Smale nave studied morse theory for a realvalued function on an infinite dimensional manifold and have given direct proofs of the main theorems, without making any use of finite dimensional ap. Nonlinear analysis and semilinear elliptic problems antonio ambrosetti and andrea malchiodi frontmatter more information. The palaissmale conditions for the yangmills functional volume 108 issue 34 d. Both morse theory on hilbert manifolds, developed by palais and smale 3, 5, and the luisternikschnirelman theory on banach manifolds due to palais 2 are several years old. Let b be a complete c3 finsler manifold with c2 partitions of unity. The second part consists of applications of morse theory over the reals, while the last part describes the basics and some applications of complex morse theory, a. Morse theory and nonlinear differential equations request pdf. Morse theory was developed in the 1920s by mathematician marston morse. Critical point theory an overview sciencedirect topics. To clarify this statement, we take a small historical digression. As an application, we give a simplified proof of a theorem of hoferviterbo on the existence of closed characteristic leaves for certain contact type. Intuitively,the relevant ends of consistof approximate solutions to the eulerlagrange equations of. Morse theory investigates the topology of a surface by looking at critical points of a function on that surface.
However, there have been important developments during the past few years which are not mentioned. We prove the morse relations for the set of all geodesics connecting two nonconjugate points on a class of globally hyperbolic lorentzian manifolds. A generalized morse theory the collected papers of stephen. Morse theory and nonminimal solutions to the yangmills. In classical morse theory, manifolds are smooth and finite dimensional, and critical points are isolated and nondegenerate. Scalability and generality attila gyulassy, peertimo bremer, member, ieee, bernd hamann, member, ieee, and valerio pascucci, member, ieee abstractthe morsesmale ms complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. Introduction in classical morse theory, manifolds are smooth and. The palaissmale condition on contact type energy levels for. In this paper, we are interested in realvalued morse functions that are induced from holomorphic morse functions. Palais, morse theory on hilbert manifolds, topology, vol. Dynamics of gradient flows in the halftransversal morse theory goda, hiroshi and pajitnov, andrei v. For further information on morse theory, the following sources are extremely useful.
Whereas morse theory tries to extract topological information from information about critical points of a function, minmax theory tries to achieve. In another context, a morse function can also mean an anharmonic oscillator see morse potential in mathematics, specifically in differential topology, morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and type. Boundedness of palaissmale sequences associated to a fourth. Morse, the calculus of variations in the large, american mathematical society, new york, 1934. Palais the term morse theory is usually understood to apply to two analagous but quite distinct bodies of mathematical theorems. Jean mawhin and michel willem dedicated, with admiration, to stephen smale abstract in 1963, palais and smale have. Morse theory to this frame and study nonlinear partial differential equations. Morse theory, higgs fields and yangmillshiggs functionals. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist edward witten relates morse theory to quantum field theory. In addition, each group contains one or two articles by world leaders on its subject which comment on the influence of smale s work, and another article by smale with his own retrospective views.
Since palais and smale 39,41,46 generalized finitedimensional morse theory 36,35 to nondegenerate c 2 functionals on infinite dimensional hilbert manifolds and used it to study multiplicity. Milnor was awarded the fields medal the mathematical equivalent of a nobel prize in 1962 for his work in differential topology. In this article, we present several new results for palais smale sequences. Both morse theory on hilbert manifolds, developed by palais and smale 3, 51, and the luisternikschnirelman theory on banach manifolds due to palais 2 are several years old. The generalization to hilbert spaces is due to richard palais and stephen smale. Be sides classical morse theory on a compact manifold, topics discussed included equivariant morse functions, and more generally nondegenerate functions having critical submanifolds, as well as morse functions on in. The proofs are based on a morse theory approach using the palaissmale condition. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form. The bott periodicity theorems,ere originally i nspi red by morse theory s ee part iv.
Lmw17a, where the nonproper morse theory of palais smale ps64 is used. Richard sheldon palais born may 22, 1931 is a mathematician working in geometry who introduced the principle of symmetric criticality, the mostowpalais theorem, the liepalais theorem, the morsepalais lemma, and the palaissmale compactness condition from 1965 to 1967 palais was a sloan fellow. Even before the appearance of 16, goresky and macpherson published two papers, strati. Let us now return to the question raised above, namely whether it is true that bounded minimizing sequences contain a convergent subsequence. Lectures on morse theory, old and new morse theory is. Despite the many applications of this theory found recently, no general information about the morse index was known. Jun 30, 2006 we prove that for a uniformly convex lagrangian system l on a compact manifold m, almost all energy levels contain a periodic orbit. Printed in great brimin morse theory on hilbert anrflds richard s. Show that a resonant morse function can be arbitrarily well approximated in the c2 topology by non. The course was geared to students who had completed a one year course in algebraic topology and had some familiarity with basic di erential geometry. Morse theory for cell complexes school of mathematics. Following palais and smale, klingenberg of course carries out everything in the infinite dimensional context of hilbertmanifolds. Smale, morse theory and a non linea r generali zat i on of the dirichlet problem, annals of mathemat ics, vol.
Morses theory of thc ca l cul us of var1ations i n the l arge. Origin and evolution of the palais smale condition in critical point theory. Stable and unstable manifolds are introduced next and the lyapunovproperty is established. We prove that for a uniformly convex lagrangian system l on a compact manifold m, almost all energy levels contain a periodic orbit. Notes for botts 1963 lectures on morse theory mathoverflow. Origin and evolution of the palaissmale condition in. So we use here the nonproper morse theory techniques developed by palais smale see also 9.
Man e in 23 can be characterized by the palais smale condition. The palaissmale condition and the existence of saddle points on a compact manifold, any continuous function assumes its minimum. In 1970 he was an invited speaker banach manifolds of fiber bundle sections at the. Matsumotos morse theory is one of few relatively recent treatments of finitedimensional morse theory, written at a level high enough to make it useful for advanced undergraduates or more likely graduate students, but with a slower pace than, say, one of milnors books. Botts topology seminar at harvard in february and march of 1963 unknown binding richard s palais author with brandeis univ, listed as the publisher. He was awarded the fields medal in 1966 and spent more than three decades on the mathematics faculty of the university of california, berkeley 19601961 and 19641995. However, mor e e l ementary proofs, which do not in. Instead we start by nontrivally restricting the set of smooth functions we will consider. Threlfall, varlationsrechnung im grossen, published in the united states by chelsea, new york, 1951. The following theorem is due to palais and smale in the case of nondegenerate critical points, and the author and tromba in the weaker cases 14, 15. Existence of three solutions for higher order bvp with parameters via morse theory mariusz jurkiewicz, bogdan przeradzki abstract. There is no hilbert space structure or a palaissmale condition to check. A generalized morse theory the collected papers of. This leads to a variant of the minimax principle in the lusternikschnirelman theory.
This is the second part of a twopart article in the notices to commemoratehislifeand work. However, as a realvalued morse function log f is almost never proper. In the context of general critical point theory, this is known today as the palaissmale property, and goes back to a famous work 34 of these two authors where they study a generalized morse theory. We prove the existence of at least three solutions to a general lidstone problem using the morse theory. Note on a function satisfying palaissmale condition.
On the one hand one considers a smooth, real valued. Smale, morse theory and a nonlinear generalization of the dirichlet problem, annals of mathematics, vol. The palaissmale condition was invented as a replacement for the compactness of sublevel sets in the in nitedimensional setting. Pdf in 196364, palais and smale have introduced a compactness condition. In 196364, palais and smale have introduced a compactness condition, namely condition c, on real functions of class c 1 defined on a riemannian manifold modeled upon a hilbert space, in order to extend morse theory to this frame and study nonlinear partial differential equations. I therefore have only added new material directly related to the themes originally covered. The only difference worth mentioning is that in morse theory 3 the compactness of is replaced by condition of palaissmale see morse function, which, besides, is not satisfied in. Morse was on the faculty of the institute for advanced study, and princeton published his topological methods in the theory of functions of a complex variable in the annals of mathematics studies series in 1947. The only difference worth mentioning is that in morse theory 3 the compactness of is replaced by condition of palaissmale see morse function, which, besides, is not satisfied in all situations of interest.
Here is a more detailed presentation of the contents. Morse theory could be very well be called critical point theory. Pdf the palaissmale condition for the hamiltonian action. Morse theory and nonminimal solutions 571 it is the effectof the map i on the relevant homotopy groups of pairs which determines whether crit. We also prove that below manes critical value of the lift of the lagrangian to the universal cover, c u l, almost all energy levels have conjugate points. Introduction in 1929, lidstone introduced a generalization of the taylor series. The dynamics of a nice enough f 2 c1m reconstructs msmoothly. Bradlow and graeme wilkin dedicated to professor dick palais on the occasion of his 80th birthday 1. The palaissmale condition on contact type energy levels.
The idea is torus provided by john milnor in his excellent book morse theory. Iff is a c3 function which satisfies palais smale condition c, then f satisfies a morse theory. In a reflexive banach space we consider a family of functionals that may admit unbounded palaissmale sequences. A critical point u with c 1pg,uq0 is called a mountain pass point. Lmw17a, where the nonproper morse theory of palaissmale ps64 is used. We in addition prove that if an energy level is of contact type, projects onto m and \m\ne\mathbb t2. Stephen smale born july 15, 1930 is an american mathematician, known for his research in topology, dynamical systems and mathematical economics. Pdf origin and evolution of the palaissmale condition in critical. We show that the hamiltonian action satisfies the palaissmale condition over a mixed regularity space of loops in cotangent bundles, namely the space of loops with regularity hs, s. A morse function is called resonant when there are two critical points with the same critical value. These techniques produce palaissmale sequences in the loop space16. A minimax theorem in the presence of unbounded palais.
A new cohomology for the morse theory of strongly indefinite functionals on hilbert spaces abbondandolo, alberto, topological methods in nonlinear analysis, 1997. This will follow from more general results to be precisely stated below. Condition c3 leads to the estimate y 1 2 kyk2 z 1 0. The bott periodicity theorems were originally inspired by morse theory see part iv. The di culty is mainly caused by the weak notions of convergence in the sense of geometric measure theory and the possibility of multiplicity. Consequently, we unify the vitali convergence theorem and many main concepts in the variational methods by lions, lientzengwang, del pinofelmer and chabrowski.
A practical approach to morsesmale complex computation. According to the basic insights of marston morse, a typical differentiable function on a. In 1961, smale proved the hcobordism theorem for smooth manifolds and hence its corollary, the poincare conjecture in dimension. The rst was to give an introduction to morse theory from a topological point of view. Nearly 50 years ago palais expanded the theory to cove. The last assumption of the above theorem is called the palaissmale condition. Smale conjectured that morse theory can be extended to banach manifolds. Richard sheldon palais born may 22, 1931 is a mathematician working in geometry who introduced the principle of symmetric criticality, the mostowpalais theorem, the liepalais theorem, the morsepalais lemma, and the palaissmale compactness condition. Then we show how the generic theorem can be used to obtain, for a given functional, a special palaissmale sequence possessing extra properties that help to ensure its convergence. Geometric heat ows have been extensively studied, yet one of the simplest examples is not yet fully understood. In addition, each group contains one or two articles by world leaders on its subject which comment on the influence of smales work, and another article by smale with his own retrospective views.
This morse theory takes on particular significance in the context of pl manifolds. Morse theory contains, in fact, morses results for nondegenerate critical points, handle attaching, deformation theorems, mountain pass lemma and results concerning the number of critical points. Morse theory and applications to variational problems. There were two basic goals in the course and in these notes. Under some structural conditions we provide a suitable deformation lemma that is obtained by modifying the classical pseudogradient flow. The palaissmale conditions for the yangmills functional.
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