Abstract
We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudo-convex pseudo-Hermitian structure \(\theta \) on the CR sphere \(\mathbb {S}^{2n+1}\subset \mathbb {C}^{n+1}\), achieves its maximum when \(\theta \) is the standard contact form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aribi, A., Dragomir, S., El Soufi, A.: On the continuity of the eigenvalues of a sublaplacian. Can. Math. Bull. 57(1), 12–24 (2014)
Aribi, A., Dragomir, S., El Soufi, A.: Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold. Differ. Geom. Appl. 39, 113–128 (2015)
Aribi, A., Dragomir, S., El Soufi, A.: A lower bound on the spectrum of the sublaplacian. J. Geom. Anal. 25(3), 1492–1519 (2015)
Aribi, A., El Soufi, A.: Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold. Calc. Var. Partial Differ. Equ. 47(3–4), 437–463 (2013)
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. Hamburg 67, 33–46 (1997)
Barletta, E., Dragomir, S.: Sublaplacians on CR manifolds. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100), 3–32 (2009)
Chang, D.-C., Li, S.-Y.: A zeta function associated to the sub-Laplacian on the unit sphere in \({\mathbb{C}}^N\). J. Anal. Math. 86, 25–48 (2002)
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 (2009)
Cowling, M.G., Klima, O., Sikora, A.: Spectral multipliers for the Kohn sublaplacian on the sphere in \({\mathbb{C}}^n\). Trans. Am. Math. Soc. 363(2), 611–631 (2011)
El Soufi, A., Ilias, S.: Le volume conforme et ses applications d’après Li et Yau. In: Séminaire de Théorie Spectrale et Géométrie, Année 1983–1984, pp. VII.1–VII.15. Univ. Grenoble I, Saint-Martin-d’Hères (1984)
El Soufi, A., Ilias, S.: Immersions minimales, première valeur propre du laplacien et volume conforme. Math. Ann. 275(2), 257–267 (1986)
Greenleaf, A.: The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold. Comm. Partial Differ. Equ. 10(2), 191–217 (1985)
Hassannezahed, A., Kokarev, G.: Sub-laplacian eigenvalue bounds on sub-riemannian manifolds. Ann. ScI. Norm. Super. Pisa (to appear)
Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A B 270, 1645–1648, (1970)
Ivanov, S., Vassilev, D.: An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence-free torsion. J. Geom. 103(3), 475–504 (2012)
Ivanov, S., Vassilev, D.: An Obata-type theorem on a three-dimensional CR manifold. Glasg. Math. J. 56(2), 283–294 (2014)
Kokarev, G.: Sub-Laplacian eigenvalue bounds on CR manifolds. Comm. Partial Differ. Equ. 38(11), 1971–1984 (2013)
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)
Niu, P., Zhang, H.: Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators. Pacific J. Math. 208(2), 325–345 (2003)
Ponge, R.S.: Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds. Mem. Am. Math. Soc. 194(906):viii+ 134 (2008)
Stanton, N.K.: Spectral invariants of CR manifolds. Michigan Math. J. 36(2), 267–288 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aribi, A., El Soufi, A. The first positive eigenvalue of the sub-Laplacian on CR spheres. Ann Glob Anal Geom 51, 1–9 (2017). https://doi.org/10.1007/s10455-016-9519-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-016-9519-z