Introduction To Global Analysis Minimal Surfaces In Riemannian Manifolds PDF Download
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Author | : John Douglas Moore |
Publisher | : American Mathematical Soc. |
Total Pages | : 385 |
Release | : 2017-12-15 |
Genre | : Mathematics |
ISBN | : 1470429500 |
Download Introduction to Global Analysis Book in PDF, ePub and Kindle
During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on by a finite-dimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinite-dimensional manifolds, and these infinite-dimensional manifolds were found useful for studying a wide variety of nonlinear PDEs. This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold, establishing Morse inequalities for perturbed versions of the energy function on the mapping space. It studies the bubbling which occurs when the perturbation is turned off, together with applications to the existence of closed minimal surfaces. The Morse-Sard theorem is used to develop transversality theory for both closed geodesics and closed minimal surfaces. This book is based on lecture notes for graduate courses on “Topics in Differential Geometry”, taught by the author over several years. The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology.
Author | : Jon T. Pitts |
Publisher | : Princeton University Press |
Total Pages | : 337 |
Release | : 2014-07-14 |
Genre | : Mathematics |
ISBN | : 1400856450 |
Download Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) Book in PDF, ePub and Kindle
Mathematical No/ex, 27 Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Author | : Min Ji |
Publisher | : American Mathematical Soc. |
Total Pages | : 63 |
Release | : 1993 |
Genre | : Mathematics |
ISBN | : 0821825607 |
Download Minimal Surfaces in Riemannian Manifolds Book in PDF, ePub and Kindle
A multiple solution theory to the Plateau problem in a Riemannian manifold is established. In [italic capital]S[superscript italic]n, the existence of two solutions to this problem is obtained. The Morse-Tompkins-Shiffman Theorem is extended to the case when the ambient space admits no minimal sphere.
Author | : Jürgen Eichhorn |
Publisher | : Nova Publishers |
Total Pages | : 664 |
Release | : 2007 |
Genre | : Mathematics |
ISBN | : 9781600215636 |
Download Global Analysis on Open Manifolds Book in PDF, ePub and Kindle
Global analysis is the analysis on manifolds. Since the middle of the sixties there exists a highly elaborated setting if the underlying manifold is compact, evidence of which can be found in index theory, spectral geometry, the theory of harmonic maps, many applications to mathematical physics on closed manifolds like gauge theory, Seiberg-Witten theory, etc. If the underlying manifold is open, i.e. non-compact and without boundary, then most of the foundations and of the great achievements fail. Elliptic operators are no longer Fredholm, the analytical and topological indexes are not defined, the spectrum of self-adjoint elliptic operators is no longer discrete, functional spaces strongly depend on the operators involved and the data from geometry, many embedding and module structure theorems do not hold, manifolds of maps are not defined, etc. It is the goal of this new book to provide serious foundations for global analysis on open manifolds, to discuss the difficulties and special features which come from the openness and to establish many results and applications on this basis.
Author | : Henri Anciaux |
Publisher | : World Scientific |
Total Pages | : 184 |
Release | : 2011 |
Genre | : Mathematics |
ISBN | : 9814291242 |
Download Minimal Submanifolds in Pseudo-Riemannian Geometry Book in PDF, ePub and Kindle
Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemannian case. For the first time, this textbook provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian geometry, only assuming from the reader some basic knowledge about manifold theory. Several classical results, such as the Weierstrass representation formula for minimal surfaces, and the minimizing properties of complex submanifolds, are presented in full generality without sacrificing the clarity of exposition. Finally, a number of very recent results on the subject, including the classification of equivariant minimal hypersurfaces in pseudo-Riemannian space forms and the characterization of minimal Lagrangian surfaces in some pseudo-Khler manifolds are given.
Author | : Tobias Holck Colding |
Publisher | : American Mathematical Society |
Total Pages | : 330 |
Release | : 2024-01-18 |
Genre | : Mathematics |
ISBN | : 1470476401 |
Download A Course in Minimal Surfaces Book in PDF, ePub and Kindle
Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.
Author | : Daniel W. Stroock |
Publisher | : American Mathematical Soc. |
Total Pages | : 290 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 0821838393 |
Download An Introduction to the Analysis of Paths on a Riemannian Manifold Book in PDF, ePub and Kindle
Hoping to make the text more accessible to readers not schooled in the probabalistic tradition, Stroock (affiliation unspecified) emphasizes the geometric over the stochastic analysis of differential manifolds. Chapters deconstruct Brownian paths, diffusions in Euclidean space, intrinsic and extrinsic Riemannian geometry, Bocher's identity, and the bundle of orthonormal frames. The volume humbly concludes with an "admission of defeat" in regard to recovering the Li-Yau basic differential inequality. Annotation copyrighted by Book News, Inc., Portland, OR.
Author | : John M. Lee |
Publisher | : Springer |
Total Pages | : 437 |
Release | : 2019-01-02 |
Genre | : Mathematics |
ISBN | : 3319917552 |
Download Introduction to Riemannian Manifolds Book in PDF, ePub and Kindle
This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Author | : Jon T. Pitts |
Publisher | : |
Total Pages | : |
Release | : 2014 |
Genre | : |
ISBN | : |
Download Existence and Regularity of Minimal Surfaces on Riemannian Manifolds Book in PDF, ePub and Kindle
Author | : Ulrich Dierkes |
Publisher | : Springer Science & Business Media |
Total Pages | : 547 |
Release | : 2010-08-16 |
Genre | : Mathematics |
ISBN | : 3642117066 |
Download Global Analysis of Minimal Surfaces Book in PDF, ePub and Kindle
Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau ́s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.