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Graduate Studies in Mathematics Ser.: Introduction to Global Analysis : Minimal Surfaces in Riemannian Manifolds by John Douglas Moore (2018, Hardcover)
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About this product
Product Identifiers
PublisherAmerican Mathematical Society
ISBN-101470429500
ISBN-139781470429508
eBay Product ID (ePID)239635956
Product Key Features
Number of Pages368 Pages
Publication NameIntroduction to Global Analysis : Minimal Surfaces in Riemannian Manifolds
LanguageEnglish
Publication Year2018
SubjectGeometry / Differential, Geometry / Algebraic
TypeTextbook
Subject AreaMathematics
AuthorJohn Douglas Moore
SeriesGraduate Studies in Mathematics Ser.
FormatHardcover
Dimensions
Item Height1 in
Item Weight28.7 Oz
Item Length10.4 in
Item Width7.3 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN2017-028964
Dewey Edition23
ReviewsThis book provides a thoughtful introduction to classical geometric applications of global analysis in the context of geodesics and minimal surfaces...it would be a good choice of textbook for a graduate topics course that provides a more classical overview of the area." - Renato G. Bettiol, Mathematical Reviews
Series Volume Number187
IllustratedYes
Volume NumberVol. 187
Dewey Decimal516.3/62
Table Of ContentInfinite-dimensional manifolds Morse theory of geodesics Topology of mapping spaces Harmonic and minimal surfaces Generic metrics Bibliography Index
SynopsisDuring 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 $M$ 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 $M$ 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., Applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. This volume then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It also treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold.
LC Classification NumberQA614.M66 2017
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