Hardcover ISBN:  9781470429508 
Product Code:  GSM/187 
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AMS Member Price:  $66.40 
Electronic ISBN:  9781470443177 
Product Code:  GSM/187.E 
List Price:  $83.00 
MAA Member Price:  $74.70 
AMS Member Price:  $66.40 

Book DetailsGraduate Studies in MathematicsVolume: 187; 2017; 368 ppMSC: Primary 58; Secondary 46; 53; 55;
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 \(M\) 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 finitedimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinitedimensional manifolds, and these infinitedimensional manifolds were found useful for studying a wide variety of nonlinear PDEs.
This book applies infinitedimensional 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 MorseSard 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.ReadershipGraduate students and researchers interested in differential geometry.

Table of Contents

Chapters

Infinitedimensional manifolds

Morse theory of geodesics

Topology of mapping spaces

Harmonic and minimal surfaces

Generic metrics


Additional Material

Reviews

This 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


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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 \(M\) 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 finitedimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinitedimensional manifolds, and these infinitedimensional manifolds were found useful for studying a wide variety of nonlinear PDEs.
This book applies infinitedimensional 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 MorseSard 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.
Graduate students and researchers interested in differential geometry.

Chapters

Infinitedimensional manifolds

Morse theory of geodesics

Topology of mapping spaces

Harmonic and minimal surfaces

Generic metrics

This 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