Degree theory on Banach manifolds

Tromba, Anthony J.

doi:10.1090/pspum/018.1/0277009

Cited by 58 publications

(33 citation statements)

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“…The reason is that the proof makes use of the Sard-Smale theorem for Fredholm maps, which requires the regularity to be strictly higher than the Fredholm index. Notice that the same problem appears in Smale degree theory for Fredholm maps [15,34]: C 2 -regularity is required, instead of a more natural C 1 -regularity. The problem is only technical and, with some additional effort, a degree theory for C 1 Fredholm maps can be obtained [21].…”

Section: Assumption (F4)mentioning

confidence: 99%

Morse homology on Hilbert spaces

Abbondandolo

Majer

2001

Comm Pure Appl Math

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Section: Assumption (F4)mentioning

confidence: 99%

Morse homology on Hilbert spaces

Abbondandolo

Majer

2001

Comm Pure Appl Math

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“…Before we describe these results we need the notion of a Fredholm map. The existing application of this theorem which has arisen in analysis [9] is where M is an open subset of a ball B in a Banach space E and the Fredholm structure on M is a restriction of the Fredholm structure on B. Thus this Fredholm structure is orientable.…”

Section: If C~ Is Homotopic To a Smooth Plane Curve Whose Interior Domentioning

confidence: 99%

“…The question of orientability is in a strong sense the heart of this paper. One could ask whether H~(M; Z2)=0 and if so invoke Theorem (1.7) and this certainly works for the class of problems considered in [9]. However one may not be able to say anything whatsoever about Hk(M; Zz) for any k. This is the case with Plateau's problem.…”

Section: If C~ Is Homotopic To a Smooth Plane Curve Whose Interior Domentioning

confidence: 99%

On the Morse number of embedded and non-embedded minimal immersions spanning wires on the boundary of special bodies in ?3

Tromba

1985

Math Z

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It is now more than fifty five years since the first paper on what is now called "Morse theory" first appeared. Since that time it has had an important impact on both topology and analysis (e.g. see the recent survey article on Morse's work by Raoul Bott in the Bull. AMS [5]). One of Morse's main goals was to apply his ideas to the Calculus of Variations in several variables and in particular to the classical problem of Plateau. This attempt was undertaken jointly by Morse and C.B. Tompkins and also independently by Max Shiffman and Richard Courant. These people had some limited success. In particular Morse and Tompkins [16] and Shiffman [-24] were able to prove that for a given wire F which admits two disc type minimal surfaces which span it, which are isolated as minimal surfaces (in some weak topology) and which are strictly (locally) area minimizing then there exists a third minimal surface spanning F which cannot be an isolated strict minimum for area.This result would, given the nature of the problem, be a type of Morse inequality. However no one could extend these methods to the situation where one knew one had two minima in a fine topology (e.g. Sobolev H S or C k'~ as opposed to M-S-TC~ Furthermore if you had two minima which are embedded (non-embedded immersions) is the third surface an embedding (nonembedded immersion)? The methods employed by the early pioneers of Analysis in the Large break down completely in attempting to answer these questions, as does those of more modern approaches to Analysis in the Large by Palais and Smale [22,26].In [4] the author and Reinhold BShme developed a global index and stability theory of minimal surfaces of disc type. It was here that notions of non-degeneracy of minimal surfaces as critical points of an energy functional were introduced and where generic non-degeneracy and generic finiteness results for minimal surfaces of disc type were proved. This was the first concrete information on the number and isolated nature of solutions to Plateau's problem to be optained, and the first necessary step in the development of any Morse theory for Plateau's problem.Using this theory as a starting point the author was able to develop [-34, 150 A. The purpose of this paper is to employ these ideas to show that under certain circumstances one can actually count the Morse number of embeddings and immersions which are not embeddings spanning a given wire F. Statement of Main ResultLet D be the unit disc in N~, (?D=S 1 and c~: S1--+IR ", n>4 an embedding of Sobolev class H ~, with F~=c~(S 1) the image wire. Let d(c~) be the space of all H ~ maps, s>s/2, r>2s+l, u: D~N", u=(u 1 .... ,u") such that for all i, Au~=O (each u ~ is harmonic) and u: S1--+F ~ is homotopic to ~.A minimal surface of disc type is an element u~X(~) satisfying and E~ is equivariant with respect to this action. Consequently if u is any critical point of E= the orbit of N through a (9,(N) consists of critical points. For such a critical point let D2E(u): T~.X(e)x T,Y(:<)--+R denote the Hessianor second deri...

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“…The first two authors introduced the notion of Fredholm structure in order to extend to infinite dimensional manifolds the classical concept of orientation of finite dimensional manifolds (on which is based the Brouwer degree) (see [4] and [5]). By a different approach, Fitzpatrick, Pejsachowicz and Rabier define a notion of orientation for Fredholm maps of index zero between real Banach spaces and then a degree for the class of oriented maps (see [7] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces

Benevieri

2001

Nonlinear Analysis and Its Applications to Differential Equations

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We define a notion of topological degree for a class of maps (called orientable), defined between real Banach spaces, which are Fredholm of index zero. We introduce first a notion of orientation for any linear Fredholm operator of index zero between two real vector spaces. This notion (which does not require any topological structure) allows to define a concept of orientability for nonlinear Fredholm maps between real Banach spaces.The degree which we present verifies the most important properties, usually taken into account in other degree theories, and it is invariant with respect to continuous homotopies of Fredholm maps.

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Degree theory on Banach manifolds

Cited by 58 publications

References 0 publications

Morse homology on Hilbert spaces

Morse homology on Hilbert spaces

On the Morse number of embedded and non-embedded minimal immersions spanning wires on the boundary of special bodies in ?3

Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces

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