Morse lemma

The statement describing the structure of the germ of a twice continuously-differentiable function. Let be a function of class , having the point as a non-degenerate critical point. Then in some neighbourhood of there is a local coordinate system (a chart) , with centre at , so that for all ,

Here the number , , is the Morse index of the critical point of . An analogue of the Morse lemma for functions is also true, namely: If is holomorphic in a neighbourhood of a non-degenerate critical point (in other terminology, a saddle point, see Saddle point method) , then in some neighbourhood of there is a local coordinate system such that

The Morse lemma also holds for functions on a separable (infinite-dimensional) Hilbert space . Let be twice (Fréchet) differentiable in some neighbourhood of a non-degenerate critical point . Then there are a convex neighbourhood of zero , a convex neighbourhood of zero and a diffeomorphism (a chart) with , such that for all ,

where is a continuous orthogonal projection and is the identity operator. Here the dimension coincides with Morse index of the critical point of and the dimension coincides with its co-index.

References

[1]	M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) MR1451874 MR1501555 MR1561686 MR1501489 MR1501428 Zbl 0011.02802 Zbl 60.0450.01

Comments

There exist generalizations of the Morse lemma to the following cases:

Equivariant Morse lemma.

Consider a holomorphic function that is invariant with respect to the linear action of a compact subgroup on . If has at a critical point with critical value , then it can be reduced to its quadratic part by a -invariant change of independent variables, biholomorphic at the point .

An analogous "equivariant Morse lemma" is true in the real-analytic and the differentiable context. Cf. [a1] and [a2].

Morse lemma depending on parameters.

Let be a real-valued differentiable function defined in a neighbourhood of . Let . Assume that and that is non-singular. Then there exist coordinates in a neighbourhood of such that

In this formula is the local solution of the equations and . The proof is a modification of that in the case without parameters. A good reference is [a3], p. 502.

References

[a1]	V.I. Arnol'd, "Wave front evolution and the equivariant Morse lemma" Comm. Pure Appl. Math. , 29 (1976) pp. 557–582
[a2]	V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001
[a3]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3. Pseudo-differential operators , Springer (1985) MR1540773 MR0781537 MR0781536 Zbl 0612.35001 Zbl 0601.35001
[a4]	R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 (1963) pp. 299–340 MR0158410 Zbl 0122.10702