Some Eigenvalues Estimate for the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="16.3568pt" version="1.1" viewBox="-0.0657574 -11.83 9.86325 16.3568" width="9.86325pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g></svg>-</span>Laplace Operator on Slant Submanifolds of Sasakian Space Forms

Li, Yanlin; Ali, Akram; Mofarreh, Fatemah; Abolarinwa, Abimbola; Ali, Rifaqat

doi:https://doi.org/10.1155/2021/6195939

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Research Article | Open Access

Volume 2021 | Article ID 6195939 | https://doi.org/10.1155/2021/6195939

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

Yanlin Li,¹Akram Ali,²Fatemah Mofarreh,³Abimbola Abolarinwa,⁴and Rifaqat Ali⁵

Academic Editor: Umair Ali

Received28 Apr 2021

Accepted08 Sept 2021

Published13 Oct 2021

Abstract

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

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 -Laplacian operators. The linearized operator of the -th anisotropic mean curvature that is an extension of the usual Laplacian operator was also studied in [12]. Let be a complete noncompact Riemannian manifold and be the compact domain in . Assume that denotes the first eigenvalue of the Dirichlet boundary value problem: where denotes the Laplace operator on . Then, the first eigenvalue is defined by 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 be compact -dimensional Riemannian manifold, and let denote the first nonzero eigenvalue of the Neumann problem: where is the outward normal on . A result of Reilly [13] reads the following.

Let be Riemannian manifold, and is the Euclidean space having dimensions and , respectively. The manifold is connected, closed, and oriented as well. The is isometrically immersed in with condition . The mean curvature of this isometric immersion is denoted by , and the first nonzero eigenvalue of the Laplacian on can be written as in the sense of Reily [13]. where the volume element of is denoted by . 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 -Laplacian. For closed manifold, the first eigenvalue of the -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 -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 -Laplacian. They used the Hessian decomposition on Kaehler manifolds having a positive Ricci curvature. A simply connected space form having a constant curvature is obtained, a well-known evaluation for the first nonzero eigenvalue of Laplacian by the immersion of submanifold in simply connected space having -dimension. This space form includes the Euclidean space , the unit sphere , and the hyperbolic space with and , respectively.

Theorem 1. [13, 19] Let be an -dimensional closed orientable submanifold in a -dimensional space form . Then, the first nonnull eigenvalue of Laplacian satisfies where is the mean curvature vector of in . The equality holds if and only if is minimally immersed in a geodesic sphere of radius of with and .

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 -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 -Laplacian which is second order quasilinear elliptic operator on (compact Riemannian manifold having -dimension), we have where to satisfy the above equation. We have the usual Laplacian for . On the other hand, the eigenvalue of has similarity with Laplacian. For instance, if a function satisfies the subsequent equation with Dirichlet boundary condition (1) (or Neumann boundary condition (2)), then (any real number) is Dirichlet eigenvalue. Similarly, the above criteria also hold for Neumann boundary condition (2).

Let us study a Riemannian manifold with no boundary. The Rayleigh-type variational characterization is observed in first nonzero eigenvalue of which is given by , 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 -Laplacian on slant submanifold. Following this opinion and motivated by the historical development in the analysis of the first nonnull eigenvalue of the -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 -Laplace operator, and we are going to provide a sharp estimate to the first eigenvalue for the -Laplacian on slant submanifold of Sasakian space form . In fact, the main finding of this paper will be announced in the following theorem.

Theorem 2. Let be an -dimensional closed orientated slant submanifold in a Sasakian space form . Then (1)The first nonnull eigenvalue of the -Laplacian satisfies(2)The equality carries in (8) and (9) if and only if and is minimally immersed in a geodesic sphere of radius of with the following equalities:

Remark 3. For , our estimate finds the corollary.

Corollary 4. Let be an -dimensional closed orientated slant submanifold in Sasakian space form . Then, satisfies the following inequality for the Laplacian: The equality’s cases are same as that in Theorem 2 (2).
This is an immediate application of Theorem 2 by using , as Sasakian space form.

Theorem 5. Let be an -dimensional closed orientated slant submanifold in Sasakian space form . Then, satisfies the following inequality for the -Laplacian: for .

Remark 6. Consider the inequality (12) and give value , 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 (see Theorem 2 in [20] and Theorem 1.5 in [21]), which are the same on the case of our Theorem 2 for and .

2. Preliminaries and Notations

An almost-contact manifold is odd-dimensional -manifold with almost-contact structure that satisfies the succeeding properties, i.e. for any belonging to .

The three parameters of almost-contact structure can be developed on its own as is a -type tensor field, whereas is the structure vector field and is dual -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 and is any vector fields on . We consider that converts into a Sasakian space form if it has -sectional constant curvature and is represented by . With all this, we can represent the curvature tensor of as for any arbitary that belong to (for more details, see [2, 31, 32]).

Assuming that is an -dimensional submanifold isometrically immersed in a Sasakian space form If and are generated connections on the tangent bundle and normal bundle of , respectively, then the Gauss and Weingarten formulas are given by for each and , where and are the second fundamental form and shape operator (analogous to the normal vector field ), respectively, for the immersion of into . They are linked as . In the whole article, is assumed to be tangential to ; otherwise is simply anti-invariant. Now for any and , we have where and are the tangential and normal components of , respectively. From (18), it is not difficult to check that for each :

A submanifold is defined to be slant submanifold if for any and for any vector field , linearly independent on , the angle between and is a constant angle that lies between zero and .

It follows the definition of slant immersions by Cabrerizo et al. [33] who obtained the necessary and sufficient condition that a submanifold is said to be a slant submanifold if and only if there exists a constant and one one tensor fled is satisfied by the following: such that Also, we have consequence of above formula:

Remark 7. It is clear that slant submanifold is generalized to invariant submanifold with slant angle .

Remark 8. Totally real submanfiold is a particular case of slant submanifold with slant angle .

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 of a Riemannian submanifold is defined by

Similarly, the length of the second fundamental form is given:

In addition, we denoted the following:

Our main motivation comes from the following example:

Example 9. (see [33]. Let denotes the Sasakian manifold with Sasakian structure: where are the coordinates system. It is easy to explain that is an almost-contact metric manifold. Now consider the -dimension submanifold in with Sasakian structure, for any such that Under above immersion, is a three-dimension minimal slant submanifold containing slant angle and scalar curvature .

Similarly, we give more examples for nonminimal submanifold.

Example 10 (see [33]). For any constant , we define an immersion: It is easy see that above immersion is a three-dimension slant submanifold with slant angle . Moreover, scalar curvature and mean curvature .

It is necessary to clarify the definition of the curvature tensor for slant submanifold in Sasakian space form and is given by

On the other hand, let be an orthonormal basis of such that , since we define

It is clear that the dimension of can be decomposed as . Then, from (22), we derive that

In similar way, we repeat that

Merging (30) and (33) implies that

2.1. Structure Equations for Slant Submanifolds

Let be a totally real embedding from to an -dimensional Riemannian manifold . Then has a generated metric . Let us consider , then pulling back in [[1] Eq. (12)] by and using [[1] Eqs (13), (14)], we obtain the Gauss equations for slant submanifold in Sasakian space form and taking into account (30).

Taking trace of the above equation and using (34), we get: where is the scalar curvature of and is the length of the second fundamental form

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 that consist a conformal metric , where . Then stands for the dual coframe of , and stands for the orthogonal frame of . The equality’s equations of are given in [[1], Eqs. (20), (21), (22) (23)] by: where is the covariant derivative of with along to , that is, .

By pulling back (33) to by , we have: from there, it is easy to get the meaningful relationship:

3. 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].

Remark 11. A simply connected Sasakian space form is a -sphere and Euclidean space with constant -sectional curvature and , respectively.

Based on the above arguments, we have the following lemma:

Lemma 12 (see [1]. Let be a slant submanifold of Sasakian space form that is close and oriented with dimension . If is embedding from into , then there is a standard conformal map such that the embedding satisfies for .

In the above Lemma 12 by the constructed test function, we produce a higher bound for in the form of the conformal functionand in comparability with Lemma 2.7 in [27].

Proposition 13. Let be an -dimensional closed orientated slant submanifold into Sasakian space form . Then, we have where stands for conformal map in Lemma 12 and for all . Identified by , the standard metric on and considered .

Proof. Considering as a test function along with Lemma 12, we derive Observe ; then . We accomplish By using , 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 , then by applying Hölder inequality, we have And the outcome we get is 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.

3.1. Proof of Theorem 2

To begin with , then . 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 . Let in previous. From (36), the Gauss equations for the embedding and the slant embedding 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 in the above equation, it implies that

Taking integration along , it is not complicated to get the following:

The above result is comparable to (8) as we desired to prove. In the case where , it is not possible to apply the Hölder inequality directly to govern by using . We did multiply both sides of (58) with the factor and then solve by using integration on (cf. [25]).

Next, it follows from the assumption . 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 . If , then or . So, there would be only one for which and , which seems to be a contradiction as the eigenvalue is nonzero. For this reason, we consider and only restricted to Laplacian case. After this, we are able to apply Theorem 1.5 from [21].

Let and equality remains valid in (9); then it shows that (49) and (50) become the equalities which indicates and condition holds for existing . It shows that is a constant value and is also equal to zero. This last result again represents a conflict that is a nonnull eigenvalue. This completes the proof of the theorem.

3.2. Proof of Theorem 5

Suppose that ; we have . Then, by the Hölder inequality, we have

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 in (8) and (9), respectively.

Corollary 14. Assume is an -dimensional closed orientated slant submanifold in -sphere . Then, satisfies the following inequality for the -Laplacian:

There is an additional corollary derived from Corollary 14 as follows.

Corollary 15. Assuming that is an -dimensional closed orientated slant submanifold in -sphere , then satisfies the following inequality for the -Laplacian: for .

Remark 16. It is noticed that Corollaries 14 and 15 are exactely same as Theorem 1.5 in [20].

3.3. Application to Invariant Submanifolds of Sasakian Space Forms

Using Remark 7 and Theorem 2, we have the following results.

Corollary 17. Let be an -dimensional closed orientated invariant submanifold in a Sasakian space form . Then, satisfies the following inequality for the -Laplacian:

From Corollary 4 for and Remark 7, we have the following.

Corollary 18. Assuming that is an -dimensional closed orientated invariant submanifold in a Sasakian space form , then satisfies the following inequality for the Laplacian:

Similarly, from Theorem 5, we obtain the following corollary.

Corollary 19. Assuming that is an -dimensional closed orientated invariant submanifold in a Sasakian space form , then satisfies the following inequality for the -Laplacian: for .

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 be an -dimensional closed orientated anti-invariant submanifold in a Sasakian space form . Then, satisfies the following inequality for the -Laplacian:

From Corollary 4 for and Remark 8, we have the following.

Corollary 21. Assuming that is an -dimensional closed orientated anti-invariant submanifold in a Sasakian space form , then satisfies the following inequality for the Laplacian:

Similarly, from Theorem 5, we obtain the following corollary.

Corollary 22. Assuming that is an -dimensional closed orientated ani-invariant submanifold in a Sasakian space form , then satisfies the following inequality for the -Laplacian: for .

Data Availability

There is no data used for this manuscript.

Conflicts of Interest

The authors declare no competing of interest.

Authors’ Contributions

All authors have equal contribution and have finalized the manuscript.

Acknowledgments

The authors would like to express their gratitude to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through the General Research Project under grant number G.R.P-316-42. This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University, through the Fast-track Research Funding Program to support publication in the top journal (grant no. 42-FTTJ-97), and the National Natural Science Foundation of China (grant no. 12101168).

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Copyright © 2021 Yanlin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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