Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere

Ibrahim Aldayel; Ibrahim Aldayel

doi:10.3934/math.20231355

AIMS Mathematics

2023, Volume 8, Issue 11: 26532-26542. doi: 10.3934/math.20231355

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Research article

Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere

Ibrahim Aldayel ^,

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P. O. Box-65892, Riyadh 11566, Saudi Arabia

Received: 18 July 2023 Revised: 22 August 2023 Accepted: 27 August 2023 Published: 18 September 2023
MSC : 47A75, 53A10, 53C42

We prove that the first nonzero eigenvalue of the Laplace-Beltrami operator of equator-like minimal submanifold embedded in the sphere $ S^{n+1} $ is equal to $ n $. The proof uses the spectral properties of the heat kernel operator corresponding to the submanifold.
- minimal submanifolds,
- Laplace operator,
- eigenvalues,
- heat kernel
Citation: Ibrahim Aldayel. Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere[J]. AIMS Mathematics, 2023, 8(11): 26532-26542. doi: 10.3934/math.20231355

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Abstract

We prove that the first nonzero eigenvalue of the Laplace-Beltrami operator of equator-like minimal submanifold embedded in the sphere $ S^{n+1} $ is equal to $ n $. The proof uses the spectral properties of the heat kernel operator corresponding to the submanifold.

References

[1]	I. Chavel, Eigenvalues in Riemannian geometry, Orlando: Academic Press, 1984.
[2]	M. A. Choudhary, First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere, Russ. Math., 58 (2014), 56–64. https://doi.org/10.3103/S1066369X14080076 doi: 10.3103/S1066369X14080076
[3]	J. Cheeger, S. T. Yau, A lower bound for the heat kernel, Commun. Pur. Appl. Math., 34 (1981), 465–480. https://doi.org/10.1002/cpa.3160340404 doi: 10.1002/cpa.3160340404
[4]	L. F. Cheung, P. F. Leung, Eigenvalues estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z., 236 (2001), 525–530. https://doi.org/10.1007/PL00004840 doi: 10.1007/PL00004840
[5]	J. Choe, R. Gulliver, Isoperimetric inequalities on minimal submanifolds of space forms, Manuscripta Math., 77 (1992), 169–189. https://doi.org/10.1007/BF02567052 doi: 10.1007/BF02567052
[6]	S. Deshmukh, First nonzero eigenvalue of a minimal hypersurface in the unit sphere, Annali di Mathematica, 191 (2012), 529–537. https://doi.org/10.1007/s10231-011-0194-1 doi: 10.1007/s10231-011-0194-1
[7]	H. Federer, Geometric measure theory, Berlin Heidelberg: Springer-Verlag, 1996. https://doi.org/10.1007/978-3-642-62010-2
[8]	A. Grigor'yan, Heat kernel and analysis on manifolds, Washington: American Mathematical Society/International Press, 2009.
[9]	H. I. Choi, A. N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differ. Geom., 18 (1983), 559–562.
[10]	M. Kotani, The first eigenvalue of homogeneous minimal hypersurfaces in a unit sphere $S^{n+1}(1)$, Tohoku Math. J., 37 (1985), 523–532. https://doi.org/10.2748/tmj/1178228592 doi: 10.2748/tmj/1178228592
[11]	P. Li, Large time behaviour of the heat equation on complete manifolds with non-negative Ricci curvature, Ann. Math., 124 (1986), 1–21. https://doi.org/10.2307/1971385 doi: 10.2307/1971385
[12]	H. Muto, Y. Ohnita, H. Urakawa, Homogeneous minimal hypersurfaces in a unit sphere and the first eigenvalue of the Laplacian, Tohoku Math. J., 36 (1984), 253–267. https://doi.org/10.2748/tmj/1178228851 doi: 10.2748/tmj/1178228851
[13]	B. Solomon, The harmonic analysis of cubic isoparametric minimal hypersurfaces Ⅰ: Dimensions 3 and 6, Am. J. Math., 112 (1990), 157–203. https://doi.org/10.2307/2374713 doi: 10.2307/2374713
[14]	B. Solomon, The harmonic analysis of cubic isoparametric minimal hypersurfaces Ⅱ: Dimensions 12 and 24, Am. J. Math., 112 (1990), 205–241. https://doi.org/10.2307/2374714 doi: 10.2307/2374714
[15]	T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385. https://doi.org/10.2969/jmsj/01840380 doi: 10.2969/jmsj/01840380
[16]	Z. Tang, W. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, J. Differ. Geom., 94 (2013), 521–540.
[17]	S. T. Yau, Seminar on differential geometry, Princeton: Princeton University Press, 1982. https://doi.org/10.1515/9781400881918

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