Introduction To Global Analysis Minimal Surfaces In Riemannian Manifolds
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Introduction To Global Analysis Minimal Surfaces In Riemannian Manifolds
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Author : John Douglas Moore
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-12-15
Introduction To Global Analysis Minimal Surfaces In Riemannian Manifolds written by John Douglas Moore and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-15 with Electronic books categories.
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.
Minimal Surfaces In Riemannian Manifolds
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Author : Min Ji
language : en
Publisher: American Mathematical Soc.
Release Date : 1993
Minimal Surfaces In Riemannian Manifolds written by Min Ji and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Mathematics categories.
This monograph studies the structure of the set of all co boundary minimal surfaces in Riemannian manifolds. The authors establish, on a solid analytical foundation, a flexible topological index theory which proves useful for the study of minimal surfaces. One of the highlights of the work is the result that for every Jordan curve on the standard $n$-sphere, there exist at least two minimal surfaces bounded by the curve.
Minimal Surfaces From A Complex Analytic Viewpoint
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Author : Antonio Alarcón
language : en
Publisher: Springer Nature
Release Date : 2021-03-10
Minimal Surfaces From A Complex Analytic Viewpoint written by Antonio Alarcón and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-03-10 with Mathematics categories.
This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.
Existence And Regularity Of Minimal Surfaces On Riemannian Manifolds
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Author : Jon T. Pitts
language : en
Publisher:
Release Date : 1981
Existence And Regularity Of Minimal Surfaces On Riemannian Manifolds written by Jon T. Pitts and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with categories.
Minimal Surfaces In Riemannian Manifolds
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Author : Min Ji
language : en
Publisher: American Mathematical Soc.
Release Date : 1990
Minimal Surfaces In Riemannian Manifolds written by Min Ji and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Mathematics categories.
This monograph studies the structure of the set of all coboundary minimal surfaces in Riemannian manifolds. The authors establish, on a solid analytical foundation, a flexible topological index theory which proves useful for the study of minimal surfaces. One of the highlights of the work is the result that for every Jordan curve on the standard $n$-sphere, there exist at least two minimal surfaces bounded by the curve.
Global Riemannian Geometry Curvature And Topology
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Author : Steen Markvorsen
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Global Riemannian Geometry Curvature And Topology written by Steen Markvorsen and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This book contains a clear exposition of two contemporary topics in modern differential geometry: distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature the application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold. It is intended for both graduate students and researchers.
Existence And Regularity Of Minimal Surfaces On Riemannian Manifolds Mn 27
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Author : Jon T. Pitts
language : en
Publisher:
Release Date : 2016-04-19
Existence And Regularity Of Minimal Surfaces On Riemannian Manifolds Mn 27 written by Jon T. Pitts and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-19 with Minimal surfaces categories.
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.
Seminar On Minimal Submanifolds Am 103 Volume 103
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Author : Enrico Bombieri
language : en
Publisher: Princeton University Press
Release Date : 2016-03-02
Seminar On Minimal Submanifolds Am 103 Volume 103 written by Enrico Bombieri and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-02 with Mathematics categories.
The description for this book, Seminar On Minimal Submanifolds. (AM-103), Volume 103, will be forthcoming.
Global Analysis Of Minimal Surfaces
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Author : Ulrich Dierkes
language : en
Publisher: Springer
Release Date : 2012-12-14
Global Analysis Of Minimal Surfaces written by Ulrich Dierkes and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-14 with Mathematics categories.
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.
Eigenfunctions Of The Laplacian On A Riemannian Manifold
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Author : Steve Zelditch
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-12-12
Eigenfunctions Of The Laplacian On A Riemannian Manifold written by Steve Zelditch and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-12 with Eigenfunctions categories.
Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.