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April  2019, 39(4): 2285-2294. doi: 10.3934/dcds.2019096

Prescribing the $ Q' $-curvature in three dimension

Pak Tung Ho

Department of Mathematics, Sogang University, Seoul 121-742, Korea

Received  September 2018 Revised  October 2018 Published  January 2019

In this note, we consider the problem of prescribing $ \overline{Q}' $-curvature on a three-dimensional pseudohermitian manifold. Given a positive CR pluriharmonic function $ f $, we construct a contact form on the three-dimensional pseudo-Einstein manifold with $ \overline{Q}' $-curvature being equal to $ f $, under some natural positivity conditions. On the other hand, we prove a Kazdan-Warner type identity for the problem of prescribing $ \overline{Q}' $-curvature on the standard CR three sphere.

Keywords: Q-prime curvature, Q-curvature, CR manifold, pseudo-Einstein manifold, Kazdan-Warner identity.
Mathematics Subject Classification: Primary: 32V20, 53A30; Secondary: 35R01, 53D10.
Citation: Pak Tung Ho. Prescribing the $ Q' $-curvature in three dimension. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2285-2294. doi: 10.3934/dcds.2019096
References:
[1]

T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math. (2), 177 (2013), 1-52.  doi: 10.4007/annals.2013.177.1.1.  Google Scholar

[2]

J. S. Case, C. Y. Hsiao and P. C. Yang, Extremal metrics for the $Q'$-curvature in three dimensions, C. R. Math. Acad. Sci. Paris, 354 (2016), 407-410.  doi: 10.1016/j.crma.2015.12.012.  Google Scholar

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J. S. Case, C. Y. Hsiao and P. C. Yang, Extremal metrics for the $Q'$-curvature in three dimensions, J. Eur. Math. Soc., (2018), accepted. Google Scholar

[4]

J. S. Case and P. C. Yang, A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.), 8 (2013), 285-322.   Google Scholar

[5]

S.-Y. A. Chang and P. C. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.  Google Scholar

[6]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.  Google Scholar

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X. Chen and X. Xu, The scalar curvature flow on $S^n$——perturbation theorem revisited, Invent. Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.  Google Scholar

[8]

J. H. Cheng, Curvature functions for the sphere in pseudo-Hermitian geometry, Tokyo J. Math., 14 (1991), 151-163.  doi: 10.3836/tjm/1270130496.  Google Scholar

[9]

S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246, Birkhäuser Boston, Boston, MA, 2006.  Google Scholar

[10]

K. Hirachi, $Q$-prime curvature on CR manifolds, Differential Geom. Appl., 14 (2014), 213-245.  doi: 10.1016/j.difgeo.2013.10.013.  Google Scholar

[11]

K. Hirachi, Scalar pseudo-Hermitian invariants and the Szegő kernel on three-dimensional CR manifolds, Complex geometry (Osaka, 1990), 67-76, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.  Google Scholar

[12]

C. Y. Hsiao and P. L. Yung, Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3, Adv. Math., 281 (2015), 734-822.  doi: 10.1016/j.aim.2015.04.028.  Google Scholar

[13]

J. M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110 (1988), 157-178.  doi: 10.2307/2374543.  Google Scholar

[14]

J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.  doi: 10.2307/2000582.  Google Scholar

show all references

References:
[1]

T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math. (2), 177 (2013), 1-52.  doi: 10.4007/annals.2013.177.1.1.  Google Scholar

[2]

J. S. Case, C. Y. Hsiao and P. C. Yang, Extremal metrics for the $Q'$-curvature in three dimensions, C. R. Math. Acad. Sci. Paris, 354 (2016), 407-410.  doi: 10.1016/j.crma.2015.12.012.  Google Scholar

[3]

J. S. Case, C. Y. Hsiao and P. C. Yang, Extremal metrics for the $Q'$-curvature in three dimensions, J. Eur. Math. Soc., (2018), accepted. Google Scholar

[4]

J. S. Case and P. C. Yang, A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.), 8 (2013), 285-322.   Google Scholar

[5]

S.-Y. A. Chang and P. C. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.  Google Scholar

[6]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.  Google Scholar

[7]

X. Chen and X. Xu, The scalar curvature flow on $S^n$——perturbation theorem revisited, Invent. Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.  Google Scholar

[8]

J. H. Cheng, Curvature functions for the sphere in pseudo-Hermitian geometry, Tokyo J. Math., 14 (1991), 151-163.  doi: 10.3836/tjm/1270130496.  Google Scholar

[9]

S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246, Birkhäuser Boston, Boston, MA, 2006.  Google Scholar

[10]

K. Hirachi, $Q$-prime curvature on CR manifolds, Differential Geom. Appl., 14 (2014), 213-245.  doi: 10.1016/j.difgeo.2013.10.013.  Google Scholar

[11]

K. Hirachi, Scalar pseudo-Hermitian invariants and the Szegő kernel on three-dimensional CR manifolds, Complex geometry (Osaka, 1990), 67-76, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.  Google Scholar

[12]

C. Y. Hsiao and P. L. Yung, Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3, Adv. Math., 281 (2015), 734-822.  doi: 10.1016/j.aim.2015.04.028.  Google Scholar

[13]

J. M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110 (1988), 157-178.  doi: 10.2307/2374543.  Google Scholar

[14]

J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.  doi: 10.2307/2000582.  Google Scholar

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