# Some Eigenvalues Estimate for theϕ-Laplace Operator on Slant Submanifolds of Sasakian Space Forms

Authors: Yanlin Li; Akram Ali; Fatemah Mofarreh; Abimbola Abolarinwa; Rifaqat Ali
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of theϕ-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ-Laplacian operator on closed oriented m-dimensional slant submanifolds in a Sasakian space form M~2k+1ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ-Laplacian on slant submanifold in a sphere S2n+1 with ε=1 and ϕ=2.


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## Body

## 1. Introduction and Statement of Main Results

Finding the bound of the eigenvalue for the Laplacian on a given manifold is a key aspect in Riemannian geometry, and there are different classes of submanifolds such as slant submanifolds, CR-submanifolds, and singular submanifolds, which motivates further exploration and attracts many researchers from different research areas [1–11]. A major objective is to study the eigenvalue that appears as solutions of the Dirichlet or Neumann boundary value problems for curvature functions. Because there are different boundary conditions on a manifold, one can take a philosophical view of the Dirichlet boundary condition, finding the upper bound for the eigenvalue as a method of investigation for the suitable bound of the Laplacian on the given manifold. In recent years, there has been increasing interest to obtain the eigenvalue for the Laplacian operator and the ϕ-Laplacian operators. The linearized operator of the r+1-th anisotropic mean curvature that is an extension of the usual Laplacian operator was also studied in [12]. Let Nm be a complete noncompact Riemannian manifold and Σ be the compact domain in Nm. Assume that Λ1Σ>0 denotes the first eigenvalue of the Dirichlet boundary value problem:
(1)Δf+Λf=0,inΣandf=0on∂Σ,where Δ denotes the Laplace operator on Nm. Then, the first eigenvalue Λ1N is defined by Λ1N=infΣΛ1Σ. The Reilly formula is solely concerned with the manifold’s intrinsic geometry, and most notably with the PDE in question. With the following example, this is easily understood. Let Nm,g be compact m-dimensional Riemannian manifold, and let Λ1 denote the first nonzero eigenvalue of the Neumann problem:
(2)Δf+Λ1f=0onNand∂f∂N=0on∂N,where N is the outward normal on ∂Nm. A result of Reilly [13] reads the following.LetNm be Riemannian manifold, and ℝk is the Euclidean space having dimensions m and k, respectively. The manifold Nm is connected, closed, and oriented as well. The Nm is isometrically immersed in ℝk with condition ∂Nm=0. The mean curvature of this isometric immersion is denoted by ℍ, and the first nonzero eigenvalue Λ1∇ of the Laplacian on Nm can be written as in the sense of Reily [13].
(3)Λ1∇≤lVolNm∫Nmℍ2dV.where the volume element of Nm is denoted by dV. It can be seen in literature that many authors prompted to work in such inequalities for different ambient spaces after the breakthrough of inequality (3). In Minkowski spaces, the upper bound for Finsler submanifold is proposed by both Zeng and He [14]. This upper bound relates to the very 1st eigenvalue of the ϕ-Laplacian. For closed manifold, the first eigenvalue of the ϕ-Laplace operator is presented by Seto and Wei [15] by using the condition of integral curvature. In the hyperbolic space, the bottom spectra of the Laplace manifold for complete and no-compact submanifold are calculated by Lin [16], and mean curvature has condition of integral pinching. In addition to this, Xiong [17] contributed his role on closed hyperspace to find the first Hodge-Laplacian eigenvalue. Moreover, Xiong worked for complete Riemannian manifold which includes the Reilly-type sharp upper bounds for the eigenvalues in product manifolds. The generalized Reilly inequality (3) and first nonzero eigenvalue of ϕ-Laplace operator are calculated by Du et.al [18]. On compact submanifold, they used the Wentzel-Laplace operator having boundary in Euclidean space. Following the same pattern, for Neumann and Dirichlet boundary restrictions, Blacker and Seto [3] evidenced the Lichnerowicz-type lower bound for the first nonzero eigenvalue of the ϕ-Laplacian. They used the Hessian decomposition on Kaehler manifolds having a positive Ricci curvature. A simply connected space form Mkε having a constant curvature ε is obtained, a well-known evaluation for the first nonzero eigenvalue of Laplacian by the immersion of submanifold Nm in simply connected space having m-dimension. This space form includes the Euclidean space ℝk, the unit sphere Sk1, and the hyperbolic space ℍ−1k with ε=0,1 and ε=−1, respectively.Theorem 1.
[13, 19] Let Nm be an m-dimensional closed orientable submanifold in a k-dimensional space form Mkε. Then, the first nonnull eigenvalue Λ1∇ of Laplacian satisfies
(4)Λ1∇≤mVolN∫Nmℍ2+εdV,where ℍ is the mean curvature vector of Nm in Mkε. The equality holds if and only if Nm is minimally immersed in a geodesic sphere of radius rε of M~kε with r0=m/Λ1Δ1/2,r1=arcsinr0 and r−1=arcsinhr0.In [20, 21], the first nonnull eigenvalue of the Laplacian is evidenced which is considered the generalization of the results in Reilly [13]. For various ambient spaces, the outcomes of different classes of Riemannian submanifolds indicate that the result of both 1st nonzero eigenvalues depict alike inequalities and ultimately have identical upper bounds [20, 22]. This result is valid for both Dirichlet and Neumann conditions. For ambient manifold, it is obvious from the literature that Laplace and ϕ-Laplace operators on Riemannian submanifolds helped a lot to acquire different breakthroughs in Riemannian geometry (see [12, 14, 23–29]) through the work of [13]. To define the ϕ-Laplacian which is second order quasilinear elliptic operator on Nm (compact Riemannian manifold Nm having m-dimension), we have
(5)Δϕf=div∇fϕ−2∇f,where ϕ>1 to satisfy the above equation. We have the usual Laplacian for ϕ=2. On the other hand, the eigenvalue of Δϕ has similarity with Laplacian. For instance, if a function f≠0 satisfies the subsequent equation with Dirichlet boundary condition (1) (or Neumann boundary condition (2)), then Λ (any real number) is Dirichlet eigenvalue. Similarly, the above criteria also hold for Neumann boundary condition (2).
(6)Δϕf=−Λfϕ−2f.Let us study a Riemannian manifoldNm with no boundary. The Rayleigh-type variational characterization is observed in first nonzero eigenvalue of Δϕ which is given by Λ1,ϕ, from (cf. [30])
(7)Λ1,ϕ=inf∫N∇fϕ∫Nfϕf∈W1,ϕNl0,∫Nfϕ−2f=0.This naturally raises the question: Is it possible to generalize the Reilly-type inequalities for submanifolds in spheres through the class almost-contact manifolds which were proved in [1, 20, 21]? In the Sasakian space form, our aim is to derive the 1st eigenvalue for the ϕ-Laplacian on slant submanifold. Following this opinion and motivated by the historical development in the analysis of the first nonnull eigenvalue of the ϕ-Laplacian on submanifold in various space forms, by using the Gauss equation and influenced by studied of [18, 20, 22], our goal is to give general view of the above Reilly’s conclusion for ϕ-Laplace operator, and we are going to provide a sharp estimate to the first eigenvalue for the ϕ-Laplacian on slant submanifold of Sasakian space form M2k+1ε. In fact, the main finding of this paper will be announced in the following theorem.Theorem 2.
LetNm be an m≥2-dimensional closed orientated slant submanifold in a Sasakian space form M~2k+1ε. Then
(1)
The first nonnull eigenvalueΛ1,ϕ of the ϕ-Laplacian satisfies(8)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ/2×∫Nmε+34+ε−143cos2θ−2m+ℍ2dVϕ2for1<ϕ≤2,(9)Λ1,ϕ≤21−ϕ/2k+1ϕ/2−1mϕ/2VolN×∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ/2dVfor2<ϕ≤m2+1(2)
The equality carries in (8) and (9) if and only if ϕ=2 and Nm is minimally immersed in a geodesic sphere of radius rε of M~2k+1ε with the following equalities:(10)r0=mΛ1Δ1/2,r1=sin−1r0,r−1=sinh−1r0.Remark 3.
Forϕ=2, our estimate finds the corollary.Corollary 4.
LetNm be an m-dimensional closed orientated slant submanifold in Sasakian space form M~2k+1ε. Then, Λ1Δ satisfies the following inequality for the Laplacian:
(11)Λ1Δ≤mVolN∫Nℍ2+ε+34+ε−143cos2θ−2mdV.
The equality’s cases are same as that in Theorem2 (2).
This is an immediate application of Theorem2 by using 1<ϕ≤2, as Sasakian space form.Theorem 5.
LetNm be an m≥2-dimensional closed orientated slant submanifold in Sasakian space form M~2k+1ε. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(12)Λ1,p≤21−ϕ/2m+11−ϕ/2mϕ/2VolNϕ−1×∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.Remark 6.
Consider the inequality (12) and give value ϕ=2, and then inequality (12) reduces to the Reilly-type inequality (11). This shows that Reilly-type calculates the first eigenvalue for the Laplace operator on slant submanifold in Euclidean sphere S2k+1 (see Theorem 2 in [20] and Theorem 1.5 in [21]), which are the same on the case of our Theorem 2 for ε=1 and ϕ=2.


## 2. Preliminaries and Notations

An almost-contact manifold is odd-dimensionalC∞-manifold M~2k+1,g with almost-contact structure ψ,ξ,η that satisfies the succeeding properties, i.e.
(13)ψ2=−I+η⊗ξ,ηξ=1,ψξ=0,η∘ψ=0,gψU2,ψV2=gU2,V2−ηU2ηV2,ηU2=gU2,ξ,for any U2,V2 belonging to M~2k+1.The three parameters of almost-contact structure can be developed on its own asψ is a 1,1-type tensor field, whereas ξ is the structure vector field and η is dual 1-form. In the perspective of the Riemannian connection, an almost-contact manifold can be a Sasakian manifold [2, 31] if
(14)∇~U2ψV2=gU2,V2ξ−ηV2U2.It indicates that(15)∇~U2ξ=−ψU2,where ∇ indicates the Riemannian connection in regard to g and U2,V2 is any vector fields on M~2k+1. We consider that M~2k+1 converts into a Sasakian space form if it has ψ-sectional constant curvature ε and is represented by M~2k+1ε. With all this, we can represent the curvature tensor R~ of M~2m+1ε as
(16)R~X2,Y2,Z2,W2=ε+34gY2,Z2gX2,W2−gX2,Z2gY2,W2+ε−14ηX2ηZ2gY2,W2+ηW2ηY2gX2,Z2−ηY2ηZ2gX2,W2−ηX2gY2,Z2ηW2+gψY2,Z2gψX2,W2−gψX2,Z2gψY2,W2+2gX2,ψY2gψZ2,W2,for any arbitary X2,Y2,Z2,W2 that belong to M~2k+1 (for more details, see [2, 31, 32]).Assuming thatNm is an m-dimensional submanifold isometrically immersed in a Sasakian space form M~2k+1ε. If ∇ and ∇⊥ are generated connections on the tangent bundle TN and normal bundle T⊥N of N, respectively, then the Gauss and Weingarten formulas are given by
(17)i∇~U2V2=∇U2V2+hU2,V2,ii∇~U2ζ=−AζU2+∇U2⊥ζ,for each U2,V2∈ΓTN and ζ∈ΓT⊥N, where h and Aζ are the second fundamental form and shape operator (analogous to the normal vector field ζ), respectively, for the immersion of Nm into M~2k+1ε. They are linked as ghU2,V2,ζ=gAζU2,V2. In the whole article, ζ is assumed to be tangential to N; otherwise N is simply anti-invariant. Now for any U∈ΓTN and N∈ΓT⊥N, we have
(18)iψU2=TU2+FU2,(19)iiψζ=tζ+fζ,where TU2tζ and FU2fζ are the tangential and normal components of ψU2ψζ, respectively. From (18), it is not difficult to check that for each U2,V2∈ΓTN:
(20)gTU2,V2=−gU2,TV2.A submanifoldNm is defined to be slant submanifold if for any x∈N and for any vector field U2∈ΓTNm, linearly independent on ξ, the angle between ψU2 and TN is a constant angle θU2 that lies between zero and π/2.It follows the definition of slant immersions by Cabrerizo et al. [33] who obtained the necessary and sufficient condition that a submanifold Nm is said to be a slant submanifold if and only if there exists a constant C∈0,π/2 and one one tensor fled T is satisfied by the following:
(21)T2=−CI−η⊗ξ,such that C=cos2θ. Also, we have consequence of above formula:
(22)gTU2,TV2=cos2θgU2,V2−ηU2ηV2.Remark 7.
It is clear that slant submanifold is generalized to invariant submanifold with slant angleθ=0.Remark 8.
Totally real submanfiold is a particular case of slant submanifold with slant angleθ=π/2.With the help of moving frame method, we explore some of the interesting features of conformal geometry and slant submanifolds. The specific convection has been applied on indices range. Though we exclude in a way the following:(23)1≤i,j,s,⋯≤m;m+1≤α,β,γ,⋯≤2k+11≤a,b,c,⋯≤2k+1.The mean curvature and squared norm of the mean curvature vectorHN of a Riemannian submanifold Nm is defined by
(24)ℍ=1m∑i=1nhei,eiandℍ2=1m2∑r=m+1k∑i=1mhiir2.Similarly, the length of the second fundamental formh is given:
(25)hijr=ghei,ej,erandS=h2=∑r=m+1k∑i,j=1nhijr2.In addition, we denoted the following:(26)T2=∑i,j=1mg2Tei,ej.Our main motivation comes from the following example:Example 9. (see [33].
Letℝ2k+1,ψ,ξ,η,g denotes the Sasakian manifold with Sasakian structure:
(27)η=12dz1−∑i=1ky1idx1i,ξ=2∂∂z1,g=η⊗η+14∑i=1kdx1i⊗dx1i⊗dy1i⊗dy1i,ψ∑i=1kXi∂∂x1i+Yi∂∂y1i+Z∂∂z1=∑i=1kYi∂∂y1i−Xi∂∂x1i+∑i=1kYiy1i∂∂z1,where x1i,y1i,z1,i=1⋯k are the coordinates system. It is easy to explain that ℝ2k+1,ψ,ξ,η,g is an almost-contact metric manifold. Now consider the 3-dimension submanifold in ℝ5 with Sasakian structure, for any θ∈0,π/2 such that
(28)ψu1,v1,t=2u1cosθ,u1sinθ,v1,0,t.
Under above immersion,N3 is a three-dimension minimal slant submanifold containing slant angle θ and scalar curvature τ=−cos2θ/3.Similarly, we give more examples for nonminimal submanifold.Example 10 (see [33]).
For any constantλ, we define an immersion:
(29)ψu1,v1,t=2eλu1cosu1cosv1,eλu1sinu1cosv1,eλu1cosu1sinv1,eλu1sinu1sinv1,t.
It is easy see that above immersion is a three-dimension slant submanifold with slant angleθ=cos−1λ/1+λ2. Moreover, scalar curvature τ=−λ2/31+λ2 and mean curvature ℍ=2e−λu1/31+λ2.It is necessary to clarify the definition of the curvature tensorR~ for slant submanifold in Sasakian space form M~2k+1ε and is given by
(30)ψR~ei,ej,ei,ej=ε+34m2−m+ε−143∑i,j=1mg2ψei,ej−2m−1.On the other hand, lete1,⋯eq,⋯em=ξ be an orthonormal basis of TxN such that e1,e2=secθTe1,⋯,e2q=secθTe2q−1,⋯e2q+1=ξ, since we define
(31)gψe1,e2=gψe1,secθTe1=secθgψe1,Te1=secθgTe1,Te1.It is clear that the dimension ofNm can be decomposed as m=2q+1. Then, from (22), we derive that
(32)gψe1,e2=cosθ.In similar way, we repeat that(33)g2ψei,ei+1=cos2θ∑i,j=1mg2ψei,ej=m−1cos2θ.Merging (30) and (33) implies that
(34)R~ei,ej,ei,ej=ε+34m2−m+ε−14m−13cos2θ−2.


### 2.1. Structure Equations for Slant Submanifolds

Letx be a totally real embedding from Nm to an 2k+1-dimensional Riemannian manifold M~,g~. Then Nm has a generated metric gN=x∗g~. Let us consider M~2k+1=M~2k+1ε, then pulling back in [[1] Eq. (12)] by x and using [[1] Eqs (13), (14)], we obtain the Gauss equations for slant submanifold in Sasakian space form M~2k+1ε and taking into account (30).
(35)Rijtl=ε+34δitδjl−δilδjt+ε−143ψei,ej−2m−1+∑αhitαhjlα−hilαhjtα.Taking trace of the above equation and using (34), we get:
(36)R=m2ℍ2−S+ε+34mm−1+ε−14m−13cos2θ−2,where R is the scalar curvature of Nm and S is the length of the second fundamental form h.


### 2.2. Conformal Relations

In this section, we’ll look at how the conformal transformation affects curvature and the second fundamental form. Although these relationships are well-known (cf. [1]), we use the moving frame method to provide a quick proof for readers’ convenience.Assume thatM~2k+1 consist a conformal metric g~=e2ρg~, where ρ∈C∞M~. Then Ω~a=eρΩa stands for the dual coframe of M~,g~, and e~a=eρea stands for the orthogonal frame of M~,g~. The equality’s equations of M~,g~ are given in [[1], Eqs. (20), (21), (22) (23)] by:
(37)Ω~ab=Ωab+ρaΩb−ρbΩa,where ρa is the covariant derivative of ρ with along to ea, that is, dρ=∑aρaea.
(38)e2ρR~ijtl=Rijtl−ρitδjl+ρjlδit−ρilδjt−ρjtδil+ρiρtδjl+ρjρlδit−ρjρtδil−ρiρlδjt−∇α2δitδjl−δilδjt.By pulling back (33) toNm by x, we have:
(39)h~ijα=e−ρhijα−ραδji,(40)ℍ~α=e−αℍα−ρα,from there, it is easy to get the meaningful relationship:
(41)e2ρS~−mℍ~2+mℍ2=S.


## 2.1. Structure Equations for Slant Submanifolds

Letx be a totally real embedding from Nm to an 2k+1-dimensional Riemannian manifold M~,g~. Then Nm has a generated metric gN=x∗g~. Let us consider M~2k+1=M~2k+1ε, then pulling back in [[1] Eq. (12)] by x and using [[1] Eqs (13), (14)], we obtain the Gauss equations for slant submanifold in Sasakian space form M~2k+1ε and taking into account (30).
(35)Rijtl=ε+34δitδjl−δilδjt+ε−143ψei,ej−2m−1+∑αhitαhjlα−hilαhjtα.Taking trace of the above equation and using (34), we get:
(36)R=m2ℍ2−S+ε+34mm−1+ε−14m−13cos2θ−2,where R is the scalar curvature of Nm and S is the length of the second fundamental form h.


## 2.2. Conformal Relations

In this section, we’ll look at how the conformal transformation affects curvature and the second fundamental form. Although these relationships are well-known (cf. [1]), we use the moving frame method to provide a quick proof for readers’ convenience.Assume thatM~2k+1 consist a conformal metric g~=e2ρg~, where ρ∈C∞M~. Then Ω~a=eρΩa stands for the dual coframe of M~,g~, and e~a=eρea stands for the orthogonal frame of M~,g~. The equality’s equations of M~,g~ are given in [[1], Eqs. (20), (21), (22) (23)] by:
(37)Ω~ab=Ωab+ρaΩb−ρbΩa,where ρa is the covariant derivative of ρ with along to ea, that is, dρ=∑aρaea.
(38)e2ρR~ijtl=Rijtl−ρitδjl+ρjlδit−ρilδjt−ρjtδil+ρiρtδjl+ρjρlδit−ρjρtδil−ρiρlδjt−∇α2δitδjl−δilδjt.By pulling back (33) toNm by x, we have:
(39)h~ijα=e−ρhijα−ραδji,(40)ℍ~α=e−αℍα−ρα,from there, it is easy to get the meaningful relationship:
(41)e2ρS~−mℍ~2+mℍ2=S.


## 3. Proof of Main Results

This section is about proving Theorem2 announced in the previous section. First of all, some fundamental formulas will be presented and some useful lemmas from [27] will be recalled to our setting. For the purpose of this paper, we will provide a significant lemma that is motivated by the review in [1, 27].Remark 11.
A simply connected Sasakian space formM2k+1 is a 2k+1-sphere S2k+1 and Euclidean space ℝ2k+1 with constant ψ-sectional curvature ε=1 and ε=−3, respectively.Based on the above arguments, we have the following lemma:Lemma 12 (see [1].
LetNm be a slant submanifold of Sasakian space form M~2k+1ε that is close and oriented with dimension m≥2. If x:Nm⟶M~2k+1ε is embedding from Nm into M~2k+1ε, then there is a standard conformal map Γ:M~2k+1ε⟶S2k+11⊂ℝ2k+2 such that the embedding ω=Γ∘x=ω1,⋯ω2k+2 satisfies
(42)∫Nmωaϕ−2ωadVN=0,a=1,⋯2k+1,for ϕ>1.In the above Lemma12 by the constructed test function, we produce a higher bound for Λ1,ϕ in the form of the conformal functionand in comparability with Lemma 2.7 in [27].Proposition 13.
LetNm be an m≥2-dimensional closed orientated slant submanifold into Sasakian space form M~2k+1ε. Then, we have
(43)Λ1,ϕVolNm≤21−ϕ2k+11−ϕ2mϕ2∫Nme2ρϕ2dV,where Γ stands for conformal map in Lemma 12 and for all ϕ>1. Identified by Yε, the standard metric on M~2k+1ε and considered Γ∗Y1=e2ρYε.Proof.
Consideringωa as a test function along with Lemma 12, we derive
(44)Λ1,ϕ∫Nmωaϕ≤∇ωaϕdV,1≤a≤2k+1.
Observe∑a=12k+2ωa2=1; then ωa≤1. We accomplish
(45)∑a=12k+2∇ωa2=∑i=1m∇eiω2=me2ρ.
By using1<ϕ≤2, then we derive
(46)ωa2≤ωaϕ.
Using the Hölder inequality along with (44)–(46), we are able to get
(47)Λ1,ϕVolN=Λ1,ϕ∑a=12k+2∫Nmωa2dV≤Λ1,ϕ∑a=12k+2∫NmωaϕdV≤Λ1,ϕ∫Nm∑a=12k+2∇ωaϕdV≤2k+21−ϕ/2∫Nm∑a=1m∇ωa2ϕ2dV=21−ϕ2k+11−ϕ2∫Nmme2ρϕ2dV.
This gives us the desired outcome (43). On the contrary, if we assume ϕ≥2, then by applying Hölder inequality, we have
(48)1=∑a=12k+2ωa2≤2k+21−2ϕ∑a=12k+2ωaϕ2ϕ.
And the outcome we get is(49)Λ1,ϕVolNm≤2k+2ϕ2−1∑a=12k+2Λ1,ϕ∫NmωaϕdV.
The Minkowski’s inequality gives(50)∑a=12k+2∇ωaϕ≤∑a=12k+2∇ωa2ϕ2=me2ρϕ2.
Hence, (43) follows from (44), (49), and (50). This completes the proof of proposition.☐We are now in the position to prove Theorem2.


### 3.1. Proof of Theorem2

To begin with1<ϕ≤2, then ϕ/2≤1. Taking help from Proposition 13 and implementing the Hölder inequality, we have
(51)Λ1,ϕVolNm≤21−ϕ2k+11−ϕ2mϕ2∫Nme2ρϕ2dV≤21−ϕ2k+11−ϕ2∣mϕ2VolN1−ϕ2∫Nme2ρdVϕ2.By using both conformal relations and Gauss equations, it is possible to calculatee2ρ. Let M~2k+1=M~2k+1ε,andg~=e−2ρYε,g~=Γ∗Y1 in previous. From (36), the Gauss equations for the embedding x and the slant embedding ω=Γ∘x are respectively
(52)R=ε+34mm−1+ε−14m−13cos2θ−2+mm−1ℍ2+mℍ2−S,(53)R~−mm−1=mm−1ℍ~2+mℍ~2−S~.Tracing (38), it can be established that
(54)e2ρR~=R−m−2m−1∇ρ2−2m−1Δρ,which replaceing together with (52) and (53) into (54) gives
(55)e2ρmm−1+mm−1ℍ~2+mℍ~2−S~=ε+34mm−1+ε−14m−13cos2θ−2+mm−1ℍ2+mℍ2−S−m−2m−1∇ρ2−2m−1Δρ.It implies the following:(56)e2ρS~−S−m−2m−1∇ρ2−2m−1Δρ=mm−1e2ρ−ε+34−ε−143cos2θ−2m+e2ρℍ~2−ℍ2+me2ρℍ~2−ℍ2.Now from (39) and (41), we derive
(57)mm−1e2ρ−ε+34−ε−143cos2θ−2n+mm−1∑αℍα−ρα2=mm−1ℍ2−m−2m−1∇ρ2−2m−1Δρ.Dividing bymm−1 in the above equation, it implies that
(58)e2ρ=ε+34+ε−143cos2θ−2m+ℍ2−2mΔρ−m−2mΔρ2−∇~ρ⊥−ℍ2.Taking integration alongdV, it is not complicated to get the following:
(59)Λ1,ϕVolNm≤21−ϕ2k+11−ϕ2∣mϕ2VolNm1−ϕ2∫Nme2ρdVϕ2≤2k+21−ϕ/2∣mϕ/2VolNϕ/2−1∫Nmε+34+ε−143cos2θ−2m+ℍ2dVϕ2.The above result is comparable to (8) as we desired to prove. In the case where ϕ>2, it is not possible to apply the Hölder inequality directly to govern ∫Ne2νϕ/2 by using ∫Ne2ρ. We did multiply both sides of (58) with the factor eϕ−2ρ and then solve by using integration on Nm (cf. [25]).
(60)∫NmeϕρdV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV−m−2−2ϕ+4m∫Neϕ−2Δρ2dV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV.Next, it follows from the assumptionm≥2ϕ−2. We apply Young’s inequality; then
(61)∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV≤2ϕ∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2dV+ϕ−2ϕ∫NmeϕρdV.From (60) and (61) we deduce the following inequality:
(62)∫NmepρdV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2dV.Now putting (62) into (43), we obtain (9). In the case of slant submanifolds, the equality case holds in (8); then considering the cases in (44) and (46), we get this:
(63)ωa2=ωaϕ,Δϕωa=−Λ1,ϕωaϕ−2ωa,for each a=1,⋯,2k+2. If 1<ϕ<2, then ωa=0 or 1. So, there would be only one a for which ωa=1 and Λ1,ϕ=0, which seems to be a contradiction as the eigenvalue is nonzero. For this reason, we consider ϕ=2 and only restricted to Laplacian case. After this, we are able to apply Theorem 1.5 from [21].Letϕ>2 and equality remains valid in (9); then it shows that (49) and (50) become the equalities which indicates
(64)ω1ϕ=⋯=ω2k+2ϕ,and condition ∇ωa=0 holds for existing a. It shows that ωa is a constant value and Λ1,ϕ is also equal to zero. This last result again represents a conflict that Λ1,ϕ is a nonnull eigenvalue. This completes the proof of the theorem.


### 3.2. Proof of Theorem5

Suppose that1<ϕ≤2; we have ϕ/2ϕ−1≥1. Then, by the Hölder inequality, we have
(65)∫Nmε+34+ε−143cos2θ−2m+ℍ2dV≤VolN1−2ϕ−1ϕ×∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2ϕ−1dV2ϕ−1ϕ.Thus, combining Equation (8) with (65), we get the desired result (12). This completes the proof of the theorem.As a result of the observations in Remark11, the next result will be specified as a special variant of Theorem 2. To be precise, we determine the following result by replacing ε=1 in (8) and (9), respectively.Corollary 14.
AssumeNm is an m≥2-dimensional closed orientated slant submanifold in 2k+1-sphere S2k+11. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(66)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNp/2∫Nm1+ℍ2dVϕ2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolN∫Nk1+ℍ2ϕ2dVfor2<ϕ≤m2+1.There is an additional corollary derived from Corollary14 as follows.Corollary 15.
Assuming thatNm is an m≥2-dimensional closed orientated slant submanifold in 2k+1-sphere S2k+11, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(67)Λ1,ϕ≤2k+21−ϕ/2mϕ/2VolNϕ−1∫Nm1+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.Remark 16.
It is noticed that Corollaries14 and 15 are exactely same as Theorem 1.5 in [20].


### 3.3. Application to Invariant Submanifolds of Sasakian Space Forms

Using Remark7 and Theorem 2, we have the following results.Corollary 17.
LetNm be an m≥2-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(68)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ/2×∫Nmε+34+ε−14m+ℍ2dVϕ/2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+1ϕ/2−1mϕ/2VolN×∫Nmε+34+ε−14m+ℍ2ϕ/2dVfor2<ϕ≤m2+1.From Corollary4 for ϕ=2 and Remark 7, we have the following.Corollary 18.
Assuming thatNm is an m-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1Δ satisfies the following inequality for the Laplacian:
(69)Λ1Δ≤mVolN∫Nℍ2+ε+34+ε−14mdV.Similarly, from Theorem5, we obtain the following corollary.Corollary 19.
Assuming thatNm is an m≥2-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(70)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ−1×∫Nmε+34+ε−14m+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.


### 3.4. Application to Anti-Invariant Submanifolds of Sasakian Space Forms

Using Remark8 and Theorem 2, we have the following results:Corollary 20.
LetNm be an m≥2-dimensional closed orientated anti-invariant submanifold in a Sasakian space form M~2k+1ε. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(71)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ/2∫Nmε+34−ε−12m+ℍ2dVp/2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+1ϕ/2−1mp/2VolN∫Nmε+34−ε−12m+ℍ2ϕ/2dVfor2<ϕ≤m2+1.From Corollary4 for ϕ=2 and Remark 8, we have the following.Corollary 21.
Assuming thatNm is an m-dimensional closed orientated anti-invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1Δ satisfies the following inequality for the Laplacian:
(72)Λ1Δ≤mVolN∫Nℍ2+ε+34−ε−12mdV.Similarly, from Theorem5, we obtain the following corollary.Corollary 22.
Assuming thatNm is an m≥2-dimensional closed orientated ani-invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(73)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ−1∫Nmε+34−ε−12m+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.


## 3.1. Proof of Theorem2

To begin with1<ϕ≤2, then ϕ/2≤1. Taking help from Proposition 13 and implementing the Hölder inequality, we have
(51)Λ1,ϕVolNm≤21−ϕ2k+11−ϕ2mϕ2∫Nme2ρϕ2dV≤21−ϕ2k+11−ϕ2∣mϕ2VolN1−ϕ2∫Nme2ρdVϕ2.By using both conformal relations and Gauss equations, it is possible to calculatee2ρ. Let M~2k+1=M~2k+1ε,andg~=e−2ρYε,g~=Γ∗Y1 in previous. From (36), the Gauss equations for the embedding x and the slant embedding ω=Γ∘x are respectively
(52)R=ε+34mm−1+ε−14m−13cos2θ−2+mm−1ℍ2+mℍ2−S,(53)R~−mm−1=mm−1ℍ~2+mℍ~2−S~.Tracing (38), it can be established that
(54)e2ρR~=R−m−2m−1∇ρ2−2m−1Δρ,which replaceing together with (52) and (53) into (54) gives
(55)e2ρmm−1+mm−1ℍ~2+mℍ~2−S~=ε+34mm−1+ε−14m−13cos2θ−2+mm−1ℍ2+mℍ2−S−m−2m−1∇ρ2−2m−1Δρ.It implies the following:(56)e2ρS~−S−m−2m−1∇ρ2−2m−1Δρ=mm−1e2ρ−ε+34−ε−143cos2θ−2m+e2ρℍ~2−ℍ2+me2ρℍ~2−ℍ2.Now from (39) and (41), we derive
(57)mm−1e2ρ−ε+34−ε−143cos2θ−2n+mm−1∑αℍα−ρα2=mm−1ℍ2−m−2m−1∇ρ2−2m−1Δρ.Dividing bymm−1 in the above equation, it implies that
(58)e2ρ=ε+34+ε−143cos2θ−2m+ℍ2−2mΔρ−m−2mΔρ2−∇~ρ⊥−ℍ2.Taking integration alongdV, it is not complicated to get the following:
(59)Λ1,ϕVolNm≤21−ϕ2k+11−ϕ2∣mϕ2VolNm1−ϕ2∫Nme2ρdVϕ2≤2k+21−ϕ/2∣mϕ/2VolNϕ/2−1∫Nmε+34+ε−143cos2θ−2m+ℍ2dVϕ2.The above result is comparable to (8) as we desired to prove. In the case where ϕ>2, it is not possible to apply the Hölder inequality directly to govern ∫Ne2νϕ/2 by using ∫Ne2ρ. We did multiply both sides of (58) with the factor eϕ−2ρ and then solve by using integration on Nm (cf. [25]).
(60)∫NmeϕρdV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV−m−2−2ϕ+4m∫Neϕ−2Δρ2dV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV.Next, it follows from the assumptionm≥2ϕ−2. We apply Young’s inequality; then
(61)∫Nmε+34+ε−143cos2θ−2m+ℍ2eϕ−2ρdV≤2ϕ∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2dV+ϕ−2ϕ∫NmeϕρdV.From (60) and (61) we deduce the following inequality:
(62)∫NmepρdV≤∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2dV.Now putting (62) into (43), we obtain (9). In the case of slant submanifolds, the equality case holds in (8); then considering the cases in (44) and (46), we get this:
(63)ωa2=ωaϕ,Δϕωa=−Λ1,ϕωaϕ−2ωa,for each a=1,⋯,2k+2. If 1<ϕ<2, then ωa=0 or 1. So, there would be only one a for which ωa=1 and Λ1,ϕ=0, which seems to be a contradiction as the eigenvalue is nonzero. For this reason, we consider ϕ=2 and only restricted to Laplacian case. After this, we are able to apply Theorem 1.5 from [21].Letϕ>2 and equality remains valid in (9); then it shows that (49) and (50) become the equalities which indicates
(64)ω1ϕ=⋯=ω2k+2ϕ,and condition ∇ωa=0 holds for existing a. It shows that ωa is a constant value and Λ1,ϕ is also equal to zero. This last result again represents a conflict that Λ1,ϕ is a nonnull eigenvalue. This completes the proof of the theorem.


## 3.2. Proof of Theorem5

Suppose that1<ϕ≤2; we have ϕ/2ϕ−1≥1. Then, by the Hölder inequality, we have
(65)∫Nmε+34+ε−143cos2θ−2m+ℍ2dV≤VolN1−2ϕ−1ϕ×∫Nmε+34+ε−143cos2θ−2m+ℍ2ϕ2ϕ−1dV2ϕ−1ϕ.Thus, combining Equation (8) with (65), we get the desired result (12). This completes the proof of the theorem.As a result of the observations in Remark11, the next result will be specified as a special variant of Theorem 2. To be precise, we determine the following result by replacing ε=1 in (8) and (9), respectively.Corollary 14.
AssumeNm is an m≥2-dimensional closed orientated slant submanifold in 2k+1-sphere S2k+11. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(66)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNp/2∫Nm1+ℍ2dVϕ2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolN∫Nk1+ℍ2ϕ2dVfor2<ϕ≤m2+1.There is an additional corollary derived from Corollary14 as follows.Corollary 15.
Assuming thatNm is an m≥2-dimensional closed orientated slant submanifold in 2k+1-sphere S2k+11, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(67)Λ1,ϕ≤2k+21−ϕ/2mϕ/2VolNϕ−1∫Nm1+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.Remark 16.
It is noticed that Corollaries14 and 15 are exactely same as Theorem 1.5 in [20].


## 3.3. Application to Invariant Submanifolds of Sasakian Space Forms

Using Remark7 and Theorem 2, we have the following results.Corollary 17.
LetNm be an m≥2-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(68)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ/2×∫Nmε+34+ε−14m+ℍ2dVϕ/2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+1ϕ/2−1mϕ/2VolN×∫Nmε+34+ε−14m+ℍ2ϕ/2dVfor2<ϕ≤m2+1.From Corollary4 for ϕ=2 and Remark 7, we have the following.Corollary 18.
Assuming thatNm is an m-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1Δ satisfies the following inequality for the Laplacian:
(69)Λ1Δ≤mVolN∫Nℍ2+ε+34+ε−14mdV.Similarly, from Theorem5, we obtain the following corollary.Corollary 19.
Assuming thatNm is an m≥2-dimensional closed orientated invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(70)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ−1×∫Nmε+34+ε−14m+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.


## 3.4. Application to Anti-Invariant Submanifolds of Sasakian Space Forms

Using Remark8 and Theorem 2, we have the following results:Corollary 20.
LetNm be an m≥2-dimensional closed orientated anti-invariant submanifold in a Sasakian space form M~2k+1ε. Then, Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(71)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ/2∫Nmε+34−ε−12m+ℍ2dVp/2for1<ϕ≤2,Λ1,ϕ≤21−ϕ/2k+1ϕ/2−1mp/2VolN∫Nmε+34−ε−12m+ℍ2ϕ/2dVfor2<ϕ≤m2+1.From Corollary4 for ϕ=2 and Remark 8, we have the following.Corollary 21.
Assuming thatNm is an m-dimensional closed orientated anti-invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1Δ satisfies the following inequality for the Laplacian:
(72)Λ1Δ≤mVolN∫Nℍ2+ε+34−ε−12mdV.Similarly, from Theorem5, we obtain the following corollary.Corollary 22.
Assuming thatNm is an m≥2-dimensional closed orientated ani-invariant submanifold in a Sasakian space form M~2k+1ε, then Λ1,ϕ satisfies the following inequality for the ϕ-Laplacian:
(73)Λ1,ϕ≤21−ϕ/2k+11−ϕ/2mϕ/2VolNϕ−1∫Nmε+34−ε−12m+ℍ2ϕ2ϕ−1dVϕ−1,for 1<ϕ≤2.



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