Abstract
In this chapter (longer than the others) we will present techniques that are close to other modern research themes: existence theorems for critical points of functions. We will move in many directions, but mainly using constructions that are taken from the techniques in symplectic geometry described up to now. In particular, those on generating functions for Lagrangian submanifolds.
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Notes
- 1.
Even though today there are ‘weak’ formulations of Morse theory extending it to degenerate critical points (maybe first in [86]), the universally known definition of Morse function is, up to today, that of functions whose critical points are all non-degenerate.
- 2.
That is, it is defined for \(t \in \mathbb{R}\).
- 3.
That is, any Cauchy sequence does converge in it.
- 4.
That are clearly Palais-Smale.
- 5.
The set of critical points is a closed set, its complement is an open set, and X b ∖ X a is in the interior of such complement: it is then easy to define two open neighborhoods A 1 and A 2 as above.
- 6.
Using Urysohn lemma, given two closed, disjoint sets cl(A 1) and X ∖ A 2, it is possible to define a C ∞-function ϕ, that has value 1 in cl(A 1) and has value 0 in X ∖ A 2. So, for example, \(\bar{Y }:= -\phi \frac{\nabla f(x)} {\vert \nabla f(x)\vert ^{2}}\).
- 7.
This means that there is no constant c > 0 such that \(\vert f^{{\prime}}(x)\vert > c\) on S.
- 8.
In other words, c cannot be an accumulation point of critical values.
- 9.
Once again, Palais-Smale condition implies this fact (Condition (C)): \(f^{-1}([c -\varepsilon _{0},c +\varepsilon _{0}])\setminus V _{0}\) is closed, so we cannot find in it sequence {x j } with df (x j ) → 0, in fact, in such a case there exists some subsequence converging to a critical point x ∗, which should belong to the closed set \(f^{-1}([c -\varepsilon _{0},c +\varepsilon _{0}])\setminus V _{0}\), absurd.
- 10.
Recall that H 0(X) represents the constant functions on the connected components of X, if X = B (ball) then \(H^{0}(B) = \mathbb{R}\), while if k > 0: H k(B) = { 0}.
- 11.
That is, with non degenerate Hessian \(f^{{\prime\prime}}\big\vert _{f^{{\prime}}=0}\).
- 12.
Observe that, from the definition of GFQI, c is uniform in x.
- 13.
- 14.
Here we denote by \(\mathcal{F}: \text{Sym}(k \times k) \times \mathbb{R}^{n} \rightarrow \text{Sym}(k \times k)\ \ \text{the map}\ \ (R,x)\mapsto R^{T}\mathit{QR} - B(x)\).
- 15.
\(f_{x}^{-\infty } = f^{-\infty },\ \forall x \in U\).
- 16.
I thank Gian Maria Dall’Ara which pointed out to me this nice fact.
- 17.
Arnol’d conjecture is itself an extension, more or less natural, of the last geometric theorem of Poincaré.
- 18.
In a non trivial way, see [119], Prop. 3.3 p. 693.
- 19.
Again, by Lusternik-Schnirelman Theorem 7.7.
- 20.
c stands for capacity, see [55].
- 21.
This is not restrictive, since by Whitney theorem it is sufficient N paracompact.
- 22.
By coordinates, \(v_{g} = \sqrt{\det g}\,\mathit{dx}^{1} \wedge \ldots \wedge \mathit{dx}^{n}\).
- 23.
By coordinates, for any scalar function Φ, the related gradient vector field X is defined by \(X^{i} = (\sharp d\varPhi )^{i} = (\nabla _{g}\varPhi )^{i} = g^{\mathit{ij}} \frac{\partial \varPhi } {\partial x^{j}}\).
- 24.
These formulas were first presented by Hamilton in the “Second essay on a general method in dynamics”, 1835. The author thanks Sergio Benenti for pointing out this remarkable fact.
- 25.
In the Section 28 of Gantmacher [66], this example is really quoted as ‘perturbation theory’.
- 26.
It is sufficient to see that contractive perturbations of the identity are bi-Lipschitz homeomorphisms: (Rem: \(\vert x - y\vert \geq \big\vert \vert x\vert -\vert y\vert \big\vert \geq \vert x\vert -\vert y\vert \)) Since f: X → X is contractive, | f(x 1) − f(x 2) | ≤ λ | x 1 − x 2 | , λ < 1, then I − f is injective: \(\vert (x_{1}-f(x_{1}))-(x_{2}-f(x_{2}))\vert = \vert (x_{1}-x_{2})-(f(x_{1})-f(x_{2}))\vert \geq \vert x_{1}-x_{2}\vert -\vert f(x_{1})-f(x_{2})\vert \geq (1-\lambda )\vert x_{1}-x_{2}\vert \). Thus the inverse g di I − f is Lipschitz with \(\mathrm{Lip}(g) = \frac{1} {1-\lambda }\). The surjectivity of I − f is gained from the fixed point: for any y ∈ X, the map x ↦ y + f(x) is obviously contractive, then there exists an unique x such that x = y + f(x), so that x − f(x) = y. □
- 27.
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Cardin, F. (2015). Notes on Lusternik-Schnirelman and Morse Theories. In: Elementary Symplectic Topology and Mechanics. Lecture Notes of the Unione Matematica Italiana, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-11026-4_7
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