Morse homology on Hilbert spaces

Abbondandolo, Alberto; Majer, Pietro

doi:10.1002/cpa.1012

Cited by 46 publications

(76 citation statements)

References 36 publications

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“…As a consequence we get that A has determinant equal to ±1 and is therefore an isomorphism. But g 1] , and this completes the proof of the Rigidity Theorem 1.19 (i). We now turn to 1.19 (ii).…”

Section: Remark 120 (A)supporting

confidence: 62%

“…Here is a result that summarizes the output of some of our constructions. All the arguments here are simple once the correct statements are formulated -similar constructions have been independently used by Abbondandolo and Majer [1]. We include the details because we shall use these constructions repeatedly later in the paper.…”

Section: Construction Of Morse-smale Cobordismsmentioning

confidence: 99%

“…To simplify notation we denote G = F 1/n , h = f 1/n with n satisfying the conclusion of 1.31. Denote by F, G respectively the chain maps F M× [0,1] and G M× [0,1] . …”

Section: Definition 128 a Morse-smale Function (G β)mentioning

confidence: 99%

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Rigidity and gluing for Morse and Novikov complexes

Cornea

Ranicki

2003

J. Eur. Math. Soc.

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Abstract. We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closedThe rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C 0 close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.

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Section: Remark 120 (A)supporting

confidence: 62%

Section: Construction Of Morse-smale Cobordismsmentioning

confidence: 99%

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Rigidity and gluing for Morse and Novikov complexes

Cornea

Ranicki

2003

J. Eur. Math. Soc.

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“…The Morse index (MI) of a critical point u 0 of φ is the dimension of the maximum negative definite subspace of φ (u 0 ) in H. When φ is of the form It is obvious that if φ is strongly indefinite, so is −φ. In this case, the Morse index of every critical point of both φ and −φ is infinite and hence provides no help for one to find those critical points [1,2]. This also implies that a strongly indefinite functional is neither bounded from above nor from below, not even modulo any finite-dimensional subspace [12].…”

Section: ∇U(x) · ∇V(x) − G(x U(x) V(x))] DXmentioning

confidence: 99%

“…Substituting (4.14) into (4.13) gives (4.12) implies that the diagonal elements of Q are negative and the determinant |Q| > a 2 …”

Section: Proof By Definition An L-⊥ Selection P(ūv) = (Tū Sv) Is mentioning

confidence: 99%

A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems

Chen

Zhou

2010

Math. Comp.

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Abstract. A local min-max-orthogonal method together with its mathematical justification is developed in this paper to solve noncooperative elliptic systems for multiple solutions in an order. First it is discovered that a noncooperative system has the nature of a zero-sum game. A new local characterization for multiple unstable solutions is then established, under which a stable method for multiple solutions is developed. Numerical experiments for two types of noncooperative systems are carried out to illustrate the new characterization and method. Several important properties for the method are explored or verified. Multiple numerical solutions are found and presented with their profiles and contour plots. As a new bifurcation phenomenon, multiple asymmetric positive solutions to the second type of noncooperative systems are discovered numerically but are still open for mathematical verification.

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Lectures on the Morse Complex for Infinite-Dimensional Manifolds

Abbondandolo

Majer

NATO Science Series II: Mathematics, Physics and Chemistry

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Morse homology on Hilbert spaces

Cited by 46 publications

References 36 publications

Rigidity and gluing for Morse and Novikov complexes

Rigidity and gluing for Morse and Novikov complexes

A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems

Lectures on the Morse Complex for Infinite-Dimensional Manifolds

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