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Characterization of Lagrangian submanifolds by geometric inequalities in complex space forms. (English) Zbl 1481.53073

Summary: In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold \(M^n\) minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere \(\mathbb{S}^n\). We also obtain Simons-type inequality for same ambient space form.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53D12 Lagrangian submanifolds; Maslov index
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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[1] Chen, B. Y.; Dillen, F., Optimal general inequalities for Lagrangian submanifolds in complex space forms, Journal of Mathematical Analysis and Applications, 379, 1, 229-239 (2011) · Zbl 1217.53082 · doi:10.1016/j.jmaa.2010.12.058
[2] Berger, M.; Gauduchon, P.; Mazet, E., Lespectre d‘une Wariete Riemanniene, 194 (1971), Springer, Berlin: Lectures Notes in Mathematics, Springer, Berlin · Zbl 0223.53034
[3] Blair, D. E., On the characterization of complex projective space by differential equations, Journal of the Mathematical Society of Japan, 27, 1 (1975) · Zbl 0355.53036 · doi:10.2969/jmsj/02710009
[4] Besse, A. L., Einstein Manifolds (2007), Springer Science & Business Media
[5] Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, Journal of the Mathematical Society of Japan, 14, 333-340 (1962) · Zbl 0115.39302 · doi:10.2969/jmsj/01430333
[6] Muto, H., The first eigenvalue of the Laplacian of an isoparametric minimal hypersurface in a unit sphere, Mathematische Zeitschrift, 197, 4, 531-549 (1988) · Zbl 0652.53039 · doi:10.1007/BF01159810
[7] Ali, A.; Laurian-Ioan, P.; Alkhaldi, A. H.; Alqahtani, L. S., Ricci curvature on warped product submanifolds of complex space forms and its applications, International Journal of Geometric Methods in Modern Physics, 16, 9, 1950142 (2019) · Zbl 07804184 · doi:10.1142/S0219887819501421
[8] Ali, R.; Mofarreh, F.; Alluhaibi, N.; Ali, A.; Ahmad, I., On differential equations characterizing Legendrian submanifolds of sasakian space forms, Mathematics, 8, 2, 150 (2020) · doi:10.3390/math8020150
[9] Ali, A.; Mofarreh, F., Geometric inequalities of bi-warped product submanifolds of nearly Kenmotsu manifolds and their applications, Mathematics, 8, 10, 1805 (2020) · doi:10.3390/math8101805
[10] Barros, A.; Bessa, G., Estimates of the first eigenvalue of minimal hypersurfaces of \(\mathbb{S}^{\text{n} + 1} \), Matematica Contemporanea, 17, 71-75 (1999) · Zbl 0978.53109
[11] Chern, S. S.; Do Carmo, M.; Kobayashi, S., Minimal Submanifolds of the Sphere with Second Fundamental Form Length (1970), Springer, New York: Functional Analysis and Related Fields, Springer, New York · Zbl 0216.44001
[12] Chavel, I., Eigenvalues in Riemannian Geometry (1984), New York: Academic Press, New York · Zbl 0551.53001
[13] Choi, H. I.; Wang, A. N., A first eigenvalue estimate for minimal hypersurfaces, Journal of Differential Geometry, 18, 559-562 (1983) · Zbl 0523.53055 · doi:10.4310/jdg/1214437788
[14] Deshmukh, S.; Al-Eid, A., Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature, Journal of Geometric Analysis, 15, 4, 589-606 (2005) · Zbl 1087.58020 · doi:10.1007/BF02922246
[15] Deshmukh, S., Conformal vector fields and eigenvectors of Laplacian operator, Mathematical Physics, Analysis and Geometry, 15, 2, 163-172 (2012) · Zbl 1255.53030 · doi:10.1007/s11040-012-9106-x
[16] Deshmukh, S.; Al-Solamy, F. R., A note on conformal vector fields on a Riemannian manifold, Colloquium Mathematicum, 136, 1, 65-73 (2014) · Zbl 1301.53036 · doi:10.4064/cm136-1-7
[17] ErkekoÄŸlu, F., Characterizing specific Riemannian manifolds by differential equations, Acta Appl. Math., 76, 2, 195-219 (2003) · Zbl 1033.53033 · doi:10.1023/A:1022987819448
[18] Kupeli, D. N.; Unal, B., Some conditions for Riemannian manifolds to be isometric with Euclidean spheres, J. Differ. Equ, 134, 2, 287-299 (2003)
[19] Barros, A., Applications of Bochner formula to minimal submanifold of the sphere, Journal of Geometry and Physics, 44, 2-3, 196-201 (2002) · Zbl 1033.53047 · doi:10.1016/S0393-0440(02)00061-X
[20] Barros, A.; Gomes, J. N.; Ernani, J. R., A note on rigidity of the almost Ricci soliton, Archiv der Mathematik, 100, 5, 481-490 (2013) · Zbl 1276.53053 · doi:10.1007/s00013-013-0524-1
[21] Deshmukh, S., Characterizing spheres and Euclidean spaces by conformal vector fields, Annali di Matematica Pura ed Applicata., 196, 6, 2135-2145 (2017) · Zbl 1378.53049 · doi:10.1007/s10231-017-0657-0
[22] Deshmukh, S., Almost Ricci solitons isometric to spheres, International Journal of Geometric Methods in Modern Physics, 16, 5 (2019) · Zbl 1422.53028 · doi:10.1142/s0219887819500737
[23] Lawson, B., Local rigidity theorems for minimal hypersurfaces, The Annals of Mathematics, 89, 1, 187-197 (1969) · Zbl 0174.24901 · doi:10.2307/1970816
[24] Leung, P. I., Minimal submanifolds in a sphere, Mathematische Zeitschrift, 183, 1, 75-86 (1983) · Zbl 0491.53045 · doi:10.1007/BF01187216
[25] Tanno, S., Some differential equations on Riemannian manifolds, Journal of the Mathematical Society of Japan, 30, 3, 509-531 (1978) · Zbl 0387.53015 · doi:10.2969/jmsj/03030509
[26] Takahashi, T., Minimal immersion of Riemannian manifolds, Journal of the Mathematical Society of Japan, 18, 380-385 (1966) · Zbl 0145.18601 · doi:10.2969/jmsj/01840380
[27] Tashiro, Y., Complete Riemannian manifolds and some vector fields, Transactions of the American Mathematical Society, 117, 251-275 (1965) · Zbl 0136.17701 · doi:10.1090/S0002-9947-1965-0174022-6
[28] Lichnerowicz, A., Geometrie des Groupes de Transformations (1958), Dunod · Zbl 0096.16001
[29] Jiancheng, L.; Zhang, Q., Simons-type inequalities for the compact submanifolds in the space of constant curvature, Kodai Mathematical Journal, 30, 3, 344-351 (2007) · Zbl 1135.53038 · doi:10.2996/kmj/1193924938
[30] Simons, J., Minimal varieties in Riemannian manifolds, The Annals of Mathematics, 88, 1, 62-105 (1968) · Zbl 0181.49702 · doi:10.2307/1970556
[31] Barbosa, J. N.; Barros, A., A lower bound for the norm of the second fundamental form of minimal hypersurfaces of \(\mathbb{S}^{\text{n} + 1} \), Archiv der Mathematik, 81, 4, 478-484 (2003) · Zbl 1058.53048 · doi:10.1007/s00013-003-4767-0
[32] Deshmukh, S.; Al-Solamy, F. R., Conformal gradient vector fields on a compact Riemannian manifold, Colloquium Mathematicum, 112, 1, 157-161 (2008) · Zbl 1135.53022 · doi:10.4064/cm112-1-8
[33] Khan, M. A.; Khan, K., Biwarped product submanifolds of complex space forms, International Journal of Geometric Methods in Modern Physics, 16 (2019) · Zbl 1422.53044 · doi:10.1142/s0219887819500725
[34] Chen, B. Y.; Ogiue, K., On totally real submanifolds, Transactions of the American Mathematical Society, 193, 257-266 (1974) · Zbl 0286.53019 · doi:10.1090/S0002-9947-1974-0346708-7
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