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Differential Geometry and Analysis on CR Manifolds

Differential Geometry and Analysis on CR Manifolds Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study ; ? A CR manifold is aC differentiable manifold endowed with a complex subbundle T (M)ofthecomplexi?edtangentbundleT(M)?CsatisfyingT (M)?T (M)= 1,0 1,0 1,0 (0) and the Frobenius (formal) integrability property ? ? ? (T (M)), (T (M)) ? (T (M)). 1,0 1,0 1,0 ? The bundle T (M) is the CR structure of M, and C maps f : M ? N of CR 1,0 manifolds preserving the CR structures (i.e., f T (M)?T (N)) areCRmaps.CR ? 1,0 1,0 manifolds and CR maps form a category containing that of complex manifolds and holomorphic maps. The most interesting examples of CR manifolds appear, however, as real submanifolds of some complex manifold. For instance, any real hypersurface n M in C admits a CR structure, naturally induced by the complex structure of the ambient space 1,0 n T (M)=T (C)? (T(M)?C). 1,0 1 n n Let(z,...,z) be the natural complex coordinates onC . Locally, in a neighborhood of each point of M, one may produce a frame{L :1?? ?n?1} ofT (M). G- ? 1,0 metrically speaking, eachL is a (complex) vector ?eld tangent to M. From the point ? of view of the theory of PDEs, the L ’s are purely tangential ?rst-order differential ? operators n; CR Manifolds.- The Fefferman Metric.- The CR Yamabe Problem.- Pseudoharmonic Maps.- Pseudo-Einsteinian Manifolds.- Pseudo-Hermitian Immersions.- Quasiconformal Mappings.- Yang-Mills Fields on CR Manifolds.- Spectral Geometry.; In fact, it will be invaluable for people working on the differential geometry of CR manifolds. –Thomas Garity, MathSciNet ; The study of CR manifolds lies at the intersection of three main mathematical disciplines: partial differential equations, complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has recently been made to understand the differential geometric side of the subject. This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. It presents the major differential geometric acheivements in the theory of CR manifolds, such as the Tanaka–Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang–Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry. Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs. ; Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study ; The study of CR manifolds lies at the intersection of three main mathematical disciplines, partial differential equations: complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has been recently made to understand the differential geometric side of the subject. This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs. ; US http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Differential Geometry and Analysis on CR Manifolds

499 pages

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Publisher
Birkhäuser Boston
Copyright
Copyright � Springer Basel AG
DOI
10.1007/0-8176-4483-0
Publisher site
See Book on Publisher Site

Abstract

Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study ; ? A CR manifold is aC differentiable manifold endowed with a complex subbundle T (M)ofthecomplexi?edtangentbundleT(M)?CsatisfyingT (M)?T (M)= 1,0 1,0 1,0 (0) and the Frobenius (formal) integrability property ? ? ? (T (M)), (T (M)) ? (T (M)). 1,0 1,0 1,0 ? The bundle T (M) is the CR structure of M, and C maps f : M ? N of CR 1,0 manifolds preserving the CR structures (i.e., f T (M)?T (N)) areCRmaps.CR ? 1,0 1,0 manifolds and CR maps form a category containing that of complex manifolds and holomorphic maps. The most interesting examples of CR manifolds appear, however, as real submanifolds of some complex manifold. For instance, any real hypersurface n M in C admits a CR structure, naturally induced by the complex structure of the ambient space 1,0 n T (M)=T (C)? (T(M)?C). 1,0 1 n n Let(z,...,z) be the natural complex coordinates onC . Locally, in a neighborhood of each point of M, one may produce a frame{L :1?? ?n?1} ofT (M). G- ? 1,0 metrically speaking, eachL is a (complex) vector ?eld tangent to M. From the point ? of view of the theory of PDEs, the L ’s are purely tangential ?rst-order differential ? operators n; CR Manifolds.- The Fefferman Metric.- The CR Yamabe Problem.- Pseudoharmonic Maps.- Pseudo-Einsteinian Manifolds.- Pseudo-Hermitian Immersions.- Quasiconformal Mappings.- Yang-Mills Fields on CR Manifolds.- Spectral Geometry.; In fact, it will be invaluable for people working on the differential geometry of CR manifolds. –Thomas Garity, MathSciNet ; The study of CR manifolds lies at the intersection of three main mathematical disciplines: partial differential equations, complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has recently been made to understand the differential geometric side of the subject. This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. It presents the major differential geometric acheivements in the theory of CR manifolds, such as the Tanaka–Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang–Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry. Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs. ; Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study ; The study of CR manifolds lies at the intersection of three main mathematical disciplines, partial differential equations: complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has been recently made to understand the differential geometric side of the subject. This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs. ; US

Published: Jun 10, 2007

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