Abstract
We state the Arnold conjecture, which gives a lower bound for the number of fixed points of certain Hamiltonian diffeomorphisms. We then identify these fixed points with periodic orbits of Hamiltonian systems and with critical points of the “action functional” a function on the space of the contractible loops on the symplectic manifold, as well as the differential equation defining the flow of the gradient of this functional, called the Floer equation. This is a partial derivatives equation because it involves both the loop’s variable and that of the gradient’s flow. We begin studying the space of solutions of this equation by showing a compactness property.
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Notes
- 1.
We saw in Remark 4.4.5 that the minimal number of critical points of a function may be less than the minimal number of critical points of a Morse function.
- 2.
This means that the base point of the loop is not fixed. Moreover, the homotopies between these loops are free.
- 3.
This is why this functional is called
. - 4.
See also Exercise 27 on p. 523.
- 5.
See also Exercise 30 on p. 524 where the nondegeneracy assumption is left out.
- 6.
See Section C.1.
- 7.
This expression refers to the ability to raise yourself, to levitate, by just pulling on your bootstraps. In general, in this book it refers to what the French call the “elliptic regularity”, where a weak solution, in the sense of distributions, of a partial differential equation is automatically a
function. See Appendix C. - 8.
As in Morse theory, the nondegeneracy translates into transversality.
- 9.
This is where the nondegeneracy is used.
- 10.
This is a classical lemma in this theory. It seems that its origin lies with Ekeland. The beginning of Chapter iv of his book [26] contains variations on this theme.
- 11.
See, for example, [64].
- 12.
See for example [45, Chapter V].
- 13.
For the theory of covers, we refer to [4].
- 14.
The maximal radius of the disks of such a fiber bundle is the injectivity radius of the Riemann manifold W.
.
function. See Appendix C.References
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Audin, M., Damian, M. (2014). The Arnold Conjecture and the Floer Equation. In: Morse Theory and Floer Homology. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5496-9_6
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