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Functional determinant on pseudo-Einstein 3-manifolds. (English) Zbl 1457.32093

Summary: Given a three-dimensional pseudo-Einstein CR manifold \((M,T^{1,0}M,\theta)\), we establish an expression for the difference of determinants of the Paneitz type operators \(A_{\theta}\), related to the problem of prescribing the \(Q'\)-curvature, under the conformal change \(\theta\mapsto e^w\theta\) with \(w\in \mathcal{P}\) the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four-dimensional Riemannian manifolds established in [T. Branson and B. Ørsted, Proc. Am. Math. Soc. 113, No. 3, 669–682 (1991; Zbl 0762.47019)]. We also provide an existence result of maximizers for the scaling invariant functional determinant as in [S.-Y. A. Chang and P. C. Yang, Ann. Math. (2) 142, No. 1, 171–212 (1995; Zbl 0842.58011)].

MSC:

32V05 CR structures, CR operators, and generalizations
32V20 Analysis on CR manifolds
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References:

[1] 10.2307/2946638 · Zbl 0826.58042 · doi:10.2307/2946638
[2] ; Branson, Compositio Math., 60, 261 (1986) · Zbl 0608.58039
[3] 10.1016/0926-2245(91)90004-S · Zbl 0785.53025 · doi:10.1016/0926-2245(91)90004-S
[4] 10.2307/2048601 · Zbl 0762.47019 · doi:10.2307/2048601
[5] 10.4007/annals.2013.177.1.1 · Zbl 1334.35366 · doi:10.4007/annals.2013.177.1.1
[6] 10.1007/BF01404456 · Zbl 0643.32006 · doi:10.1007/BF01404456
[7] ; Case, Bull. Inst. Math. Acad. Sin. (N.S.), 8, 285 (2013) · Zbl 1316.32021
[8] 10.1016/j.crma.2015.12.012 · Zbl 1381.53062 · doi:10.1016/j.crma.2015.12.012
[9] 10.2307/2118613 · Zbl 0842.58011 · doi:10.2307/2118613
[10] 10.1215/S0012-7094-90-06008-9 · Zbl 0704.53028 · doi:10.1215/S0012-7094-90-06008-9
[11] 10.4310/MRL.2003.v10.n6.a9 · Zbl 1166.53309 · doi:10.4310/MRL.2003.v10.n6.a9
[12] 10.1515/crll.2005.2005.583.1 · Zbl 1076.53048 · doi:10.1515/crll.2005.2005.583.1
[13] 10.1007/s00220-012-1535-7 · Zbl 1251.53059 · doi:10.1007/s00220-012-1535-7
[14] 10.1016/j.difgeo.2013.10.013 · Zbl 1286.32017 · doi:10.1016/j.difgeo.2013.10.013
[15] 10.3934/dcds.2019096 · Zbl 1412.32027 · doi:10.3934/dcds.2019096
[16] 10.1515/9781400881444 · Zbl 0469.47021 · doi:10.1515/9781400881444
[17] 10.1007/BF02246771 · Zbl 0852.58079 · doi:10.1007/BF02246771
[18] 10.1215/S0012-7094-02-11433-1 · Zbl 1065.58022 · doi:10.1215/S0012-7094-02-11433-1
[19] 10.2307/2661347 · Zbl 0985.58013 · doi:10.2307/2661347
[20] 10.1007/s00039-007-0636-5 · Zbl 1140.58003 · doi:10.1007/s00039-007-0636-5
[21] 10.1016/0370-2693(81)90743-7 · doi:10.1016/0370-2693(81)90743-7
[22] 10.1016/0370-2693(81)90744-9 · doi:10.1016/0370-2693(81)90744-9
[23] 10.1016/j.jfa.2007.07.001 · Zbl 1127.58024 · doi:10.1016/j.jfa.2007.07.001
[24] 10.1307/mmj/1029003949 · Zbl 0685.58033 · doi:10.1307/mmj/1029003949
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