AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem
About this Title
Stefan Ivanov, University of Sofia and Institute of Mathematics, Bulgarian Academy of Sciences, Faculty of Mathematics and Informatics, blvd. James Bourchier 5, 1164, Sofia, Bulgaria and Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany, Ivan Minchev, University of Sofia, Faculty of Mathematics and Informatics, blvd. James Bourchier 5, 1164 Sofia, Bulgaria; Department of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Str. 6, 35032 Marburg, Germany; Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 61137 Brno, Czech Republic and Dimiter Vassilev, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131-0001, and, University of California, Riverside, Riverside, California 92521
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 231, Number 1086
ISBNs: 978-0-8218-9843-7 (print); 978-1-4704-1722-2 (online)
DOI: https://doi.org/10.1090/memo/1086
Published electronically: January 31, 2014
Keywords: Yamabe equation,
quaternionic contact structures,
Einstein structures
MSC: Primary 53C17
Table of Contents
Chapters
- 1. Introduction
- 2. Quaternionic contact structures and the Biquard connection
- 3. The torsion and curvature of the Biquard connection
- 4. QC-Einstein quaternionic contact structures
- 5. Conformal transformations of a qc-structure
- 6. Special functions and pseudo-Einstein quaternionic contact structures
- 7. Infinitesimal Automorphisms
- 8. Quaternionic contact Yamabe problem
Abstract
A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolds. All conformal transformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A ‘3-Hamiltonian form’ of infinitesimal conformal automorphisms of quaternionic contact structures is presented.- Dmitri Alekseevsky and Yoshinobu Kamishima, Pseudo-conformal quaternionic CR structure on $(4n+3)$-dimensional manifolds, Ann. Mat. Pura Appl. (4) 187 (2008), no. 3, 487–529. MR 2393145, DOI 10.1007/s10231-007-0053-2
- D. V. Alekseevsky and S. Marchiafava, Quaternionic structures on a manifold and subordinated structures, Ann. Mat. Pura Appl. (4) 171 (1996), 205–273. MR 1441871, DOI 10.1007/BF01759388
- Semyon Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, Bull. Sci. Math. 127 (2003), no. 1, 1–35. MR 1957796, DOI 10.1016/S0007-4497(02)00004-0
- Semyon Alesker, Quaternionic Monge-Ampère equations, J. Geom. Anal. 13 (2003), no. 2, 205–238. MR 1967025, DOI 10.1007/BF02930695
- S. Alesker, Quaternionic plurisubharmonic functions and their applications to convexity, Algebra i Analiz 19 (2007), no. 1, 5–22; English transl., St. Petersburg Math. J. 19 (2008), no. 1, 1–13. MR 2319507, DOI 10.1090/S1061-0022-07-00982-X
- Semyon Alesker and Misha Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, J. Geom. Anal. 16 (2006), no. 3, 375–399. MR 2250051, DOI 10.1007/BF02922058
- F. Astengo, M. Cowling, and B. Di Blasio, The Cayley transform and uniformly bounded representations, J. Funct. Anal. 213 (2004), no. 2, 241–269. MR 2078626, DOI 10.1016/j.jfa.2003.12.009
- Bertrand Banos and Andrew Swann, Potentials for hyper-Kähler metrics with torsion, Classical Quantum Gravity 21 (2004), no. 13, 3127–3135. MR 2072130, DOI 10.1088/0264-9381/21/13/004
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
- Biquard, O., Métriques d’Einstein asymptotiquement symétriques, Astérisque 265 (2000).
- Olivier Biquard, Quaternionic contact structures, Quaternionic structures in mathematics and physics (Rome, 1999) Univ. Studi Roma “La Sapienza”, Rome, 1999, pp. 23–30. MR 1848655
- Charles Boyer and Krzysztof Galicki, 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 123–184. MR 1798609, DOI 10.4310/SDG.2001.v6.n1.a6
- Charles P. Boyer, Krzysztof Galicki, and Benjamin M. Mann, The geometry and topology of $3$-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183–220. MR 1293878
- M. Mamone Capria and S. M. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1 (1988), no. 4, 517–530. MR 967469
- Jingyi Chen and Jiayu Li, Quaternionic maps between hyperkähler manifolds, J. Differential Geom. 55 (2000), no. 2, 355–384. MR 1847314
- Jingyi Chen and Jiayu Li, Quaternionic maps and minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 3, 375–388. MR 2185957
- Michael Cowling, Anthony Dooley, Adam Korányi, and Fulvio Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. 8 (1998), no. 2, 199–237. MR 1705176, DOI 10.1007/BF02921641
- \leavevmode\vrule height 2pt depth -1.6pt width 23pt, $H$-type groups and Iwasawa decompositions, Adv. Math., 87 (1991), 1–41.
- Duchemin, D., Quaternionic contact hypersurfaces, math.DG/0604147.
- David Duchemin, Quaternionic contact structures in dimension 7, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 851–885 (English, with English and French summaries). MR 2266881
- Rud Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1935), no. 1, 371–378 (German). MR 1509533, DOI 10.1007/BF01199562
- Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837196
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
- G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
- Nicola Garofalo and Dimiter Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), no. 3, 411–448. MR 1813232, DOI 10.1215/S0012-7094-01-10631-5
- \leavevmode\vrule height 2pt depth -1.6pt width 23pt, Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, Math Ann., 318 (2000), no. 3, 453–516.
- C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186–225. MR 1112625, DOI 10.1016/0001-8708(91)90071-E
- Gueo Grantcharov and Yat Sun Poon, Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys. 213 (2000), no. 1, 19–37. MR 1782143, DOI 10.1007/s002200000231
- P. S. Howe and G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B 379 (1996), no. 1-4, 80–86. MR 1396267, DOI 10.1016/0370-2693(96)00393-0
- Stefan Ivanov, Geometry of quaternionic Kähler connections with torsion, J. Geom. Phys. 41 (2002), no. 3, 235–257. MR 1877929, DOI 10.1016/S0393-0440(01)00058-4
- Stefan Ivanov, Ivan Minchev, and Dimiter Vassilev, Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 1041–1067. MR 2654087, DOI 10.4171/JEMS/222
- David Jerison and John M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57–63. MR 741039, DOI 10.1090/conm/027/741039
- David Jerison and John M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13. MR 924699, DOI 10.1090/S0894-0347-1988-0924699-9
- David Jerison and John M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), no. 2, 303–343. MR 982177
- David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
- Dominic Joyce, Hypercomplex algebraic geometry, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 194, 129–162. MR 1634730, DOI 10.1093/qjmath/49.194.129
- Hiroyuki Kamada and Shin Nayatani, Quaternionic analogue of CR geometry, Séminaire de Théorie Spectrale et Géométrie, Vol. 19, Année 2000–2001, Sémin. Théor. Spectr. Géom., vol. 19, Univ. Grenoble I, Saint-Martin-d’Hères, 2001, pp. 41–52. MR 1909075
- Yoshinobu Kamishima, Geometric rigidity of spherical hypersurfaces in quaternionic manifolds, Asian J. Math. 3 (1999), no. 3, 519–555. MR 1793671, DOI 10.4310/AJM.1999.v3.n3.a1
- Toyoko Kashiwada, A note on a Riemannian space with Sasakian $3$-structure, Natur. Sci. Rep. Ochanomizu Univ. 22 (1971), 1–2. MR 303449
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
- Korányi, A., Kelvin transform and harmonic polynomials on the Heisenberg group, Adv.Math. 56 (1985), 28–38.
- John M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), no. 1, 157–178. MR 926742, DOI 10.2307/2374543
- John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
- Jiayu Li and Xi Zhang, Quaternionic maps between quaternionic Kähler manifolds, Math. Z. 250 (2005), no. 3, 523–537. MR 2179610, DOI 10.1007/s00209-005-0763-3
- Paola Matzeu and Liviu Ornea, Local almost contact metric 3-structures, Publ. Math. Debrecen 57 (2000), no. 3-4, 499–508. MR 1798730
- Jeremy Michelson and Andrew Strominger, The geometry of (super) conformal quantum mechanics, Comm. Math. Phys. 213 (2000), no. 1, 1–17. MR 1782142, DOI 10.1007/PL00005528
- O’Neill, B., Semi-Riemannian geometry, Academic Press, New York, 1883.
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- Pertici, D., Funzioni regolari di piú variabili quaternioniche, Ann. Mat. Pura Appl., Serie IV, CLI (1988), 39–65.
- Donato Pertici, Trace theorems for regular functions of several quaternion variables, Forum Math. 3 (1991), no. 5, 461–478. MR 1123820, DOI 10.1515/form.1991.3.461
- Daniel Quillen, Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 194, 163–198. MR 1634734, DOI 10.1093/qjmath/49.194.163
- S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31–55. MR 860810
- Simon Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143–171. MR 664330, DOI 10.1007/BF01393378
- E. Stiefel, On Cauchy-Riemann equations in higher dimensions, J. Research Nat. Bur. Standards 48 (1952), 395–398. MR 0048596
- A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224. MR 516081, DOI 10.1017/S0305004100055638
- Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975. Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9. MR 0399517
- Dimiter Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pacific J. Math. 227 (2006), no. 2, 361–397. MR 2263021, DOI 10.2140/pjm.2006.227.361
- \leavevmode\vrule height 2pt depth -1.6pt width 23pt, Yamabe type equations on Carnot groups, Ph. D. thesis Purdue University, 2000.
- Misha Verbitsky, HyperKähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), no. 4, 679–712. MR 1958088, DOI 10.4310/AJM.2002.v6.n4.a5
- Wei Wang, The Yamabe problem on quaternionic contact manifolds, Ann. Mat. Pura Appl. (4) 186 (2007), no. 2, 359–380. MR 2295125, DOI 10.1007/s10231-006-0010-5
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599
- Hung Hsi Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i–xii and 289–538. MR 1079031