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Yamabe type Equations on Complete Noncompact Manifolds

Yamabe type Equations on Complete  Noncompact Manifolds Author Paolo Mastrolia
ISBN-10 9783034803762
Release 2012-07-30
Pages 260
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The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem. The book also describes a wide range of methods and techniques that can be successfully applied to nonlinear differential equations in particularly challenging situations. Such situations occur where the lack of compactness, symmetry and homogeneity prevents the use of more standard tools typically used in compact situations or for the Euclidean setting. The work is written in an easy style that makes it accessible even to non-specialists. After a self-contained treatment of the geometric tools used in the book, readers are introduced to the main subject by means of a concise but clear study of some aspects of the Yamabe problem on compact manifolds. This study provides the motivation and geometrical feeling for the subsequent part of the work. In the main body of the book, it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up-to-date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.

Maximum Principles and Geometric Applications

Maximum Principles and Geometric Applications Author Luis J. Alías
ISBN-10 9783319243375
Release 2016-02-13
Pages 570
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This monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter. In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on. Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.

Differential Geometry and Analysis on CR Manifolds

Differential Geometry and Analysis on CR Manifolds Author Sorin Dragomir
ISBN-10 0817644830
Release 2007-06-10
Pages 488
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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

Nonlinear Analysis on Manifolds

Nonlinear Analysis on Manifolds Author Emmanuel Hebey
ISBN-10 9780821827000
Release 2000-10-27
Pages 290
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This volume offers an expanded version of lectures given at the Courant Institute on the theory of Sobolev spaces on Riemannian manifolds. ``Several surprising phenomena appear when studying Sobolev spaces on manifolds,'' according to the author. ``Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a central role.'' The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds. Chapter 3 presents the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5. Chapter 6 presents special types of Sobolev inequalities under constraints. Best constants problems for complete noncompact manifolds are discussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. And Chapter 9 discusses the influence of symmetries on Sobolev embeddings. An appendix offers brief notes on the case of manifolds with boundaries. This topic is a field undergoing great development at this time. However, several important questions remain open. So a substantial part of the book is devoted to the concept of best constants, which appeared to be crucial for solving limiting cases of some classes of PDEs. The volume is highly self-contained. No familiarity is assumed with differentiable manifolds and Riemannian geometry, making the book accessible to a broad audience of readers, including graduate students and researchers.

Seminar on Differential Geometry AM 102

Seminar on Differential Geometry   AM 102 Author Shing-tung Yau
ISBN-10 9781400881918
Release 2016-03-02
Pages 720
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This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincaré inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information.

A Panoramic View of Riemannian Geometry

A Panoramic View of Riemannian Geometry Author Marcel Berger
ISBN-10 9783642182457
Release 2012-12-06
Pages 824
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This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

Linear and Nonlinear Aspects of Vortices

Linear and Nonlinear Aspects of Vortices Author Frank Pacard
ISBN-10 0817641335
Release 2000-06-22
Pages 342
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Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful for the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and serves as an excellent classroom text or a valuable self-study resource.

The Riemann Legacy

The Riemann Legacy Author Krzysztof Maurin
ISBN-10 079234636X
Release 1997-07-31
Pages 717
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The study of the rise and fall of great mathematical ideas is undoubtedly one of the most fascinating branches of the history of science. It enables one to come into contact with and to participate in the world of ideas. Nowhere can we see more concretely the enormous spiritual energy which, initially still lacking clear contours, begs to be moulded and developed by mathematicians, than in Riemann (1826-1866). He perceived mathematics and physics as one discipline and thought of himself as both mathematician and physicist. His ideas as well as their contemporary descendants are the theme of this book. Audience: This volume will be useful to those interested in such diverse fields as the mathematics of physics, algebra and number theory, topology and geometry, analysis, and the history of science.

Poincar s Legacies

Poincar   s Legacies Author Terence Tao
ISBN-10 9780821848838
Release 2009
Pages 293
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There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the AMS. The posts from 2008 are being published in two volumes. This book is Part I of the second-year posts, focusing on ergodic theory, combinatorics, and number theory. Chapter 2 consists of lecture notes from Tao's course on topological dynamics and ergodic theory. By means of various correspondence principles, recurrence theorems about dynamical systems are used to prove some deep theorems in combinatorics and other areas of mathematics. The lectures are as self-contained as possible, focusing more on the ``big picture'' than on technical details. In addition to these lectures, a variety of other topics are discussed, ranging from recent developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas-Lehmer test for Mersenne primes. Some selected comments and feedback from blog readers have also been incorporated into the articles. The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.

Parabolic Geometries Background and general theory

Parabolic Geometries  Background and general theory Author Andreas Cap
ISBN-10 9780821826812
Release 2009
Pages 628
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Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CR-structures and various types of generic distributions. The characteristic feature of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold (the quotient of a semisimple Lie group by a parabolic subgroup). Background on differential geometry, with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equivalence between Cartan connections and underlying structures, including a complete proof of Kostant's version of the Bott-Borel-Weil theorem, which is used as an important tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text.

Groups and Geometric Analysis

Groups and Geometric Analysis Author Sigurdur Helgason
ISBN-10 9780821826737
Release 1984
Pages 667
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Group-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group. The introductory chapter is a self-contained account of the analysis on surfaces of constant curvature. Later chapters cover general cases of the Radon transform, spherical functions, invariant operators, compact symmetric spaces and other topics. This book, together with its companion volume, Geometric Analysis on Symmetric Spaces (AMS Mathematical Surveys and Monographs series, vol. 39, 1994), has become the standard text for this approach to geometric analysis. Sigurdur Helgason was awarded the Steele Prize for outstanding mathematical exposition for Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces.

The Scottish book

The Scottish book Author R. Daniel Mauldin
ISBN-10 3764330457
Release 1981
Pages 268
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The Scottish book has been writing in one form or another for most of life. You can find so many inspiration from The Scottish book also informative, and entertaining. Click DOWNLOAD or Read Online button to get full The Scottish book book for free.

Spectral Functions in Mathematics and Physics

Spectral Functions in Mathematics and Physics Author Klaus Kirsten
ISBN-10 9781420035469
Release 2001-12-13
Pages 400
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The literature on the spectral analysis of second order elliptic differential operators contains a great deal of information on the spectral functions for explicitly known spectra. The same is not true, however, for situations where the spectra are not explicitly known. Over the last several years, the author and his colleagues have developed new, innovative methods for the exact analysis of a variety of spectral functions occurring in spectral geometry and under external conditions in statistical mechanics and quantum field theory. Spectral Functions in Mathematics and Physics presents a detailed overview of these advances. The author develops and applies methods for analyzing determinants arising when the external conditions originate from the Casimir effect, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta function underlies all of these techniques, and the book begins by deriving its basic properties and relations to the spectral functions. The author then uses those relations to develop and apply methods for calculating heat kernel coefficients, functional determinants, and Casimir energies. He also explores applications in the non-relativistic context, in particular applying the techniques to the Bose-Einstein condensation of an ideal Bose gas. Self-contained and clearly written, Spectral Functions in Mathematics and Physics offers a unique opportunity to acquire valuable new techniques, use them in a variety of applications, and be inspired to make further advances.

Notes on Dynamical Systems

Notes on Dynamical Systems Author Jürgen Moser
ISBN-10 9780821835777
Release 2005
Pages 256
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This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jurgen Moser (1928-1999) was a professor atthe Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jurgen Moser is the author of several books, among them Stable and Random Motions in DynamicalSystems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Analysis III

Analysis III Author Sergei M. Nikol'skii
ISBN-10 9783662099612
Release 2013-03-09
Pages 221
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In the Part at hand the authors undertake to give a presentation of the historical development of the theory of imbedding of function spaces, of the internal as well as the externals motives which have stimulated it, and of the current state of art in the field, in particular, what regards the methods employed today. The impossibility to cover all the enormous material connected with these questions inevitably forced on us the necessity to restrict ourselves to a limited circle of ideas which are both fundamental and of principal interest. Of course, such a choice had to some extent have a subjective character, being in the first place dictated by the personal interests of the authors. Thus, the Part does not constitute a survey of all contemporary questions in the theory of imbedding of function spaces. Therefore also the bibliographical references given do not pretend to be exhaustive; we only list works mentioned in the text, and a more complete bibliography can be found in appropriate other monographs. O.V. Besov, v.1. Burenkov, P.1. Lizorkin and V.G. Maz'ya have graciously read the Part in manuscript form. All their critical remarks, for which the authors hereby express their sincere thanks, were taken account of in the final editing of the manuscript.

Sobolev Spaces on Riemannian Manifolds

Sobolev Spaces on Riemannian Manifolds Author Emmanuel Hebey
ISBN-10 9783540699934
Release 2006-11-14
Pages 120
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Several books deal with Sobolev spaces on open subsets of R (n), but none yet with Sobolev spaces on Riemannian manifolds, despite the fact that the theory of Sobolev spaces on Riemannian manifolds already goes back about 20 years. The book of Emmanuel Hebey will fill this gap, and become a necessary reading for all using Sobolev spaces on Riemannian manifolds. Hebey's presentation is very detailed, and includes the most recent developments due mainly to the author himself and to Hebey-Vaugon. He makes numerous things more precise, and discusses the hypotheses to test whether they can be weakened, and also presents new results.

Lie Groups and Geometric Aspects of Isometric Actions

Lie Groups and Geometric Aspects of Isometric Actions Author Marcos M. Alexandrino
ISBN-10 9783319166131
Release 2015-05-22
Pages 213
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This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.