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Author: Takeo Ohsawa
Publisher: Springer
ISBN: 4431568522
Category : Mathematics
Languages : en
Pages : 258
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Book Description
This monograph presents the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Special emphasis is put on the new precise results on the L2 extension of holomorphic functions in the past 5 years.In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and separately by Q.-A. Guan and X.-Y. Zhou. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, Guan–Zhou, and Berndtsson–Lempert. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L2 method obtained during the past 15 years.
Author: Takeo Ohsawa
Publisher: Springer
ISBN: 4431568522
Category : Mathematics
Languages : en
Pages : 258
Get Book Here
Book Description
This monograph presents the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Special emphasis is put on the new precise results on the L2 extension of holomorphic functions in the past 5 years.In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and separately by Q.-A. Guan and X.-Y. Zhou. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, Guan–Zhou, and Berndtsson–Lempert. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L2 method obtained during the past 15 years.
Author: Takeo Ohsawa
Publisher:
ISBN: 9784431557487
Category :
Languages : en
Pages :
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Book Description
The purpose of this monograph is to present the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Highlighted are the new precise results on the L2 extension of holomorphic functions. In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka-Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and by Q.-A. Guan and X.-Y. Zhou separately. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani-Yamaguchi, Berndtsson, and Guan-Zhou. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L2 method obtained during these 15 years.
Author: Takeo Ohsawa
Publisher: Springer
ISBN: 4431557474
Category : Mathematics
Languages : en
Pages : 202
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Book Description
The purpose of this monograph is to present the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Highlighted are the new precise results on the L2 extension of holomorphic functions. In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and by Q.-A. Guan and X.-Y. Zhou separately. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, and Guan–Zhou. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are discussed on the existence and classification of certain holomorphic foliations and Levi flat hypersurfaces as their stables sets. These are also applications of the L2 method obtained during these 15 years.
Author: Takeo Ōsawa
Publisher: American Mathematical Soc.
ISBN: 9780821820988
Category : Mathematics
Languages : en
Pages : 148
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Book Description
An expository account of the basic results in several complex variables that are obtained by L℗ methods.
Author: Franc Forstnerič
Publisher: Springer
ISBN: 3319610589
Category : Mathematics
Languages : en
Pages : 569
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Book Description
This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.
![Lectures on the L2-Sobolev Theory of the [d-bar]-Neumann Problem PDF](https://books.google.com/books/content/images/frontcover/OX7IaM3Cx6cC?fife=w80-h100)
Author: Emil J. Straube
Publisher: European Mathematical Society
ISBN: 9783037190760
Category : Mathematics
Languages : en
Pages : 220
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Book Description
This book provides a thorough and self-contained introduction to the $\bar{\partial}$-Neumann problem, leading up to current research, in the context of the $\mathcal{L}^{2}$-Sobolev theory on bounded pseudoconvex domains in $\mathbb{C}^{n}$. It grew out of courses for advanced graduate students and young researchers given by the author at the Erwin Schrodinger International Institute for Mathematical Physics and at Texas A & M University. The introductory chapter provides an overview of the contents and puts them in historical perspective. The second chapter presents the basic $\mathcal{L}^{2}$-theory. Following is a chapter on the subelliptic estimates on strictly pseudoconvex domains. The two final chapters on compactness and on regularity in Sobolev spaces bring the reader to the frontiers of research. Prerequisites are a solid background in basic complex and functional analysis, including the elementary $\mathcal{L}^{2}$-Sobolev theory and distributions. Some knowledge in several complex variables is helpful. Concerning partial differential equations, not much is assumed. The elliptic regularity of the Dirichlet problem for the Laplacian is quoted a few times, but the ellipticity results needed for elliptic regularization in the third chapter are proved from scratch.
Author: G.M. Khenkin
Publisher: Springer Science & Business Media
ISBN: 3642580114
Category : Mathematics
Languages : en
Pages : 292
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Book Description
This volume of the Encyclopaedia contains three contributions in the field of complex analysis; on mean periodicity and convolutionequations, Yang-Mills fields and the Radon-Penrose transform, and stringtheory. It is immensely useful to graduate students and researchers in complex analysis, differential geometry, quantum field theory, string theory and general relativity.
Author: B. A. Fuks
Publisher: American Mathematical Soc.
ISBN: 0821815644
Category : Mathematics
Languages : en
Pages : 366
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Book Description
Author: Mikael Passare
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 382
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Book Description
Author: Steven Gubkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 78
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Book Description
An "approximation theorem" usually establishes the density of one space of functions in another, with respect to some norm. Classical examples include Runge's theorem and Mergelyan's theorem from single variable complex analysis. One analogue of Runge's theorem to several complex variables is the Oka-Weil Theorem. We offer an apparently new proof of this theorem which avoids a tricky duality argument. The main new theorems in the thesis are analogues of Mergelyan's theorem, only using L2 instead of uniform norm approximations. In particular we show that if an open set U is strictly hyper convex, the space of holomorphic functions defined in a neighborhood of U is dense with respect to L2 norm in the space of square integrable holomorphic functions defined on U. We then extend this result to L2 approximation of dbar-closed (p,q)-forms.
