Difference between revisions of "Morse function"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | m0649501.png | ||
+ | $#A+1 = 32 n = 1 | ||
+ | $#C+1 = 32 : ~/encyclopedia/old_files/data/M064/M.0604950 Morse function | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
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 | + | 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) | + | 1) $ f ^ { - 1 } ( a) = V _ {0} $, |
+ | $ f ^ { - 1 } ( b) = V _ {1} $; | ||
− | 2) all critical points (cf. [[Critical point|Critical point]]) of | + | 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 | + | 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 | + | 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 | + | 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 ![]() |
[a3] | A. Wasserman, "Equivariant differential topology" Topology , 8 (1969) pp. 127–150 |
Morse function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_function&oldid=14633