Isoperimetric Inequalities in the Theory of Surfaces of Bounded External Curvature
Author: Iurii D. Burago
Publisher: Springer
Published: 1970
Total Pages: 118
ISBN-13:
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Isometric Embedding of Riemannian Manifolds in Euclidean Spaces
Author: Qing Han
Publisher: American Mathematical Soc.
Published: 2006
Total Pages: 278
ISBN-13: 0821840711
DOWNLOAD EBOOKThe question of the existence of isometric embeddings of Riemannian manifolds in Euclidean space is already more than a century old. This book presents, in a systematic way, results both local and global and in arbitrary dimension but with a focus on the isometric embedding of surfaces in ${\mathbb R}^3$. The emphasis is on those PDE techniques which are essential to the most important results of the last century. The classic results in this book include the Janet-Cartan Theorem, Nirenberg's solution of the Weyl problem, and Nash's Embedding Theorem, with a simplified proof by Gunther. The book also includes the main results from the past twenty years, both local and global, on the isometric embedding of surfaces in Euclidean 3-space. The work will be indispensable to researchers in the area. Moreover, the authors integrate the results and techniques into a unified whole, providing a good entry point into the area for advanced graduate students or anyone interested in this subject. The authors avoid what is technically complicated. Background knowledge is kept to an essential minimum: a one-semester course in differential geometry and a one-year course in partial differential equations.
Nonlinear Analysis on Manifolds. Monge-Ampère Equations
Author: Thierry Aubin
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 215
ISBN-13: 1461257344
DOWNLOAD EBOOKThis volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.
Partial Differential Relations
Author: Misha Gromov
Publisher: Springer Science & Business Media
Published: 2013-03-14
Total Pages: 372
ISBN-13: 3662022672
DOWNLOAD EBOOKThe classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book.
Two-dimensional Manifolds of Bounded Curvature
Author: Aleksandr Danilovich Aleksandrov
Publisher: American Mathematical Soc.
Published: 1967
Total Pages: 198
ISBN-13: 9780821818763
DOWNLOAD EBOOKProceedings and papers about in which the foundation of the intrinsic geometry of nonregular surfaces is developed.
Differential Geometry of Curves and Surfaces
Author: Victor Andreevich Toponogov
Publisher: Springer Science & Business Media
Published: 2006-09-10
Total Pages: 215
ISBN-13: 0817644024
DOWNLOAD EBOOKCentral topics covered include curves, surfaces, geodesics, intrinsic geometry, and the Alexandrov global angle comparision theorem Many nontrivial and original problems (some with hints and solutions) Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels
Encyclopaedia of Mathematics
Author: M. Hazewinkel
Publisher: Springer
Published: 2013-12-01
Total Pages: 967
ISBN-13: 1489937951
DOWNLOAD EBOOKMostly Surfaces
Author: Richard Evan Schwartz
Publisher: American Mathematical Soc.
Published: 2011
Total Pages: 330
ISBN-13: 0821853686
DOWNLOAD EBOOKThe goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
Metric Spaces of Non-Positive Curvature
Author: Martin R. Bridson
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 665
ISBN-13: 3662124947
DOWNLOAD EBOOKA description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.
Lectures on K3 Surfaces
Author: Daniel Huybrechts
Publisher: Cambridge University Press
Published: 2016-09-26
Total Pages: 499
ISBN-13: 1316797252
DOWNLOAD EBOOKK3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.
