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A smooth function with certain special properties. Morse functions arise and are used in [[Morse theory|Morse theory]].
 
A smooth function with certain special properties. Morse functions arise and are used in [[Morse theory|Morse theory]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649501.png" /> be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649502.png" /> is a disconnected union (possibly empty) of manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649504.png" />. A Morse function for the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649505.png" /> is a smooth (of Fréchet class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649506.png" />) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649508.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m0649509.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495010.png" />), such that:
+
Let $  W $
 +
be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary $  \partial  W $
 +
is a disconnected union (possibly empty) of manifolds $  V _ {0} $
 +
and $  V _ {1} $.  
 +
A Morse function for the triple $  ( W ;  V _ {0} , V _ {1} ) $
 +
is a smooth (of Fréchet class $  C  ^ {2} $)  
 +
function $  f :  W \rightarrow [ a , b ] $,
 +
$  - \infty < a , b < + \infty $(
 +
or $  f : W \rightarrow [ a , \infty ] $
 +
for $  V _ {1} = \emptyset $),  
 +
such that:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495012.png" />;
+
1) $  f ^ { - 1 } ( a) = V _ {0} $,  
 +
$  f ^ { - 1 } ( b) = V _ {1} $;
  
2) all critical points (cf. [[Critical point|Critical point]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495013.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495014.png" /> and are non-degenerate;
+
2) all critical points (cf. [[Critical point|Critical point]]) of $  f $
 +
lie in $  W \setminus  \partial  W = f ^ { - 1 } ( a, b ) $
 +
and are non-degenerate;
  
3) condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495016.png" /> of Palais–Smale is fulfilled (see [[#References|[2]]], [[#References|[3]]]). I.e. on any closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495017.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495018.png" /> is bounded and the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495019.png" /> is zero, there is a critical point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495020.png" />.
+
3) condition $  C $
 +
of Palais–Smale is fulfilled (see [[#References|[2]]], [[#References|[3]]]). I.e. on any closed set $  S \subset  W $
 +
where $  f $
 +
is bounded and the greatest lower bound of $  x \rightarrow \| d f ( x) \| $
 +
is zero, there is a critical point of $  f $.
  
For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495021.png" /> is a proper function, that is, all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495023.png" />, are compact (possible only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495024.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495025.png" /> satisfies condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495026.png" />. A Morse function attains a (global) minimum on each connected component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495028.png" /> is a finite-dimensional manifold, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495029.png" /> the set of Morse functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495030.png" /> is a set of the second category (and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495031.png" /> is compact, even a dense open set) in the space of all functions
+
For example, if $  f $
 +
is a proper function, that is, all sets $  f ^ { - 1 } [ c , d ] $,
 +
$  - \infty < c , d \leq  \infty $,  
 +
are compact (possible only for $  \mathop{\rm dim}  W < \infty $),  
 +
then $  F $
 +
satisfies condition $  C $.  
 +
A Morse function attains a (global) minimum on each connected component of $  W $.  
 +
If $  V $
 +
is a finite-dimensional manifold, then for $  k \geq  2 $
 +
the set of Morse functions of class $  C  ^ {k} $
 +
is a set of the second category (and, if $  W $
 +
is compact, even a dense open set) in the space of all functions
  
<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/m064950/m06495032.png" /></td> </tr></table>
+
$$
 +
f : ( W ; V _ {0} , V _ {1} )  \rightarrow  ( [ a , b ] , a , b )
 +
$$
  
in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495033.png" />-topology.
+
in the $  C  ^ {k} $-
 +
topology.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,  "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Palais,  "Morse theory on Hilbert manifolds"  ''Topology'' , '''2'''  (1963)  pp. 299–340</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Smale,  "Morse theory and a nonlinear generalization of the Dirichlet problem"  ''Ann. of Math.'' , '''80'''  (1964)  pp. 382–396</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,  "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Palais,  "Morse theory on Hilbert manifolds"  ''Topology'' , '''2'''  (1963)  pp. 299–340</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Smale,  "Morse theory and a nonlinear generalization of the Dirichlet problem"  ''Ann. of Math.'' , '''80'''  (1964)  pp. 382–396</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:01, 6 June 2020


A smooth function with certain special properties. Morse functions arise and are used in Morse theory.

Let $ W $ be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary $ \partial W $ is a disconnected union (possibly empty) of manifolds $ V _ {0} $ and $ V _ {1} $. A Morse function for the triple $ ( W ; V _ {0} , V _ {1} ) $ is a smooth (of Fréchet class $ C ^ {2} $) function $ f : W \rightarrow [ a , b ] $, $ - \infty < a , b < + \infty $( or $ f : W \rightarrow [ a , \infty ] $ for $ V _ {1} = \emptyset $), such that:

1) $ f ^ { - 1 } ( a) = V _ {0} $, $ f ^ { - 1 } ( b) = V _ {1} $;

2) all critical points (cf. Critical point) of $ f $ lie in $ W \setminus \partial W = f ^ { - 1 } ( a, b ) $ and are non-degenerate;

3) condition $ C $ of Palais–Smale is fulfilled (see [2], [3]). I.e. on any closed set $ S \subset W $ where $ f $ is bounded and the greatest lower bound of $ x \rightarrow \| d f ( x) \| $ is zero, there is a critical point of $ f $.

For example, if $ f $ is a proper function, that is, all sets $ f ^ { - 1 } [ c , d ] $, $ - \infty < c , d \leq \infty $, are compact (possible only for $ \mathop{\rm dim} W < \infty $), then $ F $ satisfies condition $ C $. A Morse function attains a (global) minimum on each connected component of $ W $. If $ V $ is a finite-dimensional manifold, then for $ k \geq 2 $ the set of Morse functions of class $ C ^ {k} $ is a set of the second category (and, if $ W $ is compact, even a dense open set) in the space of all functions

$$ f : ( W ; V _ {0} , V _ {1} ) \rightarrow ( [ a , b ] , a , b ) $$

in the $ C ^ {k} $- topology.

References

[1] M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)
[2] R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 (1963) pp. 299–340
[3] S. Smale, "Morse theory and a nonlinear generalization of the Dirichlet problem" Ann. of Math. , 80 (1964) pp. 382–396

Comments

There exist generalizations to Morse functions on stratified spaces (cf. (the editorial comments to) Morse theory and [a1]) and to equivariant Morse functions (cf. [a2] and [a3]).

References

[a1] M. Goreski, R. MacPherson, "Stratified Morse theory" , Springer (1988)
[a2] A. Wasserman, "Morse theory for -manifolds" Bull. Amer. Math. Soc. , 71 (1965) pp. 384–388
[a3] A. Wasserman, "Equivariant differential topology" Topology , 8 (1969) pp. 127–150
How to Cite This Entry:
Morse function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_function&oldid=14633
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