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On smoothness of solutions to projected differential equations
Prescribing the $ Q' $-curvature in three dimension
Department of Mathematics, Sogang University, Seoul 121-742, Korea |
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.
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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246, Birkhäuser Boston, Boston, MA, 2006. |
| [10] |
K. Hirachi,
$Q$-prime curvature on CR manifolds, Differential Geom. Appl., 14 (2014), 213-245.
doi: 10.1016/j.difgeo.2013.10.013. |
| [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. |
| [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. |
| [13] |
J. M. Lee,
Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110 (1988), 157-178.
doi: 10.2307/2374543. |
| [14] |
J. M. Lee,
The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.
doi: 10.2307/2000582. |
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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246, Birkhäuser Boston, Boston, MA, 2006. |
| [10] |
K. Hirachi,
$Q$-prime curvature on CR manifolds, Differential Geom. Appl., 14 (2014), 213-245.
doi: 10.1016/j.difgeo.2013.10.013. |
| [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. |
| [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. |
| [13] |
J. M. Lee,
Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110 (1988), 157-178.
doi: 10.2307/2374543. |
| [14] |
J. M. Lee,
The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.
doi: 10.2307/2000582. |
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