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The statement describing the structure of the [[Germ|germ]] of a twice continuously-differentiable function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649801.png" /> be a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649802.png" />, having the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649803.png" /> as a non-degenerate [[Critical point|critical point]]. Then in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649805.png" /> there is a local coordinate system (a chart) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649806.png" />, with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649807.png" />, so that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649808.png" />,
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{{MSC|57R45|37}}
 +
[[Category:Analysis]]
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m0649809.png" /></td> </tr></table>
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A lemma which describes the structure of the [[Germ|germ]] of a twice continuously-differentiable function.  
  
Here the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498011.png" />, is the [[Morse index|Morse index]] of the critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498013.png" />. An analogue of the Morse lemma for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498014.png" /> is also true, namely: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498015.png" /> is holomorphic in a neighbourhood of a non-degenerate critical point (in other terminology, a saddle point, see [[Saddle point method|Saddle point method]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498016.png" />, then in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498018.png" /> there is a local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498019.png" /> such that
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===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 $\tilde{f} (x) = f(0)- x_1^2 - \ldots - x_\lambda^2 + x_{\lambda+1}^2 + \ldots + x_n^2$.  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498020.png" /></td> </tr></table>
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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.
  
The Morse lemma also holds for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498021.png" /> on a separable (infinite-dimensional) Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498022.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498023.png" /> be twice (Fréchet) differentiable in some neighbourhood of a non-degenerate critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498024.png" />. Then there are a convex neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498025.png" />, a convex neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498026.png" /> and a diffeomorphism (a chart) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498027.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498028.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498029.png" />,
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'''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 $f (w) = f(0) + w_1^2 + \ldots + w_n^2$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498030.png" /></td> </tr></table>
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===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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498031.png" /> is a continuous orthogonal projection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498032.png" /> is the identity operator. Here the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498033.png" /> coincides with Morse index of the critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498035.png" /> and the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498036.png" /> coincides with its co-index.
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'''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 $f (x) = f (0) - \|P (\varphi (x))\|_H^2 + \|\varphi (x) - P (\varphi (x))\|_H^2$.  
  
====References====
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Here the dimension of the space ${\rm Im}\, (P)$ coincides with Morse index of the critical point $0$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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}} </TD></TR></table>
 
  
 +
====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.
  
====Comments====
+
===References===
There exist generalizations of the Morse lemma to the following cases:
+
{|
 
+
|-
===Equivariant Morse lemma.===
+
|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
Consider a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498037.png" /> that is invariant with respect to the linear action of a compact subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498040.png" /> has at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498041.png" /> a critical point with critical value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498042.png" />, then it can be reduced to its quadratic part by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498043.png" />-invariant change of independent variables, biholomorphic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498044.png" />.
+
|-
 
+
|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}}  
An analogous "equivariant Morse lemma" is true in the real-analytic and the differentiable context. Cf. [[#References|[a1]]] and [[#References|[a2]]].
+
|-
 
+
|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}}
===Morse lemma depending on parameters.===
+
|-   
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498045.png" /> be a real-valued differentiable function defined in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498046.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498047.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498048.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498049.png" /> is non-singular. Then there exist coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498050.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498051.png" /> such that
+
|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}} 
 
+
|-
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498052.png" /></td> </tr></table>
+
|valign="top"|{{Ref|Pa}}||  R.S. Palais, "Morse theory on Hilbert manifolds", ''Topology'' , '''2''' (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}}
 
+
|-
In this formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498053.png" /> is the local solution of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064980/m06498055.png" />. The proof is a modification of that in the case without parameters. A good reference is [[#References|[a3]]], p. 502.
+
|valign="top"|{{Ref|Pa2}}|| R.S. Palais, "The Morse lemma for Banach spaces",  ''Bull. Amer. Math. Soc.'', '''75''' (1969), pp. 968-971.
 
+
|-
====References====
+
|}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Wave front evolution and the equivariant Morse lemma" ''Comm. Pure Appl. Math.'' , '''29''' (1976) pp. 557–582</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R.S. Palais, "Morse theory on Hilbert manifolds" ''Topology'' , '''2''' (1963) pp. 299–340 {{MR|0158410}} {{ZBL|0122.10702}} </TD></TR></table>
 

Latest revision as of 09:32, 28 June 2014

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.
How to Cite This Entry:
Morse lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_lemma&oldid=24510
This article was adapted from an original article by M.M. PostnikovYu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article