Difference between revisions of "Morse lemma"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
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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" />. | 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" />. | ||
− | An analogous | + | An analogous "equivariant Morse lemma" is true in the real-analytic and the differentiable context. Cf. [[#References|[a1]]] and [[#References|[a2]]]. |
===Morse lemma depending on parameters.=== | ===Morse lemma depending on parameters.=== | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 16:59, 15 April 2012
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 |
Morse lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_lemma&oldid=18472