Abstract
We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang (Pac J Math 208(2):325–345, 2003 [26]) for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne–Pólya–Weinberger and Yang universal inequalities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ashbaugh M.S.: The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H. C. Yang. Proc. Indian Acad. Sci. Math. Sci. 112(1), 3–30 (2002) Spectral and inverse spectral theory (Goa, 2000)
Ashbaugh, M.S., Hermi, L.: On Harrell–Stubbe type inequalities for the discrete spectrum of a self-adjoint operator. arXiv:0712.4396v1 [math.SP] (2007)
Barletta E.: The Lichnerowicz theorem on CR manifolds. Tsukuba J. Math. 31(1), 77–97 (2007)
Barletta E., Dragomir S.: On the spectrum of a strictly pseudoconvex CR manifold. Abh. Math. Sem. Univ. Hambg 67, 33–46 (1997)
Barletta, E., Dragomir, S.: Sublaplacians on CR manifolds. Bull. Math. Soc. Sci. Math. Roum (NS) 52(100)(1), 3–32 (2009)
Barletta E., Dragomir S., Urakawa H.: Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds. Indiana Univ. Math. J. 50(2), 719–746 (2001)
Barletta, E., Dragomir, S., Urakawa, H.: Yang-Mills fields on CR manifolds. J. Math. Phys. 47(8), 083504, 41 (2006)
Chang S.-C., Chiu H.-L.: On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold. Math. Ann. 345(1), 33–51 (2001)
Chen D., Cheng Q.-M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn 60(2), 325–339 (2008)
Cheng Q.-M., Yang H.: Estimates on eigenvalues of Laplacian. Math. Ann. 331(2), 445–460 (2005)
Cheng Q.-M., Yang H.: Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 337(1), 159–175 (2007)
Danielli D., Garofalo N., Nhieu D.M.: Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv. Math. 215(1), 292–378 (2007)
Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, Volume 246 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (2006)
Eells J., Lemaire L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20(5), 385–524 (1988)
El Soufi A., Ilias S.: Une inégalité du type “Reilly” pour les sous-variétés de l’espace hyperbolique. Comment. Math. Helv. 67(2), 167–181 (1992)
El Soufi A., Harrell E.M. II, Ilias S.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Am. Math. Soc. 361(5), 2337–2350 (2009)
Greenleaf A.: The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold. Commun. Partial Differ. Equ. 10(2), 191–217 (1985)
Harrell E.M. II: Some geometric bounds on eigenvalue gaps. Commun. Partial Differ. Equ. 18(1–2), 179–198 (1993)
Harrell E.M. II: Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators. Commun. Partial Differ. Equ. 32(1–3), 401–413 (2007)
Harrell E.M. II, Michel P.L.: Commutator bounds for eigenvalues, with applications to spectral geometry. Commun. Partial Differ. Equ. 19(11–12), 2037–2055 (1994)
Harrell E.M. II, Stubbe J.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349(5), 1797–1809 (1997)
Harrell E.M. II, Stubbe J.: Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators. SIAM J. Math. Anal. 42(5), 2261–2274 (2010)
Hile G.N., Protter M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29(4), 523–538 (1980)
Li S.-Y., Luk H.-S.: The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold. Proc. Am. Math. Soc. 132(3), 789–798 (2004) (electronic)
Menikoff A., Sjöstrand J.: On the eigenvalues of a class of hypoelliptic operators. Math. Ann. 235(1), 55–85 (1978)
Niu P., Zhang H.: Payne–Polya–Weinberger type inequalities for eigenvalues of nonelliptic operators. Pac. J. Math. 208(2), 325–345 (2003)
Payne L.E., Pólya G., Weinberger H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)
Peng X.C., Chen W.Y.: Spectra of subelliptic operators on S 3. J. Math. (Wuhan) 29(3), 297–299 (2009)
Ponge, R.S.: Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds. Mem. Am. Math. Soc. 194(906), viii+ 134 (2008)
Yang, H.C.: An estimate of the difference between consecutive eigenvalues. preprint IC/91/60 of the Intl. Revised version, preprint (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Malchiodi.
Rights and permissions
About this article
Cite this article
Aribi, A., El Soufi, A. Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold. Calc. Var. 47, 437–463 (2013). https://doi.org/10.1007/s00526-012-0523-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0523-2