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The \(\bar\partial\)-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in \(\mathbb{C}P^n\) with \(n \ge 3\)

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In this paper we study the Cauchy–Riemann equation in complex projective spaces. Specifically, we use the modified weight function method to study the \(\bar\partial\)-Neumann problem on pseudoconvex domains in these spaces. The solutions are used to study function theory on pseudoconvex domains via the \(\bar\partial\)-Cauchy problem. We apply our results to prove nonexistence of Lipschitz Levi-flat hypersurfaces in complex projective spaces of dimension at least three, which removes the smoothness requirement used in an earlier paper of Siu.

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References

  1. Barrett D.E., Fornaess J.E. (1988) On the smoothness of Levi-foliations. Publ. Mat. 32: 171–177

    MATH  MathSciNet  Google Scholar 

  2. Berndtsson B., Charpentier P. (2000) A Sobolev mapping property of the Bergman kernel. Math. Zeits. 235, 1–10

    Article  MATH  MathSciNet  Google Scholar 

  3. Cao J., Shaw M.-C. (2005) A new proof of the Takeuchi theorem Proceedings of “CR Geometry and PDE’s” Trento, Italy. Interdisplinare di Mathematica. 4, 65–72

    MathSciNet  Google Scholar 

  4. Cao J., Shaw M.-C., Wang L. (2004) Estimates for the \(\bar\partial\)-Neumann problem and nonexistence of Levi-flat hypersurfaces in \(\mathbb{C}P^n\). Math. Zeit. 248, 183–221 Erratum, 223–225

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables American Math. Society-International Press, Studies in Advanced Mathematics, vol. 19. Providence, R.I. 2001

  6. Demailly J.-P. (1987) Mesures de Monge-Ampère et mesures pluriharmoniques. Math. Zeit. 194, 519–564

    Article  MATH  MathSciNet  Google Scholar 

  7. Diederich K., Fornaess J.E. (1977) Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39, 129–141

    Article  MATH  MathSciNet  Google Scholar 

  8. Evans L.E., Gariepy R.F. (1992) Measure theory and fine properties of functions. CRC press, Boca Raton

    MATH  Google Scholar 

  9. Federer H. (1959) Curvature measures. Trans. Am. Math. Soc. 93, 418–491

    Article  MATH  MathSciNet  Google Scholar 

  10. Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex. Ann. Math. Stud. 75 Princeton University Press, Princeton, (1972)

  11. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Boston (1985)

  12. Henkin G.M., Iordan A. (2003) Regularity of \(\bar\partial\) on pseudoconcave compacts and applications Asian J. Math. 4 2000 855-884 info Erratum: Asian J. Math. 7(1): 147–148

  13. Hörmander L. (1965) L 2 estimates and existence theorems for the \(\bar\partial\) operator. Acta Math. 113, 89–152

    Article  MATH  MathSciNet  Google Scholar 

  14. Hörmander L. (2004) The null space of the \(\overline\partial\)-Neumann operator. Ann. Inst. Fourier (Grenoble) 54, 1305–1369

    MathSciNet  Google Scholar 

  15. Iordan, A.: On the non-existence of smooth Levi-flat hypersurfaces in \(\mathbb{CP}^n\) will appear in the “Proceedings of the Memorial Conference of Kiyoshi Oka’s Centenial Birthday on Complex Analysis in Several Variables”, Kyoto, Nara 2001

  16. Karcher H. (1977) Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541

    MATH  MathSciNet  Google Scholar 

  17. Kerzman N., Rosay J.-P. (1981) Fonctions Plurisousharmoniques d’exhaustion bornées et domaines taut. Math. Ann. 257, 171–184

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Lions J.-L., Magenes E. (1972) Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  20. Lins Neto A. (1999) A note on projective Levi flats and minimal sets of algebraic foliations. Ann. Inst. Fourier 49, 1369–1385

    MATH  MathSciNet  Google Scholar 

  21. Milnor J.W. (1974) Characteristic classes. Princeton University Press, Princeton

    MATH  Google Scholar 

  22. Ni L., Wolfson J. (2003) The Lefschetz theorem for CR submanifolds and the nonexistence of real analytic Levi flat submanifolds. Commun. Anal. Geom. 11, 553-564

    MATH  MathSciNet  Google Scholar 

  23. Ohsawa T., Sibony N. (1998) Bounded P.S.H functions and pseudoconvexity in Kähler manifolds. Nagoya Math. J. 149, 1–8

    MATH  MathSciNet  Google Scholar 

  24. Shaw M.-C., Wang L. (2004) Hölder and L p estimates for \(\square_b\) on CR manifolds of arbitrary codimension. Math. Ann. 331, 297–343

    Article  MathSciNet  Google Scholar 

  25. Siu Y.-T. (2000) Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension \(\ge 3.\) Ann. Math. 151,1217–1243

    MATH  MathSciNet  Google Scholar 

  26. Siu Y.-T. (2002) \(\bar\partial\)-regularity for weakly pseudoconvex domains in hermitian symmetric spaces with respect to invariant metrics. Ann. Math. 156, 595–621

    MATH  MathSciNet  Google Scholar 

  27. Stein E.M. (1970) Singular Integrals and Differentiability Properties of Functions, Math. Ser. 30. Princeton University Press, Princeton

    Google Scholar 

  28. Takeuchi A. (1964) Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectifJ. Math. Soc. Jpn 16, 159–181

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jianguo Cao & Mei-Chi Shaw

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  1. Jianguo Cao
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  2. Mei-Chi Shaw
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Correspondence to Mei-Chi Shaw.

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Jianguo Cao and Mei-Chi Shaw are partially supported by NSF grants.

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Cao, J., Shaw, MC. The \(\bar\partial\)-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in \(\mathbb{C}P^n\) with \(n \ge 3\) . Math. Z. 256, 175–192 (2007). https://doi.org/10.1007/s00209-006-0064-5

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