Infimum of the spectrum of Laplace-Beltrami operator on Cartan classical domains of type Ⅲ and Ⅳ

Sujuan Long; Qiqi Zhang; Guijuan Lin; Conghui Shen; Sujuan Long; Qiqi Zhang; Guijuan Lin; Conghui Shen

doi:10.3934/math.2023999

AIMS Mathematics

2023, Volume 8, Issue 8: 19582-19594. doi: 10.3934/math.2023999

Previous Article Next Article

Research article Special Issues

Infimum of the spectrum of Laplace-Beltrami operator on Cartan classical domains of type Ⅲ and Ⅳ

1.
School of Mathematics and Data Science, Minjiang University, Fuzhou $ 350108 $, China
2.
College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China
3.
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou $ 363000 $, China

Received: 06 April 2023 Accepted: 24 May 2023 Published: 09 June 2023
MSC : 35P05, 32Q15, 47F05

Let $ R_{{\cal A}} $ be the Cartan classical domains of type Ⅲ and Ⅳ, and $ \Delta_g $ is assumed to be the Laplace-Beltrami operator associated to the Bergman metric $ g $ on $ R_{{\cal A}} $. In this paper, we derive an estimate for $ \lambda_1(\Delta_g) $, which is the bottom of the spectrum of $ \Delta_g $ on $ R_{{\cal A}} $.
- Laplace-Beltrami operator,
- Cartan classical domains,
- Bergman metric
Citation: Sujuan Long, Qiqi Zhang, Guijuan Lin, Conghui Shen. Infimum of the spectrum of Laplace-Beltrami operator on Cartan classical domains of type Ⅲ and Ⅳ[J]. AIMS Mathematics, 2023, 8(8): 19582-19594. doi: 10.3934/math.2023999

Related Papers:

Abstract

Let $ R_{{\cal A}} $ be the Cartan classical domains of type Ⅲ and Ⅳ, and $ \Delta_g $ is assumed to be the Laplace-Beltrami operator associated to the Bergman metric $ g $ on $ R_{{\cal A}} $. In this paper, we derive an estimate for $ \lambda_1(\Delta_g) $, which is the bottom of the spectrum of $ \Delta_g $ on $ R_{{\cal A}} $.

References

[1]	P. Li, A lower bound for the first eigenvalue of the Laplacian on a compact manifold, Indiana Univ. Math. J., 28 (1979), 1013–1019. http://dx.doi.org/10.1512/iumj.1979.28.28075 doi: 10.1512/iumj.1979.28.28075
[2]	P. Li, Lecture notes on geometric analysis, Seoul: Seoul National University, 1993.
[3]	P. Li, S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Proc. Symp. Pure Math., 36 (1980), 205–239.
[4]	P. Li, S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309–318. http://doi.org/10.1007/BF01213210 doi: 10.1007/BF01213210
[5]	S. Udagawa, Compact Kähler manifolds and the eigenvalues of the Laplacian, Colloq. Math., 56 (1988), 341–349. http://doi.org/10.4064/cm-56-2-341-349 doi: 10.4064/cm-56-2-341-349
[6]	S. Y. Cheng, Eigenvalue comparison theorems and its geometric application, Math. Z., 143 (1975), 289–297. http://doi.org/10.1007/BF01214381 doi: 10.1007/BF01214381
[7]	P. Li, J. P. Wang, Complete manifolds with positive spectrum, J. Differential Geom., 58 (2001), 501–534. http://doi.org/10.4310/jdg/1090348357 doi: 10.4310/jdg/1090348357
[8]	P. Li, J. P. Wang, Complete manifolds with positive spectrum Ⅱ, J. Differential Geom., 62 (2002), 143–162. http://doi.org/10.4310/jdg/1090425532 doi: 10.4310/jdg/1090425532
[9]	P. Li, J. P. Wang, Comparison theorem for Kähler manifolds and positivity of spectrum, J. Differential Geom., 69 (2005), 43–74. http://doi.org/10.4310/jdg/1121540339 doi: 10.4310/jdg/1121540339
[10]	S. Y. Li, X. D. Wang, Bottom of spectrum of Kähler manifolds with a strongly pseudoconvex boundary, Int. Math. Res. Notices, 2012 (2012), 4351-4371. http://dx.doi.org/10.1093/imrn/rnr185 doi: 10.1093/imrn/rnr185
[11]	O. Munteanu, A sharp estimate for the bottom of the spectrum of the Laplacian on Kähler manifolds, J. Differential Geom., 83 (2009), 163–187. http://doi.org/10.4310/jdg/1253804354 doi: 10.4310/jdg/1253804354
[12]	S. Y. Li, M. A. Tran, Infimum of the spectrum of Laplace-Beltrami operator on a bounded pseudoconvex domain with a Kähler metric metric of Bergman type, Comm. Anal. Geom., 18 (2010), 375–395. http://dx.doi.org/10.4310/CAG.2010.v18.n2.a5 doi: 10.4310/CAG.2010.v18.n2.a5
[13]	L. Z. Ji, P. Li, J. P. Wang, Ends of locally symmetric spaces with maximal bottom spectrum, J. Reine Angew. Math., 632 (2009), 1–35. http://doi.org/10.1515/CRELLE.2009.048 doi: 10.1515/CRELLE.2009.048
[14]	S. L. Kong, P. Li, D. T. Zhou, Spectrum of the Laplacian on quaternionic Kähler manifolds, J. Differential Geom., 78 (2008), 295–332. http://doi.org/10.4310/jdg/1203000269 doi: 10.4310/jdg/1203000269
[15]	S. Long, K. Li, Infimum of the spectrum of Laplace-Beltrami operator on classical bounded symmetric domains with Bergman metric, J. Math. Res. Appl., 39 (2019), 43–50.
[16]	L. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Soc., 1963.
[17]	Q. Lu, The classical manifolds and classical domains, Beijing: Science Press, 2011.
[18]	J. Faraut, A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 88 (1990), 64–89. http://doi.org/10.1016/0022-1236(90)90119-6 doi: 10.1016/0022-1236(90)90119-6

Reader Comments

Your name:*

Email:*
© 2023 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)