\mathit{M}})$, we have

$$\begin{array}{c}{\tilde{\Xi }}_{\alpha }=0,\&{\tilde{\Xi }}_{i\alpha }={\displaystyle \sum _j}{\tilde{h}}_{ij}^{\alpha }{\tilde{\Xi }}_j,\hfill \end{array}$$

(28)

and

$$\begin{array}{c}\left\{\begin{array}{c}d{\tilde{\Xi }}_i={\displaystyle \sum _j}{\tilde{\Xi }}_{ij}\wedge {\tilde{\Xi }}_j,{\tilde{\Xi }}_{ij}+{\tilde{\Xi }}_{ji}=0,\hfill \\ \\ d{\tilde{\Xi }}_{ij}-{\displaystyle \sum _k}{\tilde{\Xi }}_{ik}\wedge {\tilde{\Xi }}_{kj}=-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,n}}{\tilde{R}}_{ijkn}{\tilde{\Xi }}_k\wedge {\tilde{\Xi }}_n\hfill \end{array}\right.\hfill \end{array}$$

(29)

where ${\tilde{R}}_{ijkn}$ are the components of the curvature tensor of $({\mathit{M}}^{l+1},{\tilde{g}}_{\mathit{M}})$, and ${\tilde{h}}_{ij}^{\alpha }$ are the components of the second fundamental form of $({\mathit{M}}^{l+1},{\tilde{g}}_{\mathit{M}})$ in $({\tilde{\mathbb{M}}}^{2k+1},\tilde{\tilde{g}})$. From (22) and (26), we arrive at

$$\begin{array}{c}{\tilde{\Pi }}_{ab}={\Pi }_{ab}+{\nu }_a{\Pi }_b-{\nu }_b{\Pi }_a,\hfill \end{array}$$

(30)

where ${\nu }_a$ is the covariant derivative of $\nu$ with respect to $E_a;\mathrm{that}\mathrm{is},d\nu ={\displaystyle \sum _a}{\nu }_a{E}_a$. Next, formulating these from (22), (29), and (30):

$$\begin{array}{cc}{e}^{2\nu }{\tilde{R}}_{ijkn}=R_{ijkn}\hfill & -\left({\nu }_{ik}{\delta }_{jn}+{\nu }_{jn}{\delta }_{ik}-{\nu }_{in}{\delta }_{jk}-{\nu }_{jk}{\delta }_{in}\right)\hfill \\ \hfill & +\left({\nu }_i{\nu }_k{\delta }_{jn}+{\nu }_j{\nu }_n{\delta }_{ik}-{\nu }_j{\nu }_k{\delta }_{in}-{\nu }_i{\nu }_n{\delta }_{jk}\right)\hfill \\ \hfill & -{|\nabla \nu |}^2\left({\delta }_{ik}{\delta }_{jn}-{\delta }_{in}{\delta }_{jk}\right).\hfill \end{array}$$

(31)

By pulling back (30) to ${\mathit{M}}^{l+1}$ by x and using (21) and (29), we have

$$\begin{array}{c}{\tilde{h}}_{ij}^{\alpha }=e^{-\nu }(h_{ij}^{\alpha }-{\nu }_{\alpha }{\delta }_{jn}),{\tilde{\mathbb{H}}}^{\alpha }=e^{-\nu }({\mathbb{H}}^{\alpha }-{\nu }_{\alpha }).\hfill \end{array}$$

(32)

From this, it is easy to obtain a useful relation:

$$\begin{array}{c}{e}^{2\nu }(\tilde{S}-(l+1)|\tilde{\mathbb{H}}{|}^2{)=S-(l+1)|\mathbb{H}|}^2.\hfill \end{array}$$

(33)

3. Proof of Main Result

Theorem 1 announced in the preceding section will be demonstrated in this section. Prior to that, some basic formulas will be introduced, and certain pertinent lemmas from [23] will be brought back to our context. We will present a key lemma for the paper’s proposals, which is principally inspired by the research in [23].

Lemma 1. 

Let $x:{\mathit{M}}^{l+1}\to {\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$ be the immersion from an $(l+1)$-dimensional closed orientated special contact slant submanifold into a $(2k+1)$-dimensional Sasakian space form, ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$. Then, for $q>1$, there exists a regular conformal map, $\Gamma :{\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)\to {\mathbb{S}}^{2k+1}\left(1\right)\subset {\mathbb{R}}^{2k+2}$, such that the immersion ${\beta }_1=\Gamma \circ x=({\beta }_1^1,\cdots {\beta }_1^{2k+2})$ satisfies

$$\begin{array}{c}{\int }_{{\mathit{M}}^{l+1}}{|{\beta }_1^a|}^{q-2}{\beta }_1^{a}d{V}_{\mathit{M}}=0,a=1,\cdots l+2\hfill \end{array}$$

(34)

Proof. 

The main idea of Lemma 1 is inspired by the case $q=2$ (cf. [37,38]) and the case $q\ge 2$ (cf. Lemma 3.1 in [11,23]). □

We deliver an upper bound for ${\Lambda }_{1,q}$ in terms of the conformal function, compared with Lemma 2.7 in [23], in the above Lemma 1 by producing a test function.

Lemma 2. 

Let ${\mathit{M}}^{l+1}$ be an $(l+1)\ge 2$-dimensional closed orientated special contact slant submanifold in a Sasakian space form, ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$. Denote by ${{\rm Y}}_{\epsilon }$ the standard metric on ${\tilde{\mathbb{M}}}^{2k+1}\left(ϵ\right)$ and assume ${\Gamma }^{*}{{\rm Y}}_1=e^{2\nu }{{\rm Y}}_{\epsilon }$, where Γ is the conformal map in Lemma 1. Then, we have, for all $q>1$,

$$\begin{array}{c}{\Lambda }_{1,q}Vol\left({\mathit{M}}^{l+1}\right)\le 2^{|1-\frac{q}{2}|}{(k+1)}^{|1-\frac{q}{2}|}{(l+1)}^{\frac{q}{2}}{\int }_{{\mathit{M}}^{l+1}}{\left(e^{2\nu }\right)}^{\frac{q}{2}}dV.\hfill \end{array}$$

(35)

Proof. 

Considering Lemma 1, we have the choice to make ${\beta }_1^a$ the test function, so

$$\begin{array}{c}{\Lambda }_{1,q}{\int }_{{\mathit{M}}^{l+1}}|{\beta }_1^a{|}^q\le {|\nabla {\beta }_1^a|}^{q}dV,1\le a\le l+2.\hfill \end{array}$$

(36)

Note that if ${\displaystyle \sum _{a=1}^{2k+2}}{|{\beta }_1^a|}^2=1$, then $|{\beta }_1{1}^a|\le 1$. We achieve

$$\begin{array}{c}{\displaystyle \sum _{a=1}^{2k+2}}|\nabla {\beta }_1^a{|}^2={\displaystyle \sum _{i=1}^{l+1}}{|{\nabla }_{e_i}{\beta }_1|}^2=(l+1)e^{2\nu }.\hfill \end{array}$$

(37)

By applying $1<q\le 2$, we obtain

$$\begin{array}{c}|{\psi }_1^a{|}^2\le {|{\beta }_1^a|}^q.\hfill \end{array}$$

(38)

Applying the Hölder inequality together with (36)–(38), we obtain

$$\begin{array}{cc}{\Lambda }_{1,q}Vol\left(\mathit{M}\right)\hfill & ={\Lambda }_{1,q}{\displaystyle \sum _{a=1}^{2k+2}}{\int }_{{\mathit{M}}^{l+1}}|{\beta }_1^a{|}^{2}dV\le {\Lambda }_{1,q}{\displaystyle \sum _{a=1}^{2k+2}}{\int }_{{\mathit{M}}^{l+1}}{|{\beta }_1^a|}^{q}dV\hfill \\ \hfill & \le {\int }_{{\mathit{M}}^l}{\displaystyle \sum _{a=1}^{2k+2}}{|\nabla {\beta }_1^a|}^{q}dV\le {(2k+2)}^{1-q/2}{\int }_{{\mathit{M}}^{l+1}}{\left({\displaystyle \sum _{i=1}^{l+1}}{|{\nabla }_e{\beta }_1|}^2\right)}^{\frac{q}{2}}dV\hfill \\ & =2^{1-\frac{q}{2}}{(k+1)}^{1-\frac{q}{2}}{\int }_{{\mathit{M}}^{l+1}}{\left((l+1)e^{2\nu }\right)}^{\frac{q}{2}}dV.\hfill \end{array}$$

This results in what we want (35). On the other hand, if we select $q\ge 2$, then by invoking the Hölder inequality, we derive

$$\begin{array}{c}1={\displaystyle \sum _{a=1}^{2k+2}}{|{\beta }_1^a|}^2\le {(2k+2)}^{1-\frac{2}{q}}{\left({\displaystyle \sum _{a=1}^{2k+2}}{|{\beta }_1^a|}^q\right)}^{\frac{2}{q}}.\hfill \end{array}$$

(39)

And the outcome we obtain is

$$\begin{array}{c}{\Lambda }_{1,q}Vol\left({\mathit{M}}^{l+1}\right)\le {(2k+2)}^{\frac{q}{2}-1}\left({\displaystyle \sum _{a=1}^{2k+2}}{\Lambda }_{1,q}{\int }_{{\mathit{M}}^{l+1}}{|{\beta }_1^a|}^{q}dV\right).\hfill \end{array}$$

(40)

Minkowski’s inequality gives

$$\begin{array}{c}{\displaystyle \sum _{a=1}^{2k+2}}{|\nabla {\beta }_1^a|}^q\le {\left({\displaystyle \sum _{a=1}^{2k+2}}{|\nabla {\beta }_1^a|}^2\right)}^{\frac{q}{2}}={\left((l+1)e^{2\nu }\right)}^{\frac{q}{2}}.\hfill \end{array}$$

(41)

Therefore, (35) follows from (36), (40), and (41). The proof of the lemma is complete. □

This has brought us to a position to demonstrate Theorem 1.

3.1. Proof of Theorem 1

To begin with, $1<q\le 2$, so $\frac{q}{2}\le 1$. By using Lemma 2 and applying the Hölder inequality, we obtain

$$\begin{array}{cc}{\Lambda }_{1,q}Vol\left({\mathit{M}}^{l+1}\right)\le \hfill & 2^{1-\frac{q}{2}}{(k+1)}^{1-\frac{q}{2}}{(l+1)}^{\frac{q}{2}}{\int }_{{\mathit{M}}^{l+1}}{\left(e^{2\nu }\right)}^{\frac{q}{2}}dV\hfill \\ \hfill & \le 2^{1-\frac{q}{2}}{(k+1)}^{1-\frac{q}{2}}{(l+1)}^{\frac{q}{2}}{\left(Vol\left(\mathit{M}\right)\right)}^{1-\frac{q}{2}}{\left({\int }_{{\mathit{M}}^{l+1}}{e}^{2\nu }dV\right)}^{\frac{q}{2}}.\hfill \end{array}$$

$e^{2\nu }$ can be calculated using both the conformal relations and Gauss equations. Let ${\tilde{\mathbb{M}}}^{2k+1}={\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right),\mathit{and}\tilde{g}=e^{-2\nu }{{\rm Y}}_{ϵ},$ where $\tilde{g}={\Gamma }^{*}{{\rm Y}}_1$ as before. From (25), the Gauss equations for the immersion x and the special contact slant immersion ${\beta }_1=\Gamma \circ x$ are, respectively,

$$\begin{array}{cc}R=\hfill & \left(\frac{ϵ+3}{4}\right)l(l+1)+\left(\frac{ϵ-1}{4}\right)\left\{3{cos}^2\theta -2\right\}l+l{sin}^2\theta \hfill \\ \hfill & {(l+1)|\mathbb{H}|}^2+{((l+1)|\mathbb{H}|}^2-S),\hfill \end{array}$$

(42)

$$\begin{array}{c}\tilde{R}=l(l+1)+l(l+1)|\tilde{\mathbb{H}}{|}^2+((l+1)|\tilde{\mathbb{H}}{|}^2-\tilde{S}).\hfill \end{array}$$

(43)

Tracing (31), it can be found that

$$\begin{array}{c}{e}^{2\nu }\tilde{R}=R-(l-1)l{|\nabla \nu |}^2-2l\Delta \nu ,\hfill \end{array}$$

(44)

which, when jointly substituting (42) and (43) into (44), gives

$$\begin{array}{cc}{e}^{2\nu }(l(l+1)+l(l+1){|\tilde{\mathbb{H}}|}^2+\hfill & ((l+1)|\tilde{\mathbb{H}}{|}^2-\tilde{S}\left)\right)\hfill \\ \hfill =& \left(\frac{ϵ+3}{4}\right)l(l+1)+\left(\frac{ϵ-1}{4}\right)\left\{3{cos}^2\theta -2\right\}l\hfill \\ \hfill & +l{sin}^2\theta +{l(l+1)|\mathbb{H}|}^2+{((l+1)|\mathbb{H}|}^2-S)\hfill \\ \hfill & -{(l-1)l|\nabla \nu |}^2-2l\Delta \nu .\hfill \end{array}$$

From here, it follows that

$$\begin{array}{cc}l(l+1)\{\left(e^{2\nu }-\left(\frac{ϵ+3}{4}\right)\right)+\hfill & \left(\frac{ϵ-1}{4(l+1)}\right)\left\{3{cos}^2\theta -2\right\}+\left(\frac{{sin}^2\theta }{l+1}\right)+\left(e^{2\nu }|\tilde{\mathbb{H}}{|}^2-{|\mathbb{H}|}^2\right)\}\hfill \\ \hfill & +(l+1)\left(e^{2\nu }|\tilde{\mathbb{H}}{|}^2-{|\mathbb{H}|}^2\right)\hfill \\ \hfill =& e^{2\nu }\tilde{S}-S-(l-1)l{|\nabla \nu |}^2-2l\Delta \nu .\hfill \end{array}$$

(45)

From (32) and (33), we obtain

$$\begin{array}{cc}l(l+1)\left(e^{2\nu }-\left(\frac{ϵ+3}{4}\right)\right)+\hfill & \left(\frac{ϵ-1}{4(l+1)}\right)\left\{3{cos}^2\theta -2\right\}+\left(\frac{{sin}^2\theta }{l+1}\right)\hfill \\ \hfill +& l(l+1){\displaystyle \sum _{\alpha }}{({\mathbb{H}}^{\alpha }-{\nu }_{\alpha })}^2-l(l+1){|\mathbb{H}|}^2\hfill \\ \hfill & =-{(l-1)l|\nabla \nu |}^2-2l\Delta \nu .\hfill \end{array}$$

Multiplying the preceding equation by $\frac{1}{l(l+1)}$ implies that

$$\begin{array}{cc}{e}^{2\nu }=\hfill & \left(\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4(l+1)}\right)\left\{3{cos}^2\theta -2\right\}+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right)\hfill \\ \hfill & -\frac{2}{l+l}\Delta \nu -\frac{l-1}{l+1}{|\Delta \nu |}^2-{|{\left(\tilde{\nabla }\nu \right)}^{\perp }-\mathbb{H}|}^2.\hfill \end{array}$$

(46)

By integration, it is not difficult to check that

$$\begin{array}{cc}{\Lambda }_{1,q}Vol\left({\mathit{M}}^{l+1}\right)\hfill & \le 2^{1-\frac{q}{2}}{(K+1)}^{1-\frac{q}{2}}{(l+1)}^{\frac{q}{2}}{\left(Vol\left({\mathit{M}}^{l+1}\right)\right)}^{1-\frac{q}{2}}{\left({\int }_{{\mathit{M}}^{l+1}}{e}^{2\nu }dV\right)}^{\frac{q}{2}}\hfill \\ & \le \frac{{(2k+2)}^{1-\frac{q}{2}|}{(l+1)}^{\frac{q}{2}}}{{\left(Vol\left(\mathit{M}\right)\right)}^{\frac{q}{2}-1}}\times \hfill \\ \hfill & {\left\{{\int }_{{\mathit{M}}^{l+1}}\left(\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right)dV\right\}}^{\frac{q}{2}}.\hfill \end{array}$$

The goal was to demonstrate that the result above is comparable to (8). For the case where $q>2$, we cannot apply the Holder inequality directly to control ${\int }_{\mathit{M}}{\left(e^{2\nu }\right)}^{\frac{q}{2}}$ by using ${\int }_{\mathit{M}}\left(e^{2\nu }\right)$. We must multiply both sides of (46) with the factor $e^{(q-2)\nu }$ and then solve by integrating ${\mathit{M}}^{l+1}$ (cf. [17]).

$$\begin{array}{cc}{\int }_{{\mathit{M}}^l}{e}^{q\nu }dV\hfill & \le {\int }_{{\mathit{M}}^{l+1}}\left\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right\}{e}^{(q-2)\nu }dV\hfill \\ \hfill & -{\int }_{{\mathit{M}}^{l+1}}\left(\frac{l-1-2q+4}{l+1}\right)e^{(q-2)\nu }{|\Delta \nu |}^{2}dV\hfill \\ \hfill & \le {\int }_{{\mathit{M}}^{l+1}}\left\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right\}{e}^{(q-2)\nu }dV.\hfill \end{array}$$

(47)

Secondly, this follows from the fundamental idea that $l+1\ge 2q-2$. We apply Young’s inequality, and then

$$\begin{array}{cc}{\int }_{{\mathit{M}}^{l+1}}\hfill & \left\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right\}{e}^{(q-2)\nu }dV\hfill \\ \hfill & \le \frac{2}{q}{\int }_{{\mathit{M}}^{l+1}}{\left\{|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2||\right)}^{\frac{q}{2}}dV\hfill \\ \hfill & +\frac{(q-2)}{q}{\int }_{{\mathit{M}}^{l+1}}{e}^{q\nu }dV.\hfill \end{array}$$

(48)

From this, we deduce

$$\begin{array}{c}{\int }_{{\mathit{M}}^{l+1}}{e}^{q\nu }dV\le {\int }_{{\mathit{M}}^{l+1}}{\left\{|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2|\right\}}^{\frac{q}{2}}dV\hfill \end{array}$$

(49)

from (47) and (48). Putting (49) into (35), we obtain (9). The special contact slant submanifolds for which the equality in (8) holds can be determined by considering the cases in (36) and (38). From this, it follows that

$$\begin{array}{cc}|{\beta }_1^a{|}^2=\hfill & |{\beta }_1^a{|}^q\hfill \\ {\Delta }_q{\beta }_1^a=\hfill & -{\Lambda }_{1,q}{|{\beta }_1^a|}^{q-2}{\beta }_1^a\hfill \end{array}$$

for each $a=1,\cdots ,2k+2$. The case $1<q<2$ implies $|{\beta }_1^a|=0$ or 1. Then, there is only one a for which $|{\beta }_1^a|=1$ and ${\Lambda }_{1,q}=0$, which informs a contradiction as the eigenvalue is non-zero. In view of this, the case $q=2$ is considered, and we are only restricted to the Laplacian case. Consequently, we are able to apply Theorem 1.5 from [7].

For the case $q>2$, and if the equality in (9) remains valid, then it shows that (40) and (41) become the equalities that yield

$$\begin{array}{c}|{\beta }_1^1{|}^q=\cdots ={|{\beta }_1^{2k+2}|}^q\hfill \end{array}$$

and the condition $|\nabla {\beta }_1^a|=0$ holds for existing values of a. This implies that ${\beta }_1^a$ is constant and ${\Lambda }_{1,q}$ is equal to zero. This final outcome is contradictory in that the eigenvalue ${\Lambda }_{1,q}$ is non-zero. This proof is complete.

3.2. Conclusion Remarks

Remark 1. 

Setting $q=2$ in the estimate recovers the following corollary.

Corollary 1. 

Let ${\mathit{M}}^{l+1}$ be an $(l+1)$-dimensional closed orientated special contact slant submanifold in a $(2k+1)$-dimensional Sasakian space form, ${\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right)$. Then, the first non-zero eigenvalue, ${\Lambda }_1^{\Delta }$, of the Laplacian satisfies

$$\begin{array}{c}{\Lambda }_1^{\Delta }\le \frac{l+1}{Vol\left(\mathit{M}\right)}{\int }_{{\mathit{M}}^{l+1}}\left\{{|\mathbb{H}|}^2+\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)\right\}dV\hfill \end{array}$$

(50)

where $\mathbb{H}$ is the mean curvature vector of ${\mathit{M}}^{l+1}$ in ${\mathbb{M}}^{2k+l}\left(ϵ\right)$ and $Vol\left(\mathit{M}\right)$ is the volume of ${\mathit{M}}^{l+1}$. Furthermore, this equality holds in (50) if and only if it is minimally immersed in a geodesic sphere of radius $r_{ϵ}$ of ${\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right)$ with $r_0={(l+1/{\Lambda }_1^{\Delta })}^{1/2},r_1=arcsin{r}_0$, and $r_{-1}=arcsinh{r}_0$.

Remark 2. 

Assuming that $1<q\le 2$, we have $\frac{q}{2(q-1)}\ge 1$. Then, from the $H\ddot{o}lder$ inequality, we have

$$\begin{array}{cc}{\int }_{{\mathit{M}}^{l+1}}\{\left(\frac{ϵ+3}{4}\right)+\hfill & \left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\}dV\hfill \\ \hfill & \le {\left(Vol\left(\mathit{M}\right)\right)}^{1-\frac{2(q-1)}{q}}\left\{{\int }_{{\mathit{M}}^{l+1}}\right\{\left(\frac{ϵ+3}{4}\right)\hfill \\ \hfill & +\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2{\}}^{\frac{q}{2(q-1)}}dV{\}}^{\frac{2(q-1)}{q}}.\hfill \end{array}$$

This is an immediate application of Theorem 1 by using Remark 2 as a Sasakian space form.

Corollary 2. 

Let ${\mathit{M}}^{l+1}$ be an $(l+1)(\ge 2)$-dimensional closed orientated special contact slant submanifold in a $(2k+1)$-dimensional Sasakian space form, ${\tilde{\mathit{M}}}^{2k+1}\left(ϵ\right)$. The first non-zero eigenvalue, ${\Lambda }_{1,q}$, of the q-Laplacian satisfies

$$\begin{array}{cc}{\Lambda }_{1,q}\le \hfill & \frac{2^{1-\frac{q}{2}}{(k+1)}^{(1-\frac{q}{2})}{(l+1)}^{\frac{q}{2}}}{{\left(Vol\left(\mathit{M}\right)\right)}^{(q-1)}}\left\{{\int }_{{\mathit{M}}^{l+1}}\right(\left(\frac{ϵ+3}{4}\right)\hfill \\ \hfill & +\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\theta -2}{(l+1)}\right)+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2{)}^{\frac{q}{2(q-1)}}dV{\}}^{(q-1)}\hfill \end{array}$$

(51)

for $1<q\le 2$.

Remark 3. 

It is generally known that when conventional Sasakian structures are taken into account, ${\mathbb{R}}^{2k+1}(-3)$ and ${\mathbb{S}}^{2k+1}\left(1\right)$ can be viewed as canonical examples of Sasakian space forms with constant sectional curvatures, $ϵ=-3$ and $ϵ=1,$ respectively.

According to the above remark, we have another remark.

Remark 4. 

Given the value $q=2$ and reverting to the inequality (51), one sees that (51) generalizes the Reilly-type inequality (50). From this, we know that the Reilly-type inequality computes the first Laplacian eigenvalue on a special contact slant submanifold in a Euclidean sphere, ${\mathbb{S}}^{2k+1}$. For example, Theorem 1.2 in [7] and Theorem 1.3 in [11] are cases of Theorem 1 for $ϵ=1$ and $q=2$.

Based on these observations, the next result is given as a special from of Theorem 1. Specifically, we found certain results by substituting $ϵ=1$ in (8) and (9), respectively.

Corollary 3. 

Let ${\mathbb{S}}^{2k+1}\left(1\right)$ be a Sasakian space form with a ${\psi }_1$-sectional constant curvature of one, and let ${\mathit{M}}^{l+1}$ be a special contact slant submanifold in ${\mathbb{S}}^{2k+1}\left(1\right)$ of dimension $l+1\ge 2$ that is closed and orientated. Then, the connection between the first positive eigenvalue, ${\Lambda }_{1,q}$, of the q-Laplacian and the mean curvature, $\mathbb{H}$, satisfies

$$\begin{array}{cc}\hfill {\Lambda }_{1,q}\le & \frac{2^{1-\frac{q}{2}}{(k+1)}^{(1-\frac{q}{2})}{(l+1)}^{\frac{q}{2}}}{{\left(Vol\left(\mathit{M}\right)\right)}^{q/2}}{\left\{{\int }_{{\mathit{M}}^{l+1}}\left(1+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right)dV\right\}}^{\frac{q}{2}}\hfill \\ for1<q\le 2,\hfill \\ \hfill and\end{array}$$

(52)

$$\begin{array}{cc}\hfill {\Lambda }_{1,q}\le & \frac{2^{1-\frac{q}{2}}{(k+1)}^{(1-\frac{q}{2})}{(l+1)}^{\frac{q}{2}}}{Vol\left(\mathit{M}\right)}{\int }_{{\mathit{M}}^{l+1}}{\left(1+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right)}^{\frac{q}{2}}dV\hfill \\ for2<q\le \frac{l+1}{2}+1.\hfill \end{array}$$

(53)

We have one more corollary, which is constructed on Corollary 2 and given as follows.

Corollary 4. 

Let ${\mathbb{S}}^{2k+1}\left(1\right)$ be a Sasakian space form with a ${\psi }_1$-sectional constant curvature of one, and let ${\mathit{M}}^{l+1}$ be a Legendrian submanifold in ${\mathbb{S}}^{2k+1}\left(1\right)$ of dimension $l+1\ge 2$ that is closed and orientated. Then, the connection between the first positive eigenvalue, ${\Lambda }_{1,q}$, of the q-Laplacian and the mean curvature, $\mathbb{H}$, satisfies

$$\begin{array}{cc}{\Lambda }_{1,q}\le \hfill & \frac{{(2k+2)}^{(1-\frac{q}{2})}{(l+1)}^{\frac{q}{2}}}{{\left(Vol\left(\mathit{M}\right)\right)}^{(q-1)}}{\left\{{\int }_{{\mathit{M}}^{l+1}}{\left(1+\left(\frac{{sin}^2\theta }{l+1}\right)+{|\mathbb{H}|}^2\right)}^{\frac{p}{2(q-1)}}dV\right\}}^{(q-1)}\hfill \end{array}$$

(54)

for $1<q\le 2$.

4. Conclusions Remark

If a domain has Dirichlet boundary conditions, then the Dirichlet eigenvalues are the same as the Laplace eigenvalues. In different branches of mathematics, such as differential geometry, number theory, and mathematical physics, these eigenvalues have important implications. In addition, they are capable of characterizing the geometry of a given space. One Dirichlet eigenvalue represents the domain’s diameter, while higher ones reflect its curvature and its location within Euclidean space. As well as determining the eigenfunctions of the solution, Dirichlet’s eigenvalues will also determine the rate at which the solution decays. To find more motivation for our work, we refer the reader to a number of papers [39,40,41,42,43,44,45,46].