Abstract
In this paper, we first prove the CR analogue of Obata’s theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau’s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P 0. Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator.
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Chang S.-C., Cheng J.-H., Chiu H.-L.: The fourth-order Q-curvature flow on a CR 3-manifold. Indiana Univ. Math. J. 56(4), 1793–1826 (2007)
Cao J., Chang S.-C.: Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces. Indiana Univ. Math. J. 56(6), 2840–2857 (2007)
Chang S.-C., Chiu H.-L.: On the estimate of first eigenvalue of a sublaplacian on a pseudo-hermitian 3-manifold. Pac. J. Math. 232(2), 269–282 (2007)
Chiu H.-L.: The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold. Ann. Glob Anal. Geom. 30, 81–96 (2006)
Chow W.-L.: Uber system Von Lineaaren Partiellen Differentialgleichungen erster Orduung. Math. Ann. 117, 98–105 (1939)
Fukaya K.: Collapsing of Riemannian manifolds and eigenvalues of Laplacian operators. Invent. Math. 87, 517–547 (1987)
Fefferman C., Hirachi K.: Ambient metric construction of Q-curvature in conformal and CR geometries math. Res. Lett. 10(5–6), 819–831 (2003)
Gover A.R., Graham C.R.: CR invariant powers of the sub-Laplacian. J. Reine Angew. Math. 583, 1–27 (2005)
Graham C.R., Lee J.M.: Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains. Duke Math. J. 57, 697–720 (1988)
Greenleaf A.: The first eigenvalue of a Sublaplacian on a Pseudohermitian manifold. Comm. Part. Diff. Equ. 10(2)(3), 191–217 (1985)
Hirachi, K.: Scalar Pseudo-hermitian Invariants and the Szegö Kernel on 3-dimensional CR Manifolds. In: Lecture Notes in Pure and Appl. Math. 143, pp. 67–76, Dekker (1992)
Kamishima Y., Tsuboi T.: CR-Structures on Seifert manifolds. Invent. Math. 104, 149–163 (1991)
Lichnerowicz, A.: Géometrie des groupes de transformations. Dunod, Pairs (1958)
Li P., Yau S.-T.: Estimates of eigenvalues of a compact Riemannian manifold. AMS Proc. Symp. Pure Math. 36, 205–239 (1980)
Lee J.M.: Pseudo-Einstein structure on CR manifolds. Amer. J. Math. 110, 157–178 (1988)
Lee J.M.: The Fefferman metric and pseudohermitian invariants. Trans. AMS 296, 411–429 (1986)
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. Amer. Math. Soc. 132(3), 789–798 (2004)
Obata M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333–340 (1962)
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This research was supported in part by the NSC of Taiwan.
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Chang, SC., Chiu, HL. On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold. Math. Ann. 345, 33–51 (2009). https://doi.org/10.1007/s00208-009-0339-3
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DOI: https://doi.org/10.1007/s00208-009-0339-3