On differentiable functions with isolated critical points

Gromoll, Detlef; Meyer, Wolfgang

doi:10.1016/0040-9383(69)90022-6

Cited by 260 publications

(186 citation statements)

References 3 publications

Supporting

Mentioning

178

Contrasting

Unclassified

Order By: Relevance

“…We use an approach due to Gromoll and Meyer [10,22] and combine it with some ideas from the Conley index theory [6,11] (note that our cohomology is defined only on pairs of closed sets, so critical groups cannot be introduced by a formula similar to (0.1)). Let us point out that the functionals considered in [41] satisfied the Palais-Smale condition (PS) and were of the form Φ(x) = 1 2 Lx, x + ψ(x), with L linear and P F ∇ψ compact (P F is the orthogonal projector onto a certain subspace of E).…”

Section: Introductionmentioning

confidence: 99%

An infinite dimensional Morse theory with applications

Kryszewski¹,

Szulkin²

1997

Trans. Amer. Math. Soc.

103

View full text Add to dashboard Cite

Abstract. In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the onedimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.

show abstract

Section: Introductionmentioning

confidence: 99%

An infinite dimensional Morse theory with applications

Kryszewski¹,

Szulkin²

1997

Trans. Amer. Math. Soc.

103

View full text Add to dashboard Cite

show abstract

“…Recall that Y* is the gradient of the restriction of Ep to fi U ij. Using this fact we have from [2] the normal form result which uses the techniques of [31] and applies it to the splitting lemma of Gromoll-Meyer [12]. Notice that the right-hand side of this equality is just the Brouwer degree of the gradient of a smooth function about an absolute minimum.…”

Section: Gr+(e) Andgr~(e)mentioning

confidence: 99%

Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in ${\bf R}\sp n$. I. $n\geq 4$

Tromba¹

1985

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

ABSTRACT.A degree theory applicable to Plateau's problem is developed and the Morse equality for minimal surfaces spanning a contour in Rn, n > 4, is proved.In [10] the author and David Elworthy developed a theory of degree for Fredholm maps on oriented Banach manifolds, a theory which generalized the now classical Leray-Schauder degree theory.The purpose of this paper is to show how one can use a slight extension of Elworthy-Tromba theory to develop a "Morse-theory" for minimal surfaces of disc type spanning a wire in Rn. We prove, via this theory, the first and last of Morse's "inequalities", and in so doing obtain an existence theorem to the classical problem of Plateau, and simultaneously a generalization of the famous mountain pass result for minimal surfaces due to Morse, Shiftman and Tompkins.Before proceeding with a detailed statement of the results we would like to present a bit of historical background.

show abstract

“…The aim of this paper is to prove a degenerate-critical-point version of the Morse lemma as in [2] with conditions of low differentiability that, although stronger than those in [4] …”

Section: Theorem 11 If M Is a Compact Simply Connected Riemannian mentioning

confidence: 99%