-
Previous Article
Periodic homogenization of elliptic systems with stratified structure
- DCDS Home
- This Issue
-
Next Article
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. |
[1] |
Chungen Liu, Yafang Wang. Existence results for the fractional Q-curvature problem on three dimensional CR sphere. Communications on Pure and Applied Analysis, 2018, 17 (3) : 849-885. doi: 10.3934/cpaa.2018043 |
[2] |
Ali Maalaoui. Prescribing the Q-curvature on the sphere with conical singularities. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6307-6330. doi: 10.3934/dcds.2016074 |
[3] |
Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 283-299. doi: 10.3934/dcds.2015.35.283 |
[4] |
Yaiza Canzani, Dmitry Jakobson, Igor Wigman. Scalar curvature and $Q$-curvature of random metrics. Electronic Research Announcements, 2010, 17: 43-56. doi: 10.3934/era.2010.17.43 |
[5] |
Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027 |
[6] |
Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43 |
[7] |
Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure and Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033 |
[8] |
W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57 |
[9] |
E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics and Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010 |
[10] |
Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249 |
[11] |
Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial and Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 |
[12] |
Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97. |
[13] |
Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 |
[14] |
Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213 |
[15] |
Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure and Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203 |
[16] |
Vittorio Martino. On the characteristic curvature operator. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911 |
[17] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[18] |
Gokhan Yener, Ibrahim Emiroglu. A q-analogue of the multiplicative calculus: Q-multiplicative calculus. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1435-1450. doi: 10.3934/dcdss.2015.8.1435 |
[19] |
Harman Kaur, Meenakshi Rana. Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $. Electronic Research Archive, 2021, 29 (6) : 4257-4268. doi: 10.3934/era.2021084 |
[20] |
M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]