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Kohn–Rossi cohomology of spherical CR manifolds

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Abstract

We prove some vanishing theorems for the Kohn–Rossi cohomology of some spherical CR manifolds. To this end, we use a canonical contact form defined via the Patterson–Sullivan measure and Weitzenböck-type formulae for the Kohn Laplacian. We also see that our results are optimal in some cases.

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References

  1. Boutet de Monvel, L.: Intégration des équations de Cauchy–Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974–1975; équations aux derivées partielles linéaires et non linéaires, pp. Exp. No. 9, 14 (1975)

  2. Beardon, A.F.: The exponent of convergence of Poincaré series. Proc. Lond. Math. Soc. (3) 18, 461–483 (1968)

    Article  MathSciNet  Google Scholar 

  3. Bowditch, B.H.: Geometrical finiteness with variable negative curvature. Duke Math. J. 77(1), 229–274 (1995)

    Article  MathSciNet  Google Scholar 

  4. Burns, D., Jr., Shnider, S.: Spherical hypersurfaces in complex manifolds. Invent. Math. 33(3), 223–246 (1976)

    Article  MathSciNet  Google Scholar 

  5. Case, J.S.: The bigraded Rumin complex via differential forms (2021). arXiv:2108.13911. To appear in Mem. Amer. Math. Soc

  6. Corlette, K., Iozzi, A.: Limit sets of discrete groups of isometries of exotic hyperbolic spaces. Trans. Amer. Math. Soc. 351(4), 1507–1530 (1999)

    Article  MathSciNet  Google Scholar 

  7. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)

    Article  MathSciNet  Google Scholar 

  8. Corlette, K.: Hausdorff dimensions of limit sets. I. Invent. Math. 102(3), 521–541 (1990)

    Article  MathSciNet  Google Scholar 

  9. Dey, S., Kapovich, M.: A note on complex-hyperbolic Kleinian groups. Arnold Math. J. 6(3–4), 397–406 (2020)

    Article  MathSciNet  Google Scholar 

  10. Folland, G.B., Stein, E.M.: Estimates for the \(\bar{\partial }_{b}\) complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27, 429–522 (1974)

    Article  MathSciNet  Google Scholar 

  11. Garfield, P.M.: The bigraded Rumin complex, Ph.D. Thesis. Thesis (Ph.D.). University of Washington (2001)

  12. Garfield, P.M., Lee, J.M.: The Rumin complex on CR manifolds, pp. 29–36 (1998). CR Geometry and Isolated Singularities (Japanese) (Kyoto, 1996)

  13. Goldman, W.M.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs, Oxford University Press, New York (1999). (Oxford Science Publications)

    Book  Google Scholar 

  14. Izeki, H.: Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature. Proc. Amer. Math. Soc. 130(12), 3731–3740 (2002)

    Article  MathSciNet  Google Scholar 

  15. Kapovich, M.: A survey of complex hyperbolic Kleinian groups. In: In the Tradition of Thurston II. Geometry and Groups, pp. 7–51 (2022)

  16. Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of the Conference on Complex Analysis (Minneapolis, 1964), pp. 81–94 (1965)

  17. Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. of Math. (2) 81, 451–472 (1965)

    Article  MathSciNet  Google Scholar 

  18. Nayatani, S.: Patterson–Sullivan measure and conformally flat metrics. Math. Z. 225(1), 115–131 (1997)

    Article  MathSciNet  Google Scholar 

  19. Nayatani, S.: Discrete groups of complex hyperbolic isometries and pseudo-Hermitian structures. In: Analysis and Geometry in Several Complex Variables (Katata, 1997), pp. 209–237 (1999)

  20. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 136(3–4), 241–273 (1976)

    Article  MathSciNet  Google Scholar 

  21. Rumin, M.: Formes différentielles sur les variétés de contact. J. Differential Geom. 39(2), 281–330 (1994)

    Article  MathSciNet  Google Scholar 

  22. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50, 171–202 (1979)

    Article  MathSciNet  Google Scholar 

  23. Tanaka, N.: A Differential Geometric Study on Strongly Pseudo-convex Manifolds. Lectures in Mathematics, Kyoto University, Tokyo (1975)

    Google Scholar 

  24. Wang, W.: Canonical contact forms on spherical CR manifolds. J. Eur. Math. Soc. (JEMS) 5(3), 245–273 (2003)

    Article  MathSciNet  Google Scholar 

  25. Yau, S.S.T.: Kohn–Rossi cohomology and its application to the complex Plateau problem. I. Ann. of Math. (2) 113(1), 67–110 (1981)

    Article  MathSciNet  Google Scholar 

  26. Yue, C.: Webster curvature and Hausdorff dimension of complex hyperbolic Kleinian groups. In: Dynamical Systems. Proceedings of the International Conference in Honor of Professor Liao Shantao, pp. 319–328 (1999)

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Acknowledgements

The author would like to thank Wei Wang for some helpful comments on Remark 5.2. He is also grateful to the anonymous referee for valuable suggestions which led to improvements of the revised version of the paper.

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  1. Division of Mathematics, Institute of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan

    Yuya Takeuchi

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  1. Yuya Takeuchi
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This work was supported by JSPS KAKENHI Grant No. JP21K13792.

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Takeuchi, Y. Kohn–Rossi cohomology of spherical CR manifolds. Ann Glob Anal Geom 65, 22 (2024). https://doi.org/10.1007/s10455-024-09952-1

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