Some Eigenvalues Estimate for the φ-Laplace Operator on Slant Submanifolds of Sasakian Space Forms

School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China Department of Mathematics, College of Science, King Khalid University, 9004 Abha, Saudi Arabia Mathematical Science Department Faculty of Science Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia Department of Mathematics, University of Lagos, Akoka, Lagos State, Nigeria Department of Mathematics, College of Science and Arts, Muhayil King Khalid University, 61413 Abha, Saudi Arabia


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][2][3][4][5][6][7][8][9][10][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 ðr + 1Þ-th anisotropic mean curvature that is an extension of the usual Laplacian operator was also studied in [12]. Let N m be a complete noncompact Riemannian manifold and Σ be the compact domain in N m . Assume that Λ 1 ðΣÞ > 0 denotes the first eigenvalue of the Dirichlet boundary value problem: where Δ denotes the Laplace operator on N m . Then, the first eigenvalue Λ 1 ðN Þ is defined by Λ 1 ðN Þ = inf Σ Λ 1 ðΣÞ: 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 ð N m , gÞ be compact m-dimensional Riemannian manifold, and let Λ 1 denote the first nonzero eigenvalue of the Neumann problem: Δf + Λ 1 f = 0 on N and ∂f where N is the outward normal on ∂N m . A result of Reilly [13] reads the following.
Let N m be Riemannian manifold, and ℝ k is the Euclidean space having dimensions m and k, respectively. The manifold N m is connected, closed, and oriented as well. The N m is isometrically immersed in ℝ k with condition ∂N m = 0. The mean curvature of this isometric immersion is denoted by ℍ, and the first nonzero eigenvalue Λ ∇ 1 of the Laplacian on N m can be written as in the sense of Reily [13].
where the volume element of N m 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 M k ðεÞ having a constant curvature ε is obtained, a well-known evaluation for the first nonzero eigenvalue of Laplacian by the immersion of submanifold N m in simply connected space having m-dimension. This space form includes the Euclidean space ℝ k , the unit sphere S k ð1Þ, and the hyperbolic space ℍð−1Þ k with ε = 0, 1 and ε = −1, respectively. Theorem 1. [13,19] Let N m be an m-dimensional closed orientable submanifold in a k-dimensional space form M k ðε Þ. Then, the first nonnull eigenvalue Λ ∇ 1 of Laplacian satisfies where ℍ is the mean curvature vector of N m in M k ðεÞ. The equality holds if and only if N m is minimally immersed in a geodesic sphere of radius r ε ofM k ðεÞ with r 0 = ðm/Λ Δ 1 Þ 1/2 , r 1 = arcsin r 0 and r −1 = arcsinh r 0 .
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][24][25][26][27][28][29]) through the work of [13]. To define the ϕ-Laplacian which is second order quasilinear elliptic operator on N m (compact Riemannian manifold N m having m-dimension), we have where ϕ > 1 to satisfy the above equation. We have the usual Laplacian for ϕ = 2. On the other hand, the eigenvalue of Δ ϕ 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).
Let us study a Riemannian manifold N m with no boundary. The Rayleigh-type variational characterization is observed in first nonzero eigenvalue of Δ ϕ which is given by Λ 1,ϕ , from (cf. [30]) 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 M 2k+1 ðεÞ. In fact, the main finding of this paper will be announced in the following theorem.

Then
(1) The first nonnull eigenvalue Λ 1,ϕ of the ϕ-Laplacian satisfies (2) The equality carries in (8) and (9) if and only if ϕ = 2 and N m is minimally immersed in a geodesic sphere of radius r ε ofM 2k+1 ðεÞ with the following equalities: Remark 3. For ϕ = 2, our estimate finds the corollary.
The equality's cases are same as that in Theorem 2 (2). This is an immediate application of Theorem 2 by using 1 < ϕ ≤ 2, as Sasakian space form.
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 S 2k+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.
The three parameters of almost-contact structure can be developed on its own as ψ is a ð1, 1Þ-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 It indicates that where ∇ indicates the Riemannian connection in regard to g and U 2 , V 2 is any vector fields onM 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 byM 2k+1 ðεÞ.
With all this, we can represent the curvature tensorR of 3 Journal of Function Spaces M 2m+1 ðεÞ as for any arbitary X 2 , Y 2 , Z 2 , W 2 that belong toM 2k+1 (for more details, see [2,31,32]). Assuming that N m is an m-dimensional submanifold isometrically immersed in a Sasakian space formM 2k+1 ðεÞ: 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 for each U 2 , V 2 ∈ ΓðTN Þ and ζ ∈ ΓðT ⊥ N Þ, where h and A ζ are the second fundamental form and shape operator (analogous to the normal vector field ζ), respectively, for the immersion of N m intoM 2k+1 ðεÞ. They are linked as gðhð In the whole article, ζ is assumed to be tangential to N ; otherwise N is simply anti-invariant. Now for any U ∈ ΓðTN Þ and N ∈ ΓðT ⊥ N Þ, we have ii where TU 2 ðtζÞ and FU 2 ðf ζÞ are the tangential and normal components of ψU 2 ðψζÞ, respectively. From (18), it is not difficult to check that for each U 2 , V 2 ∈ ΓðTN Þ: A submanifold N m is defined to be slant submanifold if for any x ∈ N and for any vector field U 2 ∈ ΓðTN m Þ, linearly independent on ξ, the angle between ψU 2 and TN is a constant angle θðU 2 Þ 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 N m is said to be a slant submanifold if and only if there exists a constant C ∈ ½0, π/ 2 and one one tensor fled T is satisfied by the following: such that C = cos 2 θ: Also, we have consequence of above formula: Remark 7. It is clear that slant submanifold is generalized to invariant submanifold with slant angle θ = 0.
Remark 8. Totally real submanfiold is a particular case of slant submanifold with slant angle θ = 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: The mean curvature and squared norm of the mean curvature vector H N of a Riemannian submanifold N m is defined by Similarly, the length of the second fundamental form h is given: In addition, we denoted the following: Our main motivation comes from the following example: Example 9. (see [33]. Let ðℝ 2k+1 , ψ, ξ, η, gÞ denotes the Sasakian manifold with Sasakian structure: Journal of Function Spaces where ðx i 1 , y i 1 , z 1 Þ, i = 1 ⋯ k are the coordinates system. It is easy to explain that ðℝ 2k+1 , ψ, ξ, η, gÞ is an almost-contact metric manifold. Now consider the 3-dimension submanifold in ℝ 5 with Sasakian structure, for any θ ∈ ½0, ðπ/2Þ such that Under above immersion, N 3 is a three-dimension mini-mal slant submanifold containing slant angle θ and scalar curvature τ = −ðcos 2 θ/3Þ.
Similarly, we give more examples for nonminimal submanifold.
Taking trace of the above equation and using (34), we get: where R is the scalar curvature of N m and S is the length of the second fundamental form h:

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.
where ρ a is the covariant derivative of ρ with along to e a , ψ u 1 , v 1 , t ð Þ= 2 e λu 1 cos u 1 cos v 1 , e λu 1 sin u 1 cos v 1 , e λu 1 cos u 1 sin v 1 , e λu 1 sin u 1 sin v 1 , t : 5 Journal of Function Spaces that is, dρ = ∑ a ρ a e a .
By pulling back (33) to N m by x, we have: from there, it is easy to get the meaningful relationship:

Proof of Main Results
This section is about proving Theorem 2 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].
Based on the above arguments, we have the following lemma: Lemma 12 (see [1]. Let N m be a slant submanifold of Sasa- for ϕ > 1.
In the above Lemma 12 by the constructed test function, we produce a higher bound for Λ 1,ϕ in the form of the conformal functionand in comparability with Lemma 2.7 in [27]. Then, we have where Γ stands for conformal map in Lemma 12 and for all ϕ > 1. Identified by Y ε , the standard metric onM 2k+1 ðεÞ and Proof.
Considering ω a as a test function along with Lemma 12, we derive Observe ∑ 2k+2 a=1 jω a j 2 = 1; then jω a j ≤ 1. We accomplish By using 1 < ϕ ≤ 2, then we derive Using the Hölder inequality along with (44)-(46), we are able to get This gives us the desired outcome (43). On the contrary, if we assume ϕ ≥ 2, then by applying Hölder inequality, we have And the outcome we get is Journal of Function Spaces The Minkowski's inequality gives Hence, (43) follows from (44), (49), and (50). This completes the proof of proposition.☐ We are now in the position to prove Theorem 2.

Proof of Theorem 2.
To begin with 1 < ϕ ≤ 2, then ϕ/2 ≤ 1 . Taking help from Proposition 13 and implementing the Hölder inequality, we have By using both conformal relations and Gauss equations, it is possible to calculate e 2ρ . LetM 2k+1 =M 2k+1 ðεÞ, andg = e −2ρ Y ε ,g = Γ * Y 1 in previous. From (36), the Gauss equations for the embedding x and the slant embedding ω = Γ ∘ x are respectively Tracing (38), it can be established that which replaceing together with (52) and (53) into (54) gives It implies the following: Now from (39) and (41), we derive Dividing by mðm − 1Þ in the above equation, it implies that Taking integration along dV, it is not complicated to get the following: 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 Ð N ðe 2ν Þ ϕ/2 by using Ð N ðe 2ρ Þ. We did multiply both sides of (58) with the factor e ðϕ−2Þρ and then solve by using integration on N m (cf. [25]).

Journal of Function Spaces
Next, it follows from the assumption m ≥ 2ϕ − 2. We apply Young's inequality; then From (60) and (61) we deduce the following inequality: 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: for each a = 1, ⋯, 2k + 2. If 1 < ϕ < 2, then jω a j = 0 or 1. So, there would be only one a for which jω a j = 1 and Λ 1,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]. Let ϕ > 2 and equality remains valid in (9); then it shows that (49) and (50) become the equalities which indicates and condition j∇ω a j = 0 holds for existing a. It shows that ω a is a constant value and Λ 1,ϕ is also equal to zero. This last result again represents a conflict that Λ 1,ϕ is a nonnull eigenvalue. This completes the proof of the theorem.
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 Remark 11, 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 18.
Assuming that N m is an m-dimensional closed orientated invariant submanifold in a Sasakian space form M 2k+1 ðεÞ, then Λ Δ 1 satisfies the following inequality for the Laplacian: Similarly, from Theorem 5, we obtain the following corollary.
3.4. Application to Anti-Invariant Submanifolds of Sasakian Space Forms. Using Remark 8 and Theorem 2, we have the following results: Corollary 20. Let N m be an mð≥2Þ-dimensional closed orientated anti-invariant submanifold in a Sasakian space form M 2k+1 ðεÞ. Then, Λ 1,ϕ satisfies the following inequality for the ϕ-Laplacian: From Corollary 4 for ϕ = 2 and Remark 8, we have the following.

Corollary 21.
Assuming that N m is an m-dimensional closed orientated anti-invariant submanifold in a Sasakian space formM 2k+1 ðεÞ, then Λ Δ 1 satisfies the following inequality for the Laplacian: Similarly, from Theorem 5, we obtain the following corollary.

Data Availability
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Conflicts of Interest
The authors declare no competing of interest.