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
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.
Readership
Graduate students and researchers interested in
differential geometry.