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Extremals for the Sobolev inequality and the quaternionic contact Yamabe problem / Stefan P. Ivanov, Dimiter N. Vassilev.

By: Ivanov, Stefan PContributor(s): Vassilev, Dimiter NMaterial type: TextTextPublication details: Singapore ; Hackensack, NJ : World Scientific, ©2011. Description: 1 online resource (xvii, 219 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789814295710; 981429571X; 1283234653; 9781283234658; 9786613234650; 6613234656Subject(s): Geometry, Differential | Contact manifolds | Group theory | MATHEMATICS -- Geometry -- Differential | Contact manifolds | Geometry, Differential | Group theoryGenre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Extremals for the Sobolev inequality and the quaternionic contact Yamabe problem.DDC classification: 516.36 LOC classification: QA649 | .I9 2011ebOnline resources: Click here to access online
Contents:
Machine generated contents note: 1. Variational problems related to Sobolev inequalities on Carnot groups -- 1.1. Introduction -- 1.2. Carnot groups -- 1.3. Sobolev spaces and their weak topologies -- 1.4. The best constant in the Folland-Stein inequality -- 1.5. The best constant in the presence of symmetries -- 1.6. Global regularity of weak solutions -- 1.6.1. Global boundedness of weak solutions -- 1.6.2. The Yamabe equation -- Cinfinity regularity of weak solutions -- 2. Groups of Heisenberg and Iwasawa types explicit solutions to the Yamabe equation -- 2.1. Introduction -- 2.2. Groups of Heisenberg and Iwasawa types -- 2.3. The Cayley transform, inversion and Kelvin transform -- 2.3.1. The Cayley transform -- 2.3.2. Inversion on groups of Heisenberg type -- 2.3.3. The Kelvin transform -- 2.4. Explicit entire solutions of the Yamabe equation on groups of Heisenberg type -- 3. Symmetries of solutions on groups of Iwasawa type -- 3.1. Introduction -- 3.2. The Hopf Lemma.
3.3. The partially symmetric solutions have cylindrical symmetry -- 3.4. Determination of the cylindrically symmetric solutions of the Yamabe equation -- 3.5. Solution of the partially symmetric Yamabe problem -- 3.6. Applications. Euclidean Hardy-Sobolev inequalities -- 3.6.1. A non-linear equation in Rn related to the Yamabe equation on groups of Heisenberg type -- 3.6.2. The best constant and extremals of the Hardy-Sobolev inequality -- 4. Quaternionic contact manifolds -- Connection, curvature and qc-Einstein structures -- 4.1. Introduction -- 4.2. Quaternionic contact structures and the Biquard connection -- 4.3. The curvature of the Biquard connection -- 4.3.1. The first Bianchi identity and Ricci tensors -- 4.3.2. Local structure equations of qc manifolds -- 4.3.3. The curvature tensor -- 4.3.4. The flat model -- The qc Heisenberg group -- 4.4. qc-Einstein quaternionic contact structures -- 4.4.1. Examples of qc-Einstein structures -- 4.4.2. Cones over a quaternionic contact structure -- 5. Quaternionic contact conformal curvature tensor.
5.1. Introduction -- 5.2. Quaternionic contact conformal transformations -- 5.2.1. The quaternionic Cayley transform -- 5.3. qc conformal curvature -- 6. The quaternionic contact Yamabe problem and the Yamabe constant of the qc spheres -- 6.1. Introduction -- 6.2. Some background -- 6.2.1. The qc normal frame -- 6.2.2. Horizontal divergence theorem -- 6.2.3. Conformal transformations of the quaternionic Heisenberg group preserving the vanishing of the torsion -- 6.3. Constant qc scalar curvature and the divergence formula -- 6.4. Divergence formulas -- 6.5. The divergence theorem in dimension seven -- 6.6. The qc Yamabe problem on the qc sphere and quaternionic Heisenberg group in dimension seven -- 6.7. The qc Yamabe constant on the qc sphere and the best constant in the Folland-Stein embedding on the quaternionic Heisenberg group -- 7. CR manifolds -- Cartan and Chern-Moser tensor and theorem -- 7.1. Introduction -- 7.2. CR-manifolds and Tanaka-Webster connection -- 7.3. The Cartan-Chern-Moser theorem -- 7.3.1. The three dimensional case.
Summary: The aim of this book is to give an account of some important new developments in the study of the Yamabe problem on quaternionic contact manifolds. This book covers the conformally flat case of the quaternionic Heisenberg group or sphere, where complete and detailed proofs are given, together with a chapter on the conformal curvature tensor introduced very recently by the authors. The starting point of the considered problems is the well-known Folland-Stein Sobolev type embedding and its sharp form that is determined based on geometric analysis. This book also sits at the interface of the generalization of these fundamental questions motivated by the Carnot-Caratheodory geometry of quaternionic contact manifolds, which have been recently the focus of extensive research motivated by problems in analysis, geometry, mathematical physics and the applied sciences. Through the beautiful resolution of the Yamabe problem on model quaternionic contact spaces, the book serves as an introduction to this field for graduate students and novice researchers, and as a research monograph suitable for experts as well.
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Includes bibliographical references (pages 207-216) and index.

Machine generated contents note: 1. Variational problems related to Sobolev inequalities on Carnot groups -- 1.1. Introduction -- 1.2. Carnot groups -- 1.3. Sobolev spaces and their weak topologies -- 1.4. The best constant in the Folland-Stein inequality -- 1.5. The best constant in the presence of symmetries -- 1.6. Global regularity of weak solutions -- 1.6.1. Global boundedness of weak solutions -- 1.6.2. The Yamabe equation -- Cinfinity regularity of weak solutions -- 2. Groups of Heisenberg and Iwasawa types explicit solutions to the Yamabe equation -- 2.1. Introduction -- 2.2. Groups of Heisenberg and Iwasawa types -- 2.3. The Cayley transform, inversion and Kelvin transform -- 2.3.1. The Cayley transform -- 2.3.2. Inversion on groups of Heisenberg type -- 2.3.3. The Kelvin transform -- 2.4. Explicit entire solutions of the Yamabe equation on groups of Heisenberg type -- 3. Symmetries of solutions on groups of Iwasawa type -- 3.1. Introduction -- 3.2. The Hopf Lemma.

3.3. The partially symmetric solutions have cylindrical symmetry -- 3.4. Determination of the cylindrically symmetric solutions of the Yamabe equation -- 3.5. Solution of the partially symmetric Yamabe problem -- 3.6. Applications. Euclidean Hardy-Sobolev inequalities -- 3.6.1. A non-linear equation in Rn related to the Yamabe equation on groups of Heisenberg type -- 3.6.2. The best constant and extremals of the Hardy-Sobolev inequality -- 4. Quaternionic contact manifolds -- Connection, curvature and qc-Einstein structures -- 4.1. Introduction -- 4.2. Quaternionic contact structures and the Biquard connection -- 4.3. The curvature of the Biquard connection -- 4.3.1. The first Bianchi identity and Ricci tensors -- 4.3.2. Local structure equations of qc manifolds -- 4.3.3. The curvature tensor -- 4.3.4. The flat model -- The qc Heisenberg group -- 4.4. qc-Einstein quaternionic contact structures -- 4.4.1. Examples of qc-Einstein structures -- 4.4.2. Cones over a quaternionic contact structure -- 5. Quaternionic contact conformal curvature tensor.

5.1. Introduction -- 5.2. Quaternionic contact conformal transformations -- 5.2.1. The quaternionic Cayley transform -- 5.3. qc conformal curvature -- 6. The quaternionic contact Yamabe problem and the Yamabe constant of the qc spheres -- 6.1. Introduction -- 6.2. Some background -- 6.2.1. The qc normal frame -- 6.2.2. Horizontal divergence theorem -- 6.2.3. Conformal transformations of the quaternionic Heisenberg group preserving the vanishing of the torsion -- 6.3. Constant qc scalar curvature and the divergence formula -- 6.4. Divergence formulas -- 6.5. The divergence theorem in dimension seven -- 6.6. The qc Yamabe problem on the qc sphere and quaternionic Heisenberg group in dimension seven -- 6.7. The qc Yamabe constant on the qc sphere and the best constant in the Folland-Stein embedding on the quaternionic Heisenberg group -- 7. CR manifolds -- Cartan and Chern-Moser tensor and theorem -- 7.1. Introduction -- 7.2. CR-manifolds and Tanaka-Webster connection -- 7.3. The Cartan-Chern-Moser theorem -- 7.3.1. The three dimensional case.

Print version record.

The aim of this book is to give an account of some important new developments in the study of the Yamabe problem on quaternionic contact manifolds. This book covers the conformally flat case of the quaternionic Heisenberg group or sphere, where complete and detailed proofs are given, together with a chapter on the conformal curvature tensor introduced very recently by the authors. The starting point of the considered problems is the well-known Folland-Stein Sobolev type embedding and its sharp form that is determined based on geometric analysis. This book also sits at the interface of the generalization of these fundamental questions motivated by the Carnot-Caratheodory geometry of quaternionic contact manifolds, which have been recently the focus of extensive research motivated by problems in analysis, geometry, mathematical physics and the applied sciences. Through the beautiful resolution of the Yamabe problem on model quaternionic contact spaces, the book serves as an introduction to this field for graduate students and novice researchers, and as a research monograph suitable for experts as well.

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