Journal of the Mathematical Society of Japan

Inequalities for eigenvalues of the biharmonic operator with weight on Riemannian manifolds

Qiaoling WANG and Changyu XIA

Full-text: Open access

Abstract

Given a compact Riemannian manifold M with boundary (possibly empty), we consider the eigenvalues of the biharmonic operator with weight on M, proving a general inequality involving them. Using this inequality, we consider these eigenvalues when M is a compact domain of one of the following three spaces: 1) a complex projective space, 2) a minimal submanifold of a Euclidean space and 3) a minimal submanifold of a unit sphere.

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 2 (2010), 597-622.

Dates
First available in Project Euclid: 7 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1273236714

Digital Object Identifier
doi:10.2969/jmsj/06220597

Mathematical Reviews number (MathSciNet)
MR2662854

Zentralblatt MATH identifier
1200.53042

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 58G25
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
universal bounds eigenvalues biharmonic operator with weight complex projective space minimal submanifolds sphere Euclidean space

Citation

WANG, Qiaoling; XIA, Changyu. Inequalities for eigenvalues of the biharmonic operator with weight on Riemannian manifolds. J. Math. Soc. Japan 62 (2010), no. 2, 597--622. doi:10.2969/jmsj/06220597. https://projecteuclid.org/euclid.jmsj/1273236714


Export citation

References

  • M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry, Edingurgh, 1998, (eds. E. B. Davies and Yu Safalov), London Math. Soc. Lecture Notes, 273, Cambridge Univ. Press, Cambridge, 1999, pp.,95–139.
  • M. S. Ashbaugh, Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H C Yang, Proc. India Acad. Sci. Math. Sci., 112 (2002), 3–30.
  • M. S. Ashbaugh and R. D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc., 25 (1991), 19–29.
  • M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacian and extensions, Ann. of Math., 135 (1992), 601–628.
  • M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne-Pólya-Weinberger conjecture, Commun. Math. Phys., 147 (1992), 181–190.
  • Z. C. Chen and C. L. Qian, Estimates for discrete spectrum of Laplacian operator with any order, J. China Univ. Sci. Tech., 20 (1990), 259–266.
  • Q. M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331 (2005), 445–460.
  • Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces, J. Math. Soc. Japan, 58 (2006), 545–561.
  • Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc., 358 (2006), 2625–2635.
  • A. El Soufi, E. M. Harrell II and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operator on submanifolds, Tran. Amer. Math. Soc., 361 (2009), 2337–2350.
  • Evans M. Harrell, II, Some geometric bounds on eigenvalue gaps, Comm. Partial Differential Equations, 18 (1993), 179–198.
  • Evans M. Harrell, II, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Comm. Partial Differential Equations, 32 (2007), 401–413.
  • Evans M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations, 19 (1994), 2037–2055.
  • Evans M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues of some differential operators, Lecture Notes in Pure and Appl. Math., 168, Marcel Dekker, New York, 1995, pp.,235–244.
  • Evans M. Harrell, II and J. Stubbe, On trace inequalities and the universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797–1809.
  • G. N. Hile and M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J., 29 (1980), 523–538.
  • G. N. Hile and R. Z Yeh, Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math., 112 (1984), 115–133.
  • S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615–642.
  • P. F. Leung, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. (Series A), 50 (1991), 409–416.
  • P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helv., 55 (1980), 347–363.
  • L. E. Payne, G. Pólya and H. F. Weinberger, Sur le quotient de deux fréquences propres cosécutives, Comptes Rendus Acad. Sci. Paris, 241 (1955), 917–919.
  • L. E. Payne, G. Pólya and H. F. Weiberger, On the ratio of consecutive eigenvalues, J. Math. and Phys., 35 (1956), 289–298.
  • R. Schoen and S. T. Yau, Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994, v+235 pp.
  • Q. Wang and C. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Ana., 245 (2007), 334–352.
  • Q. Wang and C. Xia, Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds, Ann. Acad. Sci. Fen. Math., 33 (2008), 319–336.
  • H. C. Yang, An estimate of the difference between cosecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste, 1991.
  • P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 55–63.