Morse lemma

2020 Mathematics Subject Classification: Primary: 57R45 Secondary: 37-XX [MSN][ZBL]

A lemma which describes the structure of the germ of a twice continuously-differentiable function.

Main statement

Theorem 1 Let $f:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a non-degenerate critical point, namely $\nabla f (0) =0$ and the Hessian at $0$ has trivial kernel. Then in some neighbourhood $U$ of $0$ there is a local $C^\infty$ coordinate system, namely a $C^\infty$ diffeomorphism \[ \varphi = (x_1, \ldots , x_n): U \to V \subset \mathbb R^n\, , \] with $\varphi (0)=0$ and such that the map $\tilde{f} = f\circ \varphi^{-1}$ (namely $\varphi$ in the "$x$-coordinates") takes the form $\tilde{f} (x) = f(0)- x_1^2 - \ldots - x_\lambda^2 + x_{\lambda+1}^2 + \ldots + x_n^2$.

Here the number $\lambda$ is the Morse index of the critical point $0$ of $f$, that is the number of negative eigenvalues of the Hessian of $f$ at $0$, counted with multiplicities. The assumption $C^\infty$ may be relaxed to $C^p$ for $p\geq 2$, but in this case the change of variables $\varphi$ is in general only of class $C^{p-2}$. If instead $f$ is real analytic, then $\varphi$ is real analytic. An analogue of the Morse Lemma holds for holomorphic functions of complex variables.

Theorem 2 If $f: \mathbb C^n \to \mathbb C$ is holomorphic in a neighborhood of $0$ and $0$ is a nondegenerate critical point (namely $\frac{\partial f}{\partial z_j} (0) =0$ for every $j$ and the matrix $M_{ij} = \frac{\partial^2 f}{\partial z_i \partial z_j} (0)$ is invertible), then there is a neighborhood $U$ of $0$ with an holomorphic local chart, namely a holomorphic invertible map \[ \varphi = (w_1, \ldots , w_n) : \mathbb C^n \supset U\quad \to\quad V\subset \mathbb C^n\, , \] such that $\varphi (0) = 0$ and $\tilde{f} = f \circ \varphi^{-1}$ takes the form $f (w) = f(0) + w_1^2 + \ldots + w_n^2$.

Generalizations

Infinite-dimensional case

The Morse lemma can be generalized to the infinite-dimensional setting: it holds, for instance, on Banach spaces, see [Pa2]. On separable Hilbert spaces it takes the following form.

Theorem 3 Let $H$ be a separable Hilbert space and $f:H \to \mathbb R$ a $C^k$ function $f$ with $k\geq 3$ (in the sense of Frechet differentiability) for which $0$ is a non-degenerate critical point. Then there are convex neighborhoods $U$ and $V$ of $0$, a diffeomorphism (of class $C^{k-2}$) $\varphi: U \to V$ with $\varphi (0)=0$ and a bounded orthogonal projection $P: H \to H$ such that $f (x) = f (0) - \|P (\varphi (x))\|_H^2 + \|\varphi (x) - P (\varphi (x))\|_H^2$.

Here the dimension of the space ${\rm Im}\, (P)$ coincides with Morse index of the critical point $0$.

Equivariant Morse lemma

Consider $f$ as in Theorem 2 and assume that it is invariant with respect to the action of a compact subgroup $G$ of transformations on $\mathbb C^n$. Then the change of variables $\varphi$ can be taken $G$-invariant. An analogous "equivariant Morse lemma" is true in the real-analytic and the differentiable context. Cf. [A] and [AGV].

Morse lemma depending on parameters

Let $f: \mathbb R^n \times \R^k \ni (x,\lambda) \mapsto f (x, \lambda)\in \mathbb R$ be a real-valued $C^p$ function with $p\geq 3$. Assume that $D_x f (0,0)=0$ and that $D^2_x f (0,0)$ is nonsingular. Then there exist $C^{k-1}$ coordinates $(z, \lambda)$ in a neighbourhood of $(0,0)$ which brings $f$ to the form \[ f (z, \lambda) = f (x(\lambda), \lambda) + \frac{1}{2} \langle A z\, z\rangle\, \] where $A$ is a diagonal matrix with entries $\pm 1$ and $\lambda \mapsto x (\lambda)$ is the local (unique) solution of the equations $D_x f (x, \lambda) = 0$ (the nondegeneracy assumption on $D^2_x f (0,0)$ allows to apply the implicit function theorem and infer that $x$ depends smoothly on $\lambda$). For a proof see [H], p. 502.

References

[A]	V.I. Arnol'd, "Wave front evolution and the equivariant Morse lemma" Comm. Pure Appl. Math. , 29 (1976) pp. 557–582
[AGV]	V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) MR777682 Zbl 0554.58001
[H]	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
[M]	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
[Pa]	R.S. Palais, "Morse theory on Hilbert manifolds", Topology , 2 (1963) pp. 299–340 MR0158410 Zbl 0122.10702
[Pa2]	R.S. Palais, "The Morse lemma for Banach spaces", Bull. Amer. Math. Soc., 75 (1969), pp. 968-971.