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AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD

Published online by Cambridge University Press:  13 August 2013

S. IVANOV  and
D. VASSILEV
S. IVANOV
Affiliation:
Faculty of Mathematics and Informatics, University of Sofia, blvd. James Bourchier 5, Sofia 1164, Bulgaria e-mail: ivanovsp@fmi.uni-sofia.bg
D. VASSILEV
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131-0001, USA e-mail: vassilev@math.unm.edu
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Abstract

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We prove the CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three-dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz-type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three-dimensional unit sphere.

Keywords

53C2653C2558J6032V0532V2053C56
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 283 - 294
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Barletta, E., The Lichnerowicz theorem on CR manifolds, Tsukuba J. Math. 31 (1) (2007), 77–97.CrossRefGoogle Scholar
2.Chang, S.-C., Cheng, J.-H. and Chiu, H.-L., A fourth order curvature flow on a CR 3-manifold, Indiana Univ. Math. J. 56 (4) (2007), 1793–1826.Google Scholar
3.Chang, S.-C. and Chiu, H.-L., On the estimate of the first eigenvalue of a sublaplacian on a pseudohermitian 3-manifold, Pacific J. Math. 232 (2) (2007), 269–282.CrossRefGoogle Scholar
4.Chang, S.-C. and Chiu, H.-L., Non-negativity of the CR Paneitz operator and its application to the CR Obata's theorem, J. Geom. Anal. 19 (2009), 261–287.Google Scholar
5.Chang, S.-C. and Chiu, H.-L., On the CR analogue of Obata's s theorem in a pseudohermitian 3-manifold, Math. Ann. 345 (2009), 33–51.Google Scholar
6.Chang, S.-C. and Wu, C.-T., The entropy formulas for the CR heat equation and their applications on pseudohermitian (2n + 1)-manifolds, Pacific J. Math. 246 (1) (2010), 1–29.Google Scholar
7.Chang, S.-C. and Wu, C.-T., The diameter estimate and its application to CR Obata's Theorem on closed pseudohermitian (2n + 1)-manifolds, Trans. Amer. Math. Soc. 364 (7) (2012), 3349–3363.CrossRefGoogle Scholar
8.Chiu, H.-L., The sharp lower bound for the first positive eigenvalue of the subLaplacian on a pseudohermitian 3-manifold, Ann. Global Anal. Geom. 30 (1) (2006), 81–96.Google Scholar
9.Dragomir, S. and Tomassini, G., Differential geometry and analysis on CR manifolds, Progress in Mathematics, vol. 246 (Birkhäuser, Boston, MA, 2006).Google Scholar
10.Graham, C. R. and Lee, J. M., Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J. 57 (1988), 697–720.Google Scholar
11.Greenleaf, A., The first eigenvalue of a subLaplacian on a pseudohermitian manifold, Commun. Partial Diff. Equations 10 (2) (1985), 191–217.Google Scholar
12.Hirachi, K., Scalar pseudo-hermitian invariants and the Szegö kernel on three-dimensional CR manifolds, in Complex geometry, Proceedings of Osaka Conference 1990 Lect. Notes in Pure Applied Mathematics 143 (Marcel Dekker, New York, NY, 1993), 67–76.Google Scholar
13.Ivanov, S., Petkov, A. and Vassilev, D., The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold, J. Geom. Anal. doi:10.1007/s12220-012-9354-9.Google Scholar
14.Ivanov, S. and Vassilev, D., Extremals for the Sobolev inequality and the quaternionic contact Yamabe Problem, Imperial College Press Lecture Notes (World Scientific, Hackensack, NJ, 2011).CrossRefGoogle Scholar
15.Ivanov, S. and 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) (2012), 475–504.CrossRefGoogle Scholar
16.Ivanov, S. and Vassilev, D., An Obata-type theorem on a three-dimensional CR manifold, arXiv:1208.1240, version 1.Google Scholar
17.Ivanov, S., Vassilev, D. and Zamkovoy, S., Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata 144 (2010), 79–100.Google Scholar
18.Lee, J., Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157–178.Google Scholar
19.Li, S.-Y. and Luk, H.-S., The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold, Proc. Amer. Math. Soc. 132 (3) (2004), 789–798.CrossRefGoogle Scholar
20.Li, S.-Y. and Wang, X., An Obata-type theorem in CR geometry, arXiv:1207.4033, version 1.Google Scholar
21.Li, S.-Y. and Wang, X., An Obata-type theorem in CR geometry, arXiv:1207.4033, version 2.Google Scholar
22.Lichnerowicz, A., Géométrie des groupes de transformations, Travaux et Recherches Mathematiques, III (Dunod, Paris, France, 1958).Google Scholar
23.Obata, M., Certain conditions for a Riemannian manifold to be iosometric with a sphere, J. Math. Soc. Japan 14 (3) (1962), 333–340.Google Scholar
24.Tanaka, N., A differential geometric study on strongly pseudo-convex manifolds, Kyoto University Department of Mathematics Lectures in Mathematics No. 9 (Kinokuniya, Tokyo, Japan, 1975).Google Scholar
25.Webster, S. M., Real hypersurfaces in complex space, Thesis (University of California, Berkeley, 1975).Google Scholar
26.Webster, S. M., Pseudo-Hermitian structures on a real hypersurface, J. Diff. Geom. 13 (1979), 25–41.Google Scholar