View source for Morse lemma

{{MSC|57R45|37}}
[[Category:Analysis]]
{{TEX|done}}

A lemma which describes the structure of the [[Germ|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|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|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 function|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 {{Cite|Pa2}}. On separable [[Hilbert space|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 differential|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. {{Cite|A}} and {{Cite|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 {{Cite|H}}, p. 502.

===References===
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|valign="top"|{{Ref|A}}||  V.I. Arnol'd, "Wave front evolution and the equivariant Morse lemma" ''Comm. Pure Appl. Math.'' , '''29''' (1976) pp. 557–582  
|-
|valign="top"|{{Ref|AGV}}|| V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , '''1''' , Birkhäuser (1985) {{MR|777682}} {{ZBL|0554.58001}} 
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|valign="top"|{{Ref|H}}||  L.V. Hörmander, "The analysis of linear partial differential operators" , '''3. Pseudo-differential operators''' , Springer (1985) {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{ZBL|0612.35001}} {{ZBL|0601.35001}}
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|valign="top"|{{Ref|M}}|| M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) {{MR|1451874}} {{MR|1501555}} {{MR|1561686}} {{MR|1501489}} {{MR|1501428}} {{ZBL|0011.02802}} {{ZBL|60.0450.01}}   
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|valign="top"|{{Ref|Pa}}||  R.S. Palais, "Morse theory on Hilbert manifolds", ''Topology'' , '''2''' (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}}  
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|valign="top"|{{Ref|Pa2}}|| R.S. Palais, "The Morse lemma for Banach spaces",  "Bull. Amer. Math. Soc.", '''75''' (1969), pp. 968-971. 
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