The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

Lamia Saeed Alqahtani; Akram Ali; Lamia Saeed Alqahtani; Akram Ali

doi:10.3934/math.2024168

AIMS Mathematics

2024, Volume 9, Issue 2: 3426-3439. doi: 10.3934/math.2024168

Previous Article Next Article

Research article

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

Lamia Saeed Alqahtani ^{1
,
,},
Akram Ali ²

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia

Received: 23 August 2023 Revised: 17 December 2023 Accepted: 22 December 2023 Published: 08 January 2024
MSC : 35P15, 53C42, 58C40

In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.
- eigenvalues,
- elliptical Laplacian operators,
- complex space forms,
- slant submanifolds,
- $ \beta $-Laplcians operators
Citation: Lamia Saeed Alqahtani, Akram Ali. The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms[J]. AIMS Mathematics, 2024, 9(2): 3426-3439. doi: 10.3934/math.2024168

Related Papers:

Abstract

In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.

References

[1]	A. M. Matei, Conformal bounds for the first eigenvalue of the $p$-Laplacian, Nonlinear Anal., 80 (2013), 88–95. https://doi.org/10.1016/j.na.2012.11.026 doi: 10.1016/j.na.2012.11.026
[2]	A. Naber, D. Valtorta, Sharp estimates on the first eigenvalue of the $p$-Laplacian with negative Ricci lower bound, Math. Z., 277 (2014), 867–891. https://doi.org/10.1007/s00209-014-1282-x doi: 10.1007/s00209-014-1282-x
[3]	F. Q. Zeng, Q. He, Reilly-type inequalities for the first eigenvalue of $p$-Laplcian of submanifolds in Minkowski spaces, Mediterr. J. Math., 14 (2017), 1–9. https://doi.org/10.1007/s00009-017-1005-8 doi: 10.1007/s00009-017-1005-8
[4]	R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv., 52 (1977), 525–533. https://doi.org/10.1007/BF02567385 doi: 10.1007/BF02567385
[5]	S. Seto, G. F. Wei, First eigenvalue of the $p$-Laplacian under integral curvature condition, Nonlinear Anal., 163 (2017), 60–70. https://doi.org/10.1016/j.na.2017.07.007 doi: 10.1016/j.na.2017.07.007
[6]	H. Z. Lin, Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space, Nonlinear Anal., 148 (2017), 126–137. https://doi.org/10.1016/j.na.2016.09.015 doi: 10.1016/j.na.2016.09.015
[7]	C. W. Xiong, Eigenvalue estimates of Reilly type in product manifolds and eigenvalue comparison of strip domains, Differ. Geom. Appl., 60 (2018), 104–115. https://doi.org/10.1016/j.difgeo.2018.06.003 doi: 10.1016/j.difgeo.2018.06.003
[8]	F. Du, J. Mao, Reilly-type inequalities for $p$-Laplacian compact Riemannian manifolds, Front. Math. China, 10 (2015), 583–594. https://doi.org/10.1007/s11464-015-0422-x doi: 10.1007/s11464-015-0422-x
[9]	C. Blacker, S. Seto, First eigenvalue of the $p$-Laplacian on Kaehler manifolds, Proc. Amer. Math. Soc., 147 (2019), 2197–2206.
[10]	F. Du, Q. L. Wang, C. Y. Xia, Estimates for the eigenvalues of the Wentzell-Laplace operator, J. Geom. Phys., 129 (2018), 25–33. https://doi.org/10.1016/j.geomphys.2018.02.020 doi: 10.1016/j.geomphys.2018.02.020
[11]	A. El Soufi, S. Ilias, Une inégalité du type 'Reilly' pour les sous-variétés de l'espace hyperbolique, Comment. Math. Helv., 67 (1992), 167–181.
[12]	H. Chen, X. F. Wang, Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds, Differ. Geom. Appl., 63 (2019), 1–29. https://doi.org/10.1016/j.difgeo.2018.12.008 doi: 10.1016/j.difgeo.2018.12.008
[13]	H. Chen, G. F. Wei, Reilly-type inequalities for $p$-Laplacian on submanifolds in space forms, Nonlinear Anal., 184 (2019), 210–217. https://doi.org/10.1016/j.na.2019.02.009 doi: 10.1016/j.na.2019.02.009
[14]	F. Cavalletti, A. Mondino, Sharp geometry and functional inequalities in metric measure spaces with lower Ricci curvature bounds, Geom. Topol., 21 (2017), 603–645. https://doi.org/10.2140/gt.2017.21.603 doi: 10.2140/gt.2017.21.603
[15]	S. Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z., 143 (1975), 289–297. https://doi.org/10.1007/BF01214381 doi: 10.1007/BF01214381
[16]	Y. J. He, Reilly type inequality for the first eigenvalue of the $L_{r; F}$ operator, Differ. Geom. Appl., 31 (2013), 321–330. https://doi.org/10.1016/j.difgeo.2013.03.003 doi: 10.1016/j.difgeo.2013.03.003
[17]	A. Ali, A. H. Alkhaldi, P. Laurian-Ioan, R. Ali, Eigenvalue inequalities for the $p$-Laplacian operator on C-totally real submanifolds in Sasakian space forms, Appl. Anal., 101 (2022), 702–713. https://doi.org/10.1080/00036811.2020.1758307 doi: 10.1080/00036811.2020.1758307
[18]	A. H. Alkhaldi, M. A. Khan, M. Aquib, L. S. Alqahtani, Estimation of eigenvalues for the $\psi$-Laplace operator on Bi-slant submanifolds of Sasakian space forms, Front. Phys., 10 (2022), 252. https://doi.org/10.3389/fphy.2022.870119 doi: 10.3389/fphy.2022.870119
[19]	M. A. Khan, A. H. Alkhaldi, M. Aquib, Estimation of eigenvalues for the $\alpha$-Laplace operator on pseudo-slant submanifolds of generalized Sasakian space forms, AIMS Math., 7 (2022), 16054–16066. https://doi.org/10.3934/math.2022879 doi: 10.3934/math.2022879
[20]	Y. L. Li, A. Ali, F. Mofarreh, A. Abolarinwa, R. Ali, Some eigenvalues estimate for the ϕ-Laplace operator on slant submanifolds of Sasakian space forms. J. Funct. Spaces, 2021 (2021), 1–10. https://doi.org/10.1155/2021/6195939 doi: 10.1155/2021/6195939
[21]	Y. L. Li, F. Mofarreh, R. P. Agrawal, A. Ali, Reilly-type inequality for the Φ-Laplace operator on semislant submanifolds of Sasakian space forms. J. Inequal. Appl., 2022 (2022), 1–17. https://doi.org/10.1186/s13660-022-02838-5 doi: 10.1186/s13660-022-02838-5
[22]	Y. L. Li, A. Ali, F. Mofarreh, A. Abolarinwa, N. Alshehri, A. Ali, Bounds for eigenvalues of $q$-Laplacian on contact submanifolds of Sasakian space forms, Mathematics, 11 (2023), 1–14. https://doi.org/10.3390/math11234717 doi: 10.3390/math11234717
[23]	L. Véron, Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds, In: Colloquia Mathematica Societatis János Bolyai, 62 (1991), 317–352.
[24]	B. Y. Chen, K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc., 193 (1974), 257–266.
[25]	B. Y. Chen, Geometry of slant submanifolds, Leuven: Katholieke Universiteit Leuven, 1990.
[26]	A. Ali, J. W. Lee, A. H. Alkhaldi, The first eigenvalue for the $p$-Laplacian on Lagrangian submanifolds in complex space forms, Int. J. Math., 33 (2022), 2250016. https://doi.org/10.1142/S0129167X22500161 doi: 10.1142/S0129167X22500161
[27]	B. Y. Chen, Geometry of submanifolds (pure and applied mathematics, 22), New York: M. Dekker, 1973.
[28]	B. Y. Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital., 10 (1974), 380–385.
[29]	P. Li, S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269–291. https://doi.org/10.1007/BF01399507 doi: 10.1007/BF01399507
[30]	A. El Soufi, S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature, Commun. Math. Phys., 208 (2000), 761–770. https://doi.org/10.1007/s002200050009 doi: 10.1007/s002200050009
[31]	D. Chen, H. Z. Li, Second eigenvalue of Paneitz operators and mean curvature, Commun. Math. Phys., 305 (2011), 555–562. https://doi.org/10.1007/s00220-011-1281-2 doi: 10.1007/s00220-011-1281-2
[32]	C. D. Sogge, Hangzhou lectures on eigenfunctions of the Laplacian, Princeton University Press, 2014.
[33]	B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Berlin, Heidelberg: Springer, 2005. https://doi.org/10.1007/b104762
[34]	P. Lindqvist, A nonlinear eigenvalue problem, In: Topics in mathematical analysis, World Scientific, 2008,175–203. https://doi.org/10.1142/9789812811066_0005
[35]	R. Steven, The Laplacian on a Riemannian manifold, Cambridge: Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511623783
[36]	D. Valtorta, Sharp estimates on the first eigenvalue of the $p$-Laplacian, Nonlinear Anal., 75 (2012), 4974–4994. https://doi.org/10.1016/j.na.2012.04.012 doi: 10.1016/j.na.2012.04.012

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)